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Idler-Free Multi-Channel Discrimination via Multipartite Probe States

The characterisation of Quantum Channel Discrimination (QCD) offers critical insight for future quantum technologies in quantum metrology, sensing and communications. The task of multi-channel discrimination creates a scenario in which the discrimination of multiple quantum channels can be equated to the idea of pattern recognition, highly relevant to the tasks of quantum reading, illumination and more. Whilst the optimal quantum strategy for many scenarios is an entangled idler-assisted protocol, the extension to a multi-hypothesis setting invites the exploration of discrimination strategies based on unassisted, multipartite probe states. In this work, we expand the space of possible quantum enhanced protocols by formulating general classes of unassisted multi-channel discrimination protocols which are not assisted by idler modes. Developing a general framework for idler-free protocols, we perform an explicit investigation in the bosonic setting, studying prominent Gaussian channel discrimination problems for real world applications. Our findings uncover the existence of strongly quantum advantageous, idler-free protocols for the discrimination of bosonic loss and environmental noise. This circumvents the necessity for idler assistance to achieve quantum advantage in some of the most relevant discrimination settings, significantly loosening practical requirements for prominent quantum sensing applications.

The fundamental task of Quantum Channel Discrimination (QCD) models many of these applications. In QCD, a user is tasked with classifying an ensemble of quantum channels through the use of an input quantum state (probe state) and a discriminatory measurement. Locating an optimal discrimination protocol is very difficult, as it embodies a double optimisation problem of both the probe state and the output measurement. Nonetheless, significant progress has been made in recent years in a variety of contexts [37,38].
Until recently, QCD has been mostly limited to the problem of binary classification. However, advances in multi-channel discrimination and the formulation of Channel Position Finding (CPF) [39,40] have brought with them new insight and opportunities for more complex multi-hypothesis classification problems. These multi-channel discrimination problems are highly relevant in a number of fascinating settings, such as datareadout from optical memories, quantum enhanced optical/thermal pattern recognition [41,42], and target detection [43].
Within these applications (and many more in quantum sensing) the assistance of idler modes has been a crucial * cth528@york.ac.uk † stefano.pirandola@york.ac.uk feature in order to attain quantum enhanced performance [39,44]. Idler-modes refer to perfectly preserved, ancillary quantum systems which share entanglement with input probe states throughout a sensing protocol. In the bosonic setting, these protocols consist of using one mode of a Two Mode Squeezed Vacuum (TMSV) state to probe a target, whilst the remaining mode (the idler) is kept by the user. Idler-assisted protocols have been shown to be optimal for a number of important discrimination tasks, and offer significant advantage for many more. Yet, the necessity for idler-modes is problematic, due to the requirement that they need to be perfectly protected. In practice this is not possible, as some decoherence will always be imparted on the idler while the probe mode is interacting with a target. To combat this, idlers are either contained in delay lines (e.g. very low-loss fibre optics) or stored in quantum memories until required for measurement. This preservation requirement causes serious practical difficulties due to the challenging nature of creating stable quantum memories with adequate storage time [45][46][47][48]. In some settings, it may be much more practical to use unassisted protocols limited to signalonly probe modes, especially for near term quantum technologies.
Research on unassisted protocols has been primarily limited to single channel sensing problems, motivating the use of coherent states to formulate classical benchmarks, and even to search for quantum enhancements beyond entanglement [49,50]. However the multichannel discrimination picture invites us to explore different unassisted protocols. In particular, it is now possible to construct protocols that distribute entanglement over multiple quantum channels using multipartite entangled states. Without additional idler modes to defend entanglement, input states must be cleverly designed to preserve quantum correlations in the face of increased decoherence. Recently, Pereira et al. [51] have explored the use of a block protocol with entangled bosonic states arXiv:2010.12354v4 [quant-ph] 6 Sep 2021 for discriminating small collections of Gaussian quantum channels, proving that there do exist idler-free protocols capable of exceeding the best known classical strategies.
Motivated by this, we arrive at our key research objectives: To generalise the theory of unassisted protocols for multi-channel discrimination, and to ask: Can we design unassisted multi-channel discrimination protocols that achieve significant quantum advantage?
Hence, in this work, we construct general classes of unassisted protocols for multi-channel discrimination. These are block protocols which utilise (generally entangled) multipartite quantum states as probe states. Multipartite states (and thus entanglement) can now be distributed across multiple quantum channels in many inequivalent ways, leading to two distinct, broad classes of discrimination protocols. Via multi-mode entanglement and carefully designed probe distributions, we present unassisted protocols that are able to attain performances on par with that of idler-assistance. This circumvents the necessity for idler-assistance in some of the most relevant discrimination settings, loosening practical requirements for quantum-enhanced pattern recognition.
This paper is structured as follows: In Results, after first reviewing the model of quantum pattern recognition, we present our main findings. We introduce the general framework of block protocols using unassisted multipartite quantum probe states. We then identify two distinct classes of unassisted protocol which emerge from this framework, discuss their operational interpretations, and devise a diagrammatic language for describing such protocols. We corroborate these general findings by demonstrating the efficacy of idler-free protocols for the discrimination of multiple bosonic Gaussian quantum channels. In Discussion, our results are summarised and we identify future investigative paths. Finally, the Methods section contains a number of useful theoretical tools and insights used within this research.

Quantum Pattern Recognition
In this work, we study the discrimination of quantum multi-channels which we call quantum channel patterns. A binary channel pattern is defined as an m-length sequence of quantum channels, such that each channel in the sequence admits the properties of a target channel E T or background channel E B (identified by the labels T, B respectively). It is useful to convert this sequence into a multi-set of binary variables which represents the channel pattern i = {i 1 , i 2 , . . . , i m }, where i j ∈ {B, T } for all j ∈ {1, . . . , m}. We can then more precisely denote an m-length channel pattern as tensor product Throughout this work, we refer to a channel pattern simply by its binary string i, unless E i is formally required. Background and target channels can be used to encode physical properties of a multipartite system. For instance, one can associate each pixel of an m-pixel binary thermal image with a cold (background) or hot (target) temperature. Quantum mechanically, one may attribute each pixel to a quantum channel that describes how a quantum probe may interact with either pixel. A channel pattern i represents only a single instance of a possible binary arrangement. More generally, these instances belong to a larger space of multi-channels we may call an image space. We label an arbitrary Nelement image space as the set U = {i 1 , i 2 , . . . , i N } containing N unique channel patterns. Since we are considering binary patterns, the most general image space we can consider is the set of all m-length binary strings U m = {i 1 , i 2 , . . . , i 2 m }, of which all other binary image spaces are a subset. Image spaces can be used to specify important, physical problem settings such as those defined by the task of Channel Position Finding (CPF), which is concerned with locating target channels hidden amongst collections of background channels (see Methods for more details).
The challenge of multi-channel discrimination may now be presented: Consider an m-length pattern of unidentified quantum channels. Suppose that the sequence of channels belongs to a pattern from a known image space U. Each pattern in the image space possesses a unique probability of existing, π i . The task of discrimination then consists of distinguishing between all the multi-channels in the statistical ensemble {π i ; E i } i∈U , which describes an ensemble of multi-channels {E i } i∈U distributed according to the classical probability distribution {π i } i∈U .
The most general multi-channel discrimination protocol is a general adaptive protocol, P. This is best described by a quantum comb [52][53][54]; a quantum circuit board with an arbitrary number of registers, with M slots in which channel patterns E i are placed. There is no limit to the amount of entanglement that can used to construct a quantum comb, and a general adaptive protocol can make use of adaptive operations and feedback based state preparation. Due to their generality, these protocols are very difficult to characterise and optimise. Therefore it is often much more beneficial to consider simpler protocols.
Of such, block protocols B represent a very important class of non-adaptive discrimination strategy. Channel patterns are probed using M identical and independent copies of some input probe state, ρ ⊗M → ρ ⊗M i := E i (ρ) ⊗M . After M pattern interactions, an optimised POVM {Π i } i∈U is used to perform the classification. Given an image space U with the pattern probability distribution {π i } i∈U , the average error probability of misclassification is given by where this sum runs over all pairs of unequal channel patterns throughout the image space. In order to benchmark this discrimination performance without specifying precise measurements, the following fidelity based bounds from can be used [55,56], where F (ρ, σ) = Tr √ ρ σ √ ρ denotes the Bures fidelity. These bounds are completely general, and do not depend on the channel dimension. Hence, they may be utilised for both finite and infinite dimensional input states (provided that we use energy-constrained quantum states). These non-adaptive block protocols have been shown to offer high performance in a number of discrimination settings, and in some cases are optimal [37]. If a block protocol makes use of entangled, ancillary quantum systems (idlers) then it is known as a block-assisted protocol B a . Idler-based entanglement can induce quantumenhancements in many different discrimination settings [41,42]. Without additional idler modes, we are left with an unassisted block protocol, B u . Much less is known about unassisted protocols in a multi-channel setting, which we rectify in the following sections.

Fixed Unassisted Block Protocols
Consider an image space i ∈ U of m-length multichannels each of which occur with probability π i , generating the channel pattern ensemble {π i ; E i } i∈U . Unassisted discrimination involves developing a strategy for accurately distinguishing patterns from the image space without utilising entangled idler-modes or ancillary quantum systems. Unlike in an assisted protocol, entanglement is now only permitted between probe modes. We proceed in this practical direction by investigating how inter-probe entanglement can play a role in constructing quantum enhanced, unassisted block protocols.
Consider an m-length channel pattern. An unassisted block protocol B u using multipartite states will assign an M -copy, n ≤ m multi-mode state to interact with some region of the channel pattern, defined by a set of channel labels s = {s 1 , . . . , s n } for s i ∈ {1, . . . , m}. This channel region s, which we aptly call a probedomain, defines a sub-pattern of the total channel pattern over which a multipartite state σ ⊗M s can be irradiated. Hence, a probe-domain s is a sub-set of channel labels s ⊆ {1, . . . , m} which designates a region of the channel pattern over which probe modes are permitted to be entangled. Input modes which are incident in the domain s can be entangled but are fully separable with respect to any modes outside of this region. Furthermore, each M -copy probe state σ ⊗M s can be different for its respective probe-domain.
In order to completely interact with all m-channels in the pattern it is necessary to define a discrete probedomain distribution. This is a collection of distinct channel pattern sub-regions {s 1 , s 2 , . . . , s N } over which an associated N length collection M -copy multipartite states {σ ⊗M s1 , σ ⊗M s2 , . . . , σ ⊗M s N } are irradiated. More precisely, we can define a probe-domain distribution as ∃ j such that i ∈ s j , ∀i ∈ {1, 2, . . . , m}.
In Eq. (6), we demand that every channel index 1, . . . , m is accounted for in at least one subset s ∈ S, so that no channels are left un-probed. Using S we can define a global probe state irradiated over a channel pattern, constructed as the tensor product of all the local substates. It is not immediately clear how one should design this probe-domain distribution. However, the most intuitive way to construct S is to devise a distribution such that each channel is only associated with a single probedomain. A probe-domain distribution disjoint if it satisfies this property. Suppose one constructs a N -partite probe-domain distribution that is disjoint, S d . Formally, we can define this as, {s j }, such that s j ∩ s k = ∅, ∀j, k, (7) where disjointedness is demanded on the RHS of this equation, such that no two probe-domains s j and s k are permitted to share the same channel label, for all j, k. Again, we demand that all channels 1, . . . , m are accounted for in this distribution, as in Eq. (6). We may then choose an N -element set of multipartite probe states in accordance with this disjoint structure {σ sj } N j=1 , where each σ sj can be unique. Assuming M -copies of each sub-state, we can define a global probe state In this way, each channel in the pattern is probed exactly M times per total round of discrimination. Furthermore, since all probe-domains are disjoint, there are no overlaps between any multipartite states; each channel in the pattern is always probed within the same probe-domain and within the same collection of channels.
From an operational point of view, the disjointedness of S d and lack of probe-domain overlaps means that each sub-state σ ⊗M sj can interact simultaneously with the multi-channel. As such, each probe state can be considered to be static (or fixed ) over a sub-region of the channel pattern throughout the entire discrimination protocol. For this reason, we describe an unassisted protocol using a disjoint probe-domain distribution as a fixed block protocol, B u fix (see Fig. 1(a) for an example). Fixed block protocols are very intuitive thanks to their simple, static format. Indeed, classical block protocols can inherently be considered to be fixed protocols, where separable collections of coherent states are irradiated upon a channel pattern. Using our previous formalism and considering m-length channel patterns, one may define a trivial probe-domain distribution S d = {{1}, {2}, . . . , {m}} and a corresponding set of single-mode coherent states {α j } m j=1 which produces the global state σ ⊗M S d = m j=1 α ⊗M j . Larger probe-domains invite the potential for entangled probe states over fixed probe-domains, and can provide an easy route for potential quantum enhancements in many settings. In general, the performance of fixed block protocols can always be assessed through the average error probability by substituting σ S d into Eq. (2).

Dynamic Unassisted Block Protocols
Interestingly, we need not restrict ourselves to probedomain distributions which are disjoint. Departing the rigidity of disjoint probe-domain distributions offers a fascinating route for quantum-enhanced, unassisted protocols. While this path is less intuitive, it unveils a rich and flexible class of discrimination protocols with rewarding features.
Consider now a non-disjoint, N -partite probe-domain distribution S nd = N j=1 {s j }, meaning that probedomains are free to overlap and share similar channel labels, i.e. the overlap of two probe-domains is no longer the empty set s j ∩ s k = ∅. This renders a much larger, and more general space of possible distributions. A global quantum probe state σ S nd associated with such a distribution is again found as the tensor product of all local sub-states, however its interpretation is much less obvious. We begin by describing the physical interpretation of a non-disjoint probe-domain distribution within a discrimination protocol.
Any non-disjoint discrete distribution S nd can be decomposed into a sequence of r disjoint distributions, where S k d is a disjoint sub-collection of probe-domains in accordance with Eq. (7). In this case, each S k d need not contain all the channel labels, but all m channels must be accounted for in the global distribution S nd . This allows us to rewrite the global, single-copy probe state in a more meaningful way  That is, it is the tensor product of r disjointly distributed multipartite input states. Therefore, the utilisation of a non-disjoint probedomain distribution corresponds to a block protocol with r rounds of disjoint pattern interaction. At each round, the user interacts with the channel pattern by irradiating unassisted multipartite states, and over the course of r rounds the probe-domain distribution "moves" around the channel pattern. For this reason, it can be intuitively called a dynamic block protocol, B u dy . Fig. 1(b) depicts an m = 4 × 6 = 24 channel pattern which is being non-disjointly probed. The dynamic "movement" of probe-domains throughout its r = 4 rounds of disjoint pattern interaction is visualised in Fig. 2(a).
The number of disjoint rounds r required to construct a dynamic protocol depends on the number of overlaps that occur within the decomposition in Eq. (9). An overlap simply refers to an instance of a channel label that is contained in more than one probe-domain. We can define the number of overlaps m ov as the total number of additional channel labels contained in the non-disjoint distribution If there are many probe-domain overlaps then r may be very large; if there are no overlaps, then r = 1 and we return to a fixed protocol. In order to fairly compare dynamic and fixed block protocols, one must also be careful when distributing the number of probe copies M ; a dynamic protocol with r rounds of disjoint pattern interaction and M -copy input states will clearly use more than M total probe modes. It is useful to define a resource metric known as the average channel use,M which describes average number of probe copies applied per channel within a dynamic block protocol. When comparing the performance of fixed/dynamic block protocols, we must ensure they have the same average channel use.  Dynamic to Fixed Block Protocol Transformation: (a) Non-disjoint probe-domain distributions can be decomposed into multiple rounds of disjoint pattern interaction, generating a dynamic discrimination protocol. This dynamic protocol can be equivalently represented by a fixed block protocol on a modified image space, as shown in (b). The original (6 × 4)-channel pattern i is transformed into a (8 × 4)-pattern ν i which has been appropriately modified in accordance with the non-disjoint probe-domain distribution S nd using Eq. (13). Here we have assumed the use of single-copy probe states, M = 1.

Dynamic/Fixed Block Protocol Transformation
Consider a dynamic block protocol which follows a non-disjoint probe-domain distribution S nd . Now, any channel E ij within the global pattern E i may be probed as part of several different multipartite domains. This more general scenario requires a mathematical model that allows us to quantitatively investigate the performance of dynamic protocols.
To achieve this, we find a simple relationship between dynamic and fixed block protocols, corresponding to an appropriate transformation on a channel pattern image space, U. When two probe-domains overlap, the overlapping channels are probed twice, but by independent probe states. Therefore, we attribute a unique Hilbert space to each independent probe mode and channel in each disjoint round throughout the protocol, whilst retaining the characteristics of the original channels. This can be done by considering a modified channel pattern which has been concatenated with copies of the channels that are overlapped. Fig. 2 depicts how this pattern modification takes place. Given that S nd contains m ov overlapping channels, an originally m-length channel pattern i can be mapped to a (m + m ov )-length pattern, where the additional copies of overlapping channels are concatenated with the multi-channel. These copy channels directly obey the behaviour of their originals. In this way, a dynamic protocol over m-length channel patterns can be equivalently studied as a fixed block protocol over an appropriately modified (m + m ov )-length channel pattern space.
Let us more precisely express this transformation. A S nd dynamic protocol invokes the following transformation on a generic m-length channel pattern i into an extended channel pattern ν i . Since ν i contains repeated elements it is formally treated as a multi-set, rather than a traditional set [57]. Then the transformation can be explicitly written as where is the multi-set union operator which concatenates each subset of channel labels, e.g. if we consider m = 3 length channel patterns and a probe- From a channel perspective, this transformation can be equivalently portrayed as By iterating this concatenation process over all patterns in an image space {ν i } i∈U , one can easily convert a dynamic protocol into a fixed representation. Furthermore, it is expedient to write the global output states of these protocols in this format, such that This transformation greatly simplifies the complication of overlapping probe-domains, and allows for an investigation of error probabilities. By abstracting our set of discriminatory POVMs to the modified image space {Π ν i } i∈U , and using an M -copy global probe state, then the average error probability of classification can be succinctly given by Without explicit knowledge of these POVMs, we can simply utilise the fidelity bounds from Eqs. (3) and (4). These fidelity-based bounds can be readily computed thanks to this fixed protocol transformation, by iterating over all unequal channel patterns in the modified image space (see Methods for more details).

Correspondence with Error Correction
This dynamic to fixed block protocol mapping identifies a fascinating feature. In essence, a dynamic protocol invokes an encoding of quantum channel patterns, wherein m-length patterns from some image space i ∈ U are encoded into an extended counterpart {ν i } i∈U . This modified image space is a function of the non-disjoint probe-domain distribution S nd . Thus, we make the crucial observation: Using entangled probe states, one can design a dynamic block protocol which encodes a quantum image space into a more distinguishable form.
Consider a single channel E iov within a larger-channel pattern, which happens to fall within the domain of two entangled sub-states of a global probe, σ S nd = σ s ⊗ σ s . Because of this, the probe states must be applied at different disjoint rounds in a dynamic protocol. In one round, the probe state σ s is being used to determine the classification of all the channels E i[s] := {E i k } k∈s . In another round, the probe state σ s is being used to classify the channels in the region E i[s ] = {E i k } k∈s . Because these probe sub-states are entangled over their domains, then the distinguishability of their output states E i[s] (σ s ) and E i[s ] (σ s ) are correlated with the precise collection of quantum channels in each region. Dependent upon the size of entangled probe domains and the physical setting of discrimination, some collections of channels are easier to discriminate than others.
We arrive at the key insight: Since E iov is contained in both probe-domains, we are able to gather two potentially unique opinions on its classification; one from the perspective of σ s in the pattern region s, and another from σ s in its region s . On their own, these states may not be very effective at discriminating the channel E iov , i.e. one of the output states may not be very distinguishable from from other potential output states in that region. But by probing E iov in conjunction with two different probing domains, it is more likely that at least one of the sub-regions will be a more distinguishable collection of channels; thus providing a greater chance of correct classification.
In this way, dynamic block protocols implicitly possess a form of error-correcting behaviour. By varying the spatial probe-domain distributions throughout the protocol, channels are probed from various perspectives, correlated with different sub-regions of the channel pattern. Poorly distinguishable channels in one sub-region may be significantly more distinguishable when probed within a different sub-region. Over the course of r disjoint rounds of pattern interaction, each entangled multipartite sub-state can help to correct errors that would arise if only fixed probe-domains were used. Exploiting this behaviour, dynamic protocols can indeed encode channel patterns into more easily discriminated image spaces.
This is a remarkable property of dynamic block protocols, one that depends strongly on the choice of entangled quantum probes and the quantum channel pat-terns. Explicit examples of this behaviour and physical/mathematical intuition are elucidated in the Methods section.

Designing Unassisted Block Protocols
Given an m-length channel pattern discrimination problem, there are clearly an enormous number of ways in which one can design a (generally non-disjoint) probedomain distribution. Let us provide a diagrammatic approach to constructing these protocols.
An m-length channel pattern E i can be represented by an m-pixel grid. Each pixel is used to represent an individual channel E ij , and the grid can adopt any preferred the height, width and shape. In order to create a tidy language that allows one to convey probe-domains which contain both local and non-local channels, we provide two ways to portray a probe-domain. Firstly, a probe-domain can be indicated by means of a coloured box which covers a number of local channels. The size and position of the coloured-probing domains indicate the regions of a channel pattern that are irradiated by a multipartite input state. This is particularly useful for illustrating fixed block protocols with local probe-domains, as shown in Fig. 3(a), which can be intuitively interpreted.
It is also useful to possess a convention for probedomains which is more convenient with non-local channels, or when are a number of overlapping domains in close proximity. Hence, we can equivalently illustrate a probe-domain via connective lines between coloured single-pixel boxes. A probe-domain is indicated by means of a continuous (unbroken) connecting line between a number of pixels. Dashed connective lines through channel boxes indicate a lack of entanglement, used to bypass certain channels while illustrating nonlocal domains. A clear example of this is shown in Fig. 3(b). Here, we describe a distribution of probedomains where each domain is of size |s| = 2, i.e. S nd = {{1, 2}, {2, 3}, . . . , {8, 9}, {9, 1}}. A dashed connective line is used to connect the non-local channel labels 1 and 9 so that 5 is not included in the probe-domain.

Bosonic Gaussian Channel Patterns
We wish to corroborate the construction of these classes of unassisted discrimination protocols and demonstrate their efficacy. To do so, we will focus on the discrimination of bosonic Gaussian Phase Insensitive (GPI) channels. This is a family of very important channels within CV quantum communications, sensing and computation [34], and can be used to model a vast array of physically significant scenarios. Crucially, a GPI channel maintains the Gaussianity of its input state. Hence, the transformation of a Gaussian state (with zero first moments) through under the action of a GPI channel can be fully characterised using its covariance matrix V , Disjoint Distribution S d assuming zero first moments (see Methods for the explicit transformations). The overall quantum channel can be denoted as E τ,ν and is defined with respect to a transmissivity parameter 0 ≤ τ ≤ 1 describing attenuation/amplification properties and an induced noise parameter ν ≥ 0. When τ = 1 and ν = 0 we regain the identity channel.
Binary GPI channel patterns then consist of a sequence of m GPI channels with unique target/background transmissivities τ B , τ T and noise properties ν B , ν T . Generally, we may write the channel pattern Let us identify some essential GPI channels: Setting τ = η, such that 0 < η < 1 and ν = (1 − η)/2 then we have the single parameter bosonic pure-loss channel E η . This describes the interaction of bosonic mode with a zero-temperature bath. This is an essential channel model for the description of optical fibres, and shortrange optical target detection known as quantum reading. The multi-hypothesis setting of discrimination pureloss channel patterns has also be equated to the task of optical imaging, pattern recognition and classical datareadout from optical memories [25,41]. Hence, the discrimination of bosonic loss poses a key problem setting for our work.
Alternatively, we may study thermal-loss channels E τ,ν such that the transmissivity satisfies 0 < τ < 1, or thermal-amplifier channels where τ > 1. In both cases, the induced thermal noise is connected to the number of thermal photons in the channel environment N env , such that ν = (N env + 1 2 )|1−τ |. In the idealised absence of loss, we have a Gaussian additive-noise channel E ν , where the transmissivity satisfies τ = 1 but we have non-zero noise ν > 0. The discrimination of thermal multi-channels is known as environment localisation, and has been used to model fascinating scenarios within target detection and thermal pattern recognition [42,43]. In this work, we focus on the discrimination of additive-noise binary channel patterns, since the performance of this task will always be an upper bound for multi-channels with non-trivial transmissivity.

Unassisted Bosonic Quantum Pattern Recognition
In order to devise fixed/dynamic unassisted block protocols for the discrimination of GPI channel patterns, we must specify a class of multipartite probe state. Here, we make use of the Gaussian analogue of the entangled GHZ state known as a CV-GHZ state which is designed as the extension of a TMSV state to many modes. Consider a probe-domain s which describes a collection of |s| channels over which an input probe state is irradiated. A CV-GHZ state defined over this probe-domain is a |s|mode, fully symmetric state denoted by Φ µ s . It can be completely characterised by its covariance matrix (CM) with zero first moments [58], Here, µ denotes the energy (squeezing) of the state for shot noise 1/2 and mean photon number (or signal energy) N S , and I denotes the 2×2 identity matrix. Therefore, V µ s is a 2|s| × 2|s| real matrix which is fully symmetric. In order to capture maximal correlations at finite squeezing, we set the correlation parameters See the Methods section for more details on this state. Hence, we may construct unassisted, global quantum probe states from CV-GHZ sub-states. Given an arbitrary N -partite probe-domain distribution S, and assuming that all sub-states are of the same energy µ, the global M -copy input state is given by As seen in Eq. (19), the magnitude of the correlations held by CV-GHZ states c max has a reciprocal dependence on the number of modes m in the state. This implies that the quantum correlations become "thinner" as the number of modes increase, demanding more energy in order to maintain a high degree of entanglement. It is therefore beneficial to consider probe-domain distributions of shorter range CV-GHZ states in order to make the best use of the enhanced distinguishability that entanglement can provide. Motivated by this, we can design specific probe-domain distributions that exclusively use unassisted TMSV entangled states, rather than wider-spread CV-GHZ states with weaker quantum correlations, i.e. the probe-domain of each sub-state is simply |s j | = 2, ∀j. By employing TMSV states in conjunction with dynamic block protocols, we can combine the enhanced distinguishability of entangled input states with the implicit error-correcting behaviour available through dynamic probing.
To systematically access both of these features, we introduce a nearest-neighbour probe-domain distribution. This defines a non-disjoint probe-domain distribution which probes neighbouring channels using twomode probe states (defining neighbouring channels on a closed 1-dimensional lattice). The non-disjoint partition set takes the form, where mod denotes the modulo operation. For example, if m = 4, the probe-domain distribution is sim- In this way, each channel in the global pattern is probed exactly twice per single round of a dynamic block protocol (the average channel use is simplyM = 2M ). Diagrammatically, this distribution is illustrated in Fig. 3(b). The nearestneighbour protocol is conveniently designed, as it allows us to develop non-disjoint probing structures in a consistent way, and can be applied to channel patterns of any size (for more detailed arguments and motivations surrounding this protocol, see the Methods section).

Numerical Results
In this section, we collect numerical results to benchmark the performance of both fixed and dynamic unassisted block protocols for the discrimination of bosonic pure-loss channel patterns (quantum reading) and Gaussian additive-noise channel patterns (environment localisation). We investigate a number of pattern recognition scenarios: CPF, k-CPF, and arbitrary binary pattern classification (or barcode decoding). In each setting, we consider the worst-case discrimination scenario such that all patterns within an image space occur with a uniform probability, i.e. we consider the pattern probability distribution In all cases we employ unassisted CV-GHZ states in accordance with various disjoint/non-disjoint probedomain distributions. The average error probability associated with these protocols can be accurately upper and lower bounded using the fidelity bounds in Eqs. (3) and (4) for which a variety of numerical and analytical techniques can be used for arbitrary multipartite distributions (see Methods for details on the numerical computations).
These unassisted protocols can be compared to the best known classical and quantum assisted protocols in order to critically benchmark their efficacy (details can be found in Methods). A sufficient condition for quantum advantage occurs when the upper bound for the average error probability associated with a quantum enhanced protocol p q,U err is less than a lower bound on the error probability associated with an optimal classical protocol p cl,L err . Hence, we may qualify guaranteed quantum advantage when We use this quantity ∆p err to identify when an unassisted quantum protocol can certifiably obtain quantum advantage over all classical protocols.

Discrimination of Bosonic Loss
This can be used to describe a basic imaging setting, in which pixels are described by pure-loss channels of different transmissivity/reflectivity η j for j ∈ {B, T }. As explored in [41], Banchi et al. showed that major quantum advantage can be obtained using an idler-assisted approach. This advantage is particularly useful in a low energy regime, where the number of probe copies required to achieve high precision is dramatically reduced. Here we report that quantum advantage can be similarly guaranteed using a range of unassisted protocols. Moreover, it is possible to achieve unassisted performances comparable with that of full idler-assistance via dynamic block protocols.
Figs. 4 (a)-(c) depicts error upper and lower bounds for the multi-channel discrimination of bosonic loss (upper bounds are plotted as dashed lines, lower bounds are solid). We consider m = 9 binary channel patterns such that background channel possess transmissivity η B = 0.99, while target channels possess η T = 0.97. In each panel (a)-(c) we consider a different image space: CPF, (k = 3)-CPF and barcode pattern recognition respectively. Within each setting, we construct fixed and dynamic unassisted block protocols using CV-GHZ substates, each with mean photon energy N S = 20. The precise probe-domain distributions are identified diagrammatically in the legend. Fig. 4(a) shows results for CPF. While one can eventually confirm quantum advantage using a block protocol with a single m = 9 mode CV-GHZ state (as studied in Ref. [51]), this is only certifiably advantageous using a very large average channel use,M ≈ 3000, compared to the idler-assisted protocolM ≈ 30. Furthermore, for the larger image spaces it quickly becomes too costly to achieve guaranteed quantum advantage, such as for 3-CPF and barcode discrimination. Instead, one may use a dynamic protocol to achieve error rates on par with the idler-assisted performance. Using the nearest-neighbour dynamic protocol as per the probe-domain distribution in Eq. (21), one may readily obtain guaranteed quantum advantage regardless of the image space. This dynamic protocol not only outperforms the optimal classical protocol, but also quickly provides guaranteed advantage over the best fixed unassisted block protocols also, achieving performance on par with idler-assistance. Fig. 5(a)-(c) displays the minimum guaranteed quantum advantage ∆p err associated with the use of nearestneighbour dynamic protocols. Here we plot the difference between the quantum upper bound and the optimal classical lower bound for m = 9 channel pattern discrimination. This is carried out for η T = 1,M = 100, a variety of signal energies N S , and background transmissivities η B . The difference in advantage achieved by the idler-assisted protocol and the nearest-neighbour dynamic protocol is too small to be displayed; emphasising that we can not only achieve quantum advantage without idlers, but effectively match the performance of idler-assistance.

Environment Localisation
We now consider environment localisation. Here the task is to classify channel patterns in which each channel possesses background or target noise properties, ν B /ν T . Note that we focus on additive noise channels as an idealised scenario for thermal-loss/amplifier channels since the inclusion of loss τ = 1 will only degrade the performance of our unassisted protocols. It has recently been proven that the ultimate error bounds for this pattern recognition setting are non-adaptively achieved by idlerassisted TMSV states [42,43].
In Figs. 4(d)-(c) we report the performance of a number of different fixed/dynamic unassisted protocols for the task of environment localisation. Again, we consider m = 9 length channel patterns for a trio of image spaces, CPF, (k = 3)-CPF and barcode pattern recognition. Each channel is characterised as an additive noise channels with ν B = 0.02 or ν T = 0.01, and our probe states again have mean photon number N S = 20. It is immediately clear that unassisted, fixed block protocols in this setting are ineffective, as shown by the very poor lower bounds in these results. Without idlers, the output distinguishability of disjointly distributed probestates is extremely poor, and degrades further with increasing probe-domain size.  Yet, performance can be redeemed via dynamic protocols. By overlapping entangled probe-domains over channel patterns, we increase the opportunity of interacting with distinguishable channel regions. Indeed, the use of the nearest-neighbour dynamic protocol allows for guaranteed quantum advantage to be obtained in a number of discrimination settings where fixed protocols are unable to even match the classical performance (see Methods for more nuanced insight to this result). Interestingly, alternative non-disjoint probe-domain distributions can be seen to achieve quantum advantage also, in some cases outperforming the nearest-neighbour protocol as shown in Fig. 4(d). The question of identifying optimal dynamic protocols is thus highly non-trivial and very interesting.
Finally, Figs. 5(d)-(f) compare the guaranteed quantum advantage ∆p err associated with idler assisted protocols with that of the nearest-neighbour dynamic protocol in this discrimination setting, for ν T = 0.01 and a variety of resource/environmental parameters. The most significant guaranteed advantage is observed for 1-CPF, as shown in both Fig. 4(d) and Fig. 5(d). While it is clear that unassisted protocols are more sensitive to noisy, thermal environments, quantum advantage is still achievable without the use of idlers. These results emphasise the achievability of quantum-enhanced, idler-free protocols for short-range environment localisation tasks.

DISCUSSION
We have formalised the construction of unassisted, quantum enhanced discrimination protocols in a multichannel setting, using multipartite quantum states. We identified two distinct classes of block protocols, fixed and dynamic, which differ in how they distribute multipartite entanglement across channel patterns. The operational interpretations of these protocols were discussed, along with their relationship with one another. Furthermore, we formulated a logical correspondence between dynamic protocols and error correction; variable probedomains throughout discrimination help to correct errors that fixed probe-domains cannot.
In order to explicitly study the efficacy of these protocols, we designed unassisted protocols for the discrimination of bosonic Gaussian channel patterns. These protocols were based based on the use of entangled, multi-mode CV-GHZ states. Through analytical and numerical investigation, we showed that these unassisted protocols can provide significant advantage over the optimal classical strategies for the discrimination of both bosonic loss and environmental noise. In some cases, idler-free approaches can achieve performance on par with idlerassistance.
These results strongly encourage the further investigation of dynamic block protocols. Motivated by the problem setting and chosen class of probe-states, we were able to engineer high performance, quantum-enhanced unassisted protocols. However, determining the optimal unassisted protocol for specific multi-channel discrimination tasks is now an open question. It is of interest to explore more sophisticated versions of these protocols based on the optimisation of probing configurations over specific image spaces, and adaptive protocols that modify probe-domain distributions on the fly. Such studies could reveal high performance, unassisted discrimination strategies tailored to realistic applications.
Since this research was conducted in the CV picture based on Gaussian entangled states, this makes it particularly relevant to near term quantum sensing applications. The exploration of alternative entangled probe states is an immediate path of interest, as the employment of popular non-Gaussian entangled states could provide further enhancements to these unassisted protocols. Furthermore, the translation of this research for finite-dimensional channels is also an important topic, in which similar unassisted protocols may display strong quantum advantage.
Investigating the space of unassisted discrimination protocols is of importance for near term quantum technologies. The insights and results of this work significantly loosen the resource constraints on realisable quantum technologies that rely on pattern recognition, emphasising that idler-assistance is not always a necessity.

Quantum Barcode Decoding
The most general pattern recognition task for binary channel patterns is quantum barcode decoding [41]. This is a multi-hypothesis discrimination task of identifying a channel pattern from the entire space of binary channel patterns. All other image spaces are necessarily a subset of U m , hence quantum barcode decoding embodies the most challenging multi-channel discrimination problem in this setting. It represents a scenario in which we have no a priori information that can narrow the space of potential quantum channel patterns, and all i ∈ U m must be considered within the ensemble.

Channel Position Finding
The task of Channel Position Finding (CPF) describes the multi-hypothesis discrimination task of locating a single target channel E T hidden amongst an array of background channels E B . An m-channel CPF problem is associated with the image space U CPF which is the set of all m-length multi-channels which contain exactly one target channel.
Let us define a function which constructs an m-length channel pattern with one target channel in the x th position of the set More generally, we may investigate k-CPF, where the number of targets channel that occur within each channel pattern is precisely k < m, hidden amongst m − k background channels. We denote this image space U k CPF . Let us define a more general function which generates an m-length channel pattern with precisely k-target labels in the positions indicated by the unique indices Here, each permutation operator P 1...k:x1...x k swaps all of the target channel labels from positions 1, . . . , k with the channel labels at the positions x 1 , . . . , x k . Then we can construct any k-CPF image space by iterating over all unique permutations of the target channel labels, For an m-channel k-CPF problem, U k CPF contains exact C k m channel patterns, where C k m = m!/(k!(m − k)!) is the binomial coefficient. For example, if m = 3, k = 2, then the image space is Clearly when k = 1 we regather the previous single CPF image space. Both CPF and k-CPF find a number of fundamental settings within target-detection, quantum enhanced classical data-readout and environment localisation. They provide a valuable platform for studying multi-channel discrimination; if we can understand how to attain quantum enhancements in the readily analysable CPF framework, then we can learn to extract and apply these enhancements in more complex settings.

Bosonic Gaussian Channel Patterns
Under the action of a single-mode GPI quantum channel, an input Gaussian state described completely via its covariance matrix (CM) V in with zero first moments undergoes the transformation where I is a 2 × 2 identity matrix. The overall quantum channel can be denoted as E τ,ν and is defined with respect to a transmissivity parameter 0 ≤ τ ≤ 1 describing attenuation/amplification properties and an induced noise parameter ν ≥ 0. Binary GPI channel patterns then consist of a sequence of m-GPI channels with unique target/background transmissivities τ B , τ T and noise properties ν B , ν T . Consider now a m-mode Gaussian state with CM V in and zero first moments. Let the following be a matrix function of a general variable x which depends on a position k in a channel pattern i, Then a multi-mode Gaussian state which is transformed according to a GPI binary channel pattern ρ → E i (ρ) undergoes the following transformation on its CM in phase space Therefore, it is easy to study the CMs of multi-mode Gaussian probe states interacting with GPI channel patterns.

Bosonic CV-GHZ States
As discussed in Results, a CV-GHZ state can be constructed as the extension of a TMSV state to many modes. Indeed, consider a CV-GHZ state defined over an m-length probe-domain, Φ µ {1,...,m} . This m-mode state can be completely characterised by its CM (with zero first moments) as given in Eq. (18). Here we show why maximum correlations are satisfied at c 1 = −c 2 = c max .
The symplectic spectrum of the CV-GHZ state takes the form, such that ν + is (m − 1)-degenerate. In order to capture maximal correlations (at finite squeezing), we use the bona fide condition ν ± ≥ 1 2 to state that Hence, this leads to the notion of maximal symmetric correlations when the correlation parameters satisfy c 1 = −c 2 = c max = (m − 1) −1 µ 2 − 1/4. CV-GHZ states can then be readily used to construct probe states in conjunction with a probe-domain distribution, where it can be used as a building block for more general multipartite states. This construction can be equivalently represented in phase space, where tensor products over sub-states become direct sums over sub-CMs. More precisely, given an N -partite probe-domain distribution S (disjoint or non-disjoint) we can equivalently express the global CV-GHZ input state Φ µ S via its CM, where V µ sj is the CM of a |s j |-mode CV-GHZ state irradiated over the modes contained in the probe-domain s j .

Numerical Computation of Error Bounds
Consider two unique, m-length Gaussian channel patterns E i and E i which are probed by two identical, mmode CV-GHZ states Φ µ {1,...,m} . We can conveniently write the output states from these interactions, Now consider the fidelity between these two output states F (Φ µ i , Φ µ i ). Thanks to the Gaussianity of CV-GHZ states and GPI multi-channels, the fidelity between these states can be computed exactly using only their phase space representations using the formulae from [59], where we denote F G as the Gaussian fidelity function.
In summary, we have a way to represent the input probe states of unassisted block protocols, through V µ S ; the ability to describe output states by transforming input states according to GPI multi-channels E i as in Eq. (32), and the means to compute the fidelity between any two output states. Given these techniques, and a multichannel discrimination problem {π i ; E i } i∈U , we can readily compute the fidelity-based error probability lower and upper bounds, To study the error bounds of dynamic block protocols, we need only invoke the dynamic/fixed protocol transformation discussed in the Results section. In this way, we modify the channel patterns according to the probedomain distribution i → ν i . By computing the fidelity between the outputs of CV-GHZ states irradiated over the modified patterns error bounds can be readily computed for dynamic block protocols. The numerical methods presented here can always be used for fixed or dynamic block protocols, and more generally using any Gaussian input states.

Classical Performance
Here we collect expressions for the classical fidelities using optimal coherent states for the multi-channel discrimination settings explicitly studied in this work. These can then be used to derive exact error-bounds, and benchmark quantum advantage.
The best classical protocol for discriminating a single pure-loss channel is achieved by a block protocol using coherent states. Indeed, the optimal (energy constrained) M -copy, single-mode coherent state given by [41], where N S is the mean photon number of the signal state. If a single pure-loss channel possesses transmissivity η B or η T , the fidelity between the two possible single-copy output states is given by For m-length multi-channels, we simply use m single mode coherent states to discriminate each channel independently.
For additive noise channels, τ = 1 and the task is to discriminate between background/target noise ν B , ν T > 0. The optimal classical input state is just the m-copy vacuum state |0 ⊗m , since displacements or phase shifts have no impact on the output states from the channel. The single probe copy fidelity between the potential single-mode output states can be written as [43], Using these fidelity expressions within the error probability bounds from Eqs. (3) and (4), we can provide effective classical benchmarks for both multi-channel discrimination settings.

Idler-Assisted Performance
We can similarly collect expressions for idler-assisted block protocols in the context of bosonic pure-loss channel patterns and environment localisation.
For the discrimination of bosonic loss, one can employ an idler-assisted protocol in which each channel is probed with one mode from an M -copy TMSV state, Φ µ , where µ = N S + 1/2 is the level squeezing. Consider a single pure-loss channel E j which may have transmissivity η j ∈ {η B , η T }. While one mode interacts with the channel, the other mode is perfectly protected (in a quantum memory for instance) and thus undergoes the action of an identity channel. The output state is then simply the finite-energy Choi state Φ µ Ej = E j ⊗ I(Φ µ ). The fidelity between the output states Φ µ E B and Φ µ E T is [41] F loss idler = By extending this to m-channels using M -copy probes, we can easily bound performance of the idler-assisted block protocol. We can perform an identical analysis for environment localisation by computing the fidelity between possible output states of a pair of additive-noise channels with target noise or background noise ν B /ν T . In this case, it is convenient to utilise the parameter µ = N S + 1/2, where the output fidelity reads [42] Once again extending these fidelities to consider mchannels and using M -copy probes states, we can easily bound the performance of idler-assisted block protocols.

Fidelity Degeneracies of CV-GHZ States
Consider an arbitrary, N -element image space U = {i 1 , i 2 , . . . , i N } which generate m-length quantum channel patterns, and the associated multi-channel ensemble {π i ; E i } i∈U . Now, consider the use of an unassisted block protocol using m-mode CV-GHZ states. As we know from Results, we can benchmark the performance of this protocol via the fidelity-based error probability bounds in Eqs. (3) and (4).
While this can be achieved numerically, it can become inefficient. The total number of ways that we can choose unequal pairs of channel patterns is N (N − 1). This means that in general, there exist N (N − 1) potentially unique, non-unit fidelities that one needs to compute in order to calculate the error bounds. For large pattern spaces this can be enormous, making it difficult to analytically keep track of all possible output fidelities, or numerically perform these sums.
However, thanks to their symmetry, when using CV-GHZ states as quantum probes the number of unique fidelities that may occur is dramatically reduced. The CV-GHZ state symmetry causes many of the unique output fidelities within Eqs. (3) and (4) to be highly degenerate. Indeed, fidelity degeneracy tells us that if there are exactly g fid unique output fidelities, typically g fid N (N − 1). Let us be more precise: Consider a pair of image spaces of m-length, binary channel patterns: one is the k-CPF image space U k CPF , such that each pattern contains precisely k-target channels, and the other is a l-CPF image space U l CPF such that each pattern contains precisely (l = k)-target channels. Take two identical m-mode CV-GHZ states Φ µ {1,...,m} which interact with the multichannels E i and E i , resulting in two unique output states Φ µ i and Φ µ i . Now consider the fidelity, We find that this fidelity is equivalent for all pairs of channel patterns i, i which have the same Hamming distance. That is, for all For a rigorous proof of this, please see [60]. Fidelity degeneracies are extremely useful, and can help to not only improve numerical efficiency, but reveal analytical insights.
In Fig. 6 we have numerically investigated the fidelity degeneracy properties of a number of different unassisted dynamic/disjoint protocols for the discrimination of pure-loss channel patterns. Here we observe two clear points; CV-GHZ states lose distinguishability when we widen their domain size as expected, due to weakening quantum correlations (discussed in the Results). This can be seen by comparing the output fidelity spectrum of the m = 10 mode CV-GHZ probe state (green) compared to the other probe-domain distributions.
Furthermore, non-disjoint probe domain distributions are able to "spread out" the degeneracies involved with disjoint probing protocols. In Fig. 6, we compare a fixed block protocol using a disjoint, exclusively two-mode distribution of probe domains (orange). We then use a dynamic nearest neighbour protocol with the exact same  Figure 6. CV-GHZ Fidelity Degeneracies: Histograms of fidelity degeneracies for unequal m = 10 pure-loss channel patterns over the complete image space of binary channel patterns U10, for a selection of probe-domain distributions: m-mode CV-GHZ state (green), disjointly distributed TMSV states (orange) and nearest neighbour distributed TMSV states (blue). The fixed block protocols are using M = 2 probe-copies, while the dynamic nearest neighbour protocol is using M = 1 copies. In this way, the average channel use of all the protocols is the same,M = 2.
number of probe modes (blue). While the output fidelity distributions possess a similar spread in values, the variation in probe-domains raises many of the degenerate fidelities. In doing so, it flattens the overall distribution, and gives rise to more distinguishable output fidelities.

Fidelity Degeneracies of TMSV states
We will now focus on the case of using m = 2 CV-GHZ states, i.e. TMSV states. As discussed in Results, these states maximise their entanglement content with respect to input energy, since quantum correlations do not need to be spread across many modes. Furthermore, they offer the simplest test case for analytically investigating fidelity degeneracies. This will help to unveil concrete reasons for the discrepancy between fixed and dynamic protocols.
We wish to identify all the possible, unique output fidelities associated with TMSV states irradiated over m = 2 length binary channel patterns. We can summarise this image space easily as it is very small, Thanks to the fidelity degeneracy properties discussed in the previous section it turns out that there are only four unique sub-fidelities that can occur when one irradiates two-mode binary channel patterns with unassisted TMSV states. Here, we define a sub-fidelity as a single output fidelity that occurs between specific pairs of channel patterns. These sub-fidelities are completely determined by the number of the target channels, k and l, contained within the considered channel pair, i and i respectively. Hence we will denote each sub-fidelity in the form F k:l where k (l) indicates the number of target channels in the channel pattern i (i ). Doing so, we can write all the unique, two-mode sub-fidelities The Bures fidelity is a symmetric function, therefore the order of i and i is irrelevant. These are the only fidelities that can occur when using TMSV states over pairs of m = 2 length channel patterns. Furthermore, the fidelity is multiplicative, meaning that Hence, when using collections of exclusively two-mode states following some probe-domain distribution (such as in the nearest neighbour protocol), then all of their unique output fidelities will always be a specific product of these sub-fidelities in Eq. (52). Hence, these subfidelities can be used to completely characterise any unassisted discrimination protocol using exclusively TMSV states (see Ref. [60] for more details). While this may seem like an unnecessary level of detail, the investigation of these sub-fidelities helps to reveal critical features of dynamic block protocols. Each of the sub-fidelities in Eq. (52) is a unique function that can be analytically characterised via the Gaussian fidelity formulae from [59]. They each possess a unique behaviour dependent upon the multi-channel discrimination setting that we are considering. If one of the sub-fidelities is typically very large, this means that the specific pair of channel patterns that is refers to are very difficult to discriminate. If a sub-fidelity is very small, then the pair of channels it refers to are very easy to discriminate. For example, if F 0:2 F 1:1 in a particular problem setting, then it is much easier to discriminate the patterns {B, T } from {T, B}, rather than {B, B} from {T, T }.
Most importantly, when utilising unassisted discrimination protocols, there is an inconsistency of distinguishability between different collections of quantum channels. This inconsistency leads to the corrective behaviour that dynamic protocols can provide. We will convey this inconsistency by considering the settings studied in the Results section. channels with target noise ν T and background noise ν B . When we look closely at the sub-fidelities concerned with this setting, we notice a glaring inconsistency. Three of these sub-fidelities (F 0:1 , F 0:2 and F 1:2 ) all assume their minimum in the limit of infinite squeezing, That is, by increasing the energy of our input states, we can expect to improve the discrimination of the appropriate pairs of channel patterns. This sub-fidelity behaviour is displayed in Fig. 7(a). However, this is not the case for the sub-fidelity F 1:1 (µ), which is concerned with the discrimination of the pattern {B, T } from {T, B} (and vice versa). This sub-fidelity explicitly takes the form where we define the quantities θ = 2ν B ν T + 1 + 2µ(ν B + ν T ), In the limit of infinite squeezing, we find that lim µ→∞ F 1:1 (µ) = 1, meaning that in the limit of infinite probe state energy, the pair of channel patterns {B, T } and {T, B} become completely indistinguishable. Clearly, this will have a hugely detrimental effect on discrimination performance which cannot be remedied by increasing the input probe energy. A similar effect can be observed within pure-loss channel patterns. Let us consider the case of probing m = 2 length binary channel patterns such that each channel is a pure-loss channel with either a target transmissivity η T or background transmissivity η B . Now, we find that the two mode sub-fidelities F 0:1 , F 1:1 , and F 1:2 tend to zero in the limit of infinite squeezing, min F k:l (µ) = lim µ→∞ F k:l (µ) = 0.
This means that in the limit of infinite energy and maximum entanglement, the channel pairs that characterise each of these sub-fidelities become perfectly distinguishable. However, we may focus on the quantity F 0:2 (µ) which defines the distinguishability of the patterns {B, B}, {T, T }. This sub-fidelity takes a relatively compact form given by, where we use µ = N S + 1/2 as before and we define the quantities In the limit of infinite squeezing, this sub-fidelity F 0:2 instead finds the finite quantity which is non-zero when either η j = 1. Therefore, even when using infinitely squeezed input states, the patterns i = {B, B} and i = {T, T } are not perfectly distinguishable via unassisted input states. The behaviour of these sub-fidelities are displayed in Fig. 7(b).
It is now clear why dynamic protocols are so effective at redeeming the performance of these multi-channel discrimination tasks. When using unassisted, multipartite entangled probe states, the distinguishability of the output states can vary considerably, dependent upon the collection of quantum channels that they interact with. By overlapping probe-domains, channels can be probed in conjunction with different collections of channels in the pattern. In doing so, we are increasing the likelihood of probing a more a distinguishable collection. These more distinguishable channel regions are then able to correct the errors invoked by probes interacting with poorer regions.
As an example, let us take the discrimination of Gaussian additive-noise channels with TMSV states according to some probe-domain distribution. Consider an m = 4 length channel pattern, that we wish to discriminate. Here, we first consider a fixed block protocol that follows the probe-domain distribution S d = {{1, 2}, {3, 4}}. As a result of this distribution, we will possess the following TMSV states which irradiate specific channel sub-patterns, As discussed earlier in this section, the sub-fidelity F 1:1 is very poor, meaning that the sub-patterns {B, T } and {T, B} are very difficult to distinguish from one another. Therefore, if we irradiate a sub-pattern {B, T } or {T, B} directly with a TMSV state, our overall discrimination ability will be very ineffective, as we will struggle to determine which is the true pattern. It is highly desirable to avoid instances of this kind of pattern interaction, but obviously we cannot know prior to interaction where these pairs of channel patterns arise (this would defeat the purpose of discrimination). The embodies a critical weakness of fixed block protocols. If we unwittingly choose a probe-domain distribution which irradiates input states over poorly distinguishable collections of patterns, the ability to discriminate the overall channel pattern will be compromised.
We now see why varying the probe-domains is so effective. Consider a non-disjoint probe-domain distribution, S nd = {{1, 2}, {2, 3}, {3, 4}, {1, 4}} (this is in fact the nearest neighbour protocols discussed in Results). Consider the same m = 4 length channel pattern, i = {B, T, T, B}. As a result of this distribution, we will possess the following TMSV states which irradiate the specific channel sub-patterns, While we are still unfortunately irradiating the poorly distinguishable collections of channels {i 1 , i 2 } = {B, T } and {i 3 , i 4 } = {T, B} with two of our input probes, we now also apply probe states to the sub-patterns {i 2 , i 3 } = {T, T } and {i 1 , i 4 } = {B, B}. These collections of channels are much more distinguishable, and invoke the stronger sub-fidelities F 0:1 , F 0:2 , and F 1:2 . Hence, by varying our probe domains we are able to gather different, more distinguishable "opinions" of regions of the channel pattern. The stronger distinguishability of the regions {i 2 , i 3 } and {i 1 , i 4 } can help to correct the contribution of the poorly distinguishable channel collections.
It is important to note that this improved performance is not connected to an increased number of probe modes. Recall that we can fairly compare dynamic/fixed block protocols with the same average channel useM . With equivalent resources, the dynamic protocol will outperform the fixed version thanks to its variable probedomains.

ACKNOWLEDGMENTS
C.H acknowledges funding from the EPSRC via a Doctoral Training Partnership (EP/R513386/1). S.P acknowledges funding from the European Union's Horizon 2020 Research and Innovation Action under grant agree-


extfieldsofstudy: Physics Mathematics

text:
Large Deviations in Random Sequential Adsorption

In a random sequential adsorption process, objects are deposited randomly, irreversibly, and sequentially; if an attempt to add an object results in an overlap with previously deposited objects, the attempt is discarded. The process continues until the system reaches a jammed state when no further additions are possible. Exact analyses have been performed only in one-dimensional models, and the average number of absorbed particles has been computed in a few solvable situations. We analyze a process in which landing on an empty site is allowed when at least $b$ neighboring sites on the left and the right are empty. For the minimal model ($b=1$), we compute the full counting statistics of the occupation number.

I. INTRODUCTION
Random sequential adsorption (RSA) is a toy model mimicking the irreversible deposition of suspended particles onto substrates. The RSA model postulates that the deposition events are random: If the new particle is sufficiently far away from already deposited ones, it sticks to the substrate; otherwise, the deposition event is discarded. In the two-dimensional setting, the RSA models have been applied to modeling chemisorption on singlecrystal surfaces and adsorption in colloidal systems [1][2][3][4][5]. There are also applications in nanotechnology, see [6][7][8][9]. The RSA models have been also used in high dimensions, e.g. in the context of packing problems [10].
The first RSA model with adsorption of dimers was introduced by Flory for the description of reactions along a long polymer chain [11]. Another beautiful RSA model with adsorption on a continuous one-dimensional line was introduced by Rényi [12] as a toy model of car parking. The RSA type of models has been also used in several other one-dimensional settings, e.g., in modeling polymer translocation [13,14] and describing zero-temperature dynamics of Ising chains [15][16][17].
The RSA models mimic the generation Rydberg excitations. In experiments, a laser excites ultra-cold atoms on a lattice from the ground state to a Rydberg state (see e.g. [18][19][20][21]). Interactions between Rydberg atoms cause the blockage forbidding the excitation of atoms sufficiently close to a Rydberg atom. When the radiative decay of the Rydberg atoms can be ignored, this RSA model mimics certain features of the excitation process [22], although it disregards features like the non-ergodic quantum dynamics of Rydberg-blockaded chains (see [23][24][25][26] and references therein).
We investigate a model in which particles are absorbed randomly and irreversibly onto an interval with L sites. We assume that each particle attaches to a single site. The blockage effect is modeled by the requirement that adjacent particles are separated by at least b vacant sites. In experimental realizations with Rydberg atoms where excitation plays the role of deposition, these systems typically occupy L ∼ 50 lattice sites. For such relatively small systems, fluctuations are significant and they can be probed experimentally [21,27]. The minimal model with blockage radius b = 1 is especially popular in the realm of laser-driven Rydberg lattice gases [23][24][25][26]28].
In Sec. II, we compute the average occupation number and a few other basic properties of the minimal model. The average occupation number was already established by Flory in a mathematically isomorphic model [11], our goal is to outline the tools which we employ later. The minimal model is further studied in Sec. III where we compute the full counting statistics of the occupation number. In the general case when the blockage radius is arbitrary (Sec. IV), the average occupation number is still easily computable; the generating function encoding all the cumulants satisfies a Riccati equation which appears unsolvable for any b ≥ 2. Cumulants can still be extracted one by one via more and more cumbersome calculations, so we only determine the variance (the computations are relegated to Appendix A). The extreme probabilities of maximally sparse and maximally dense jammed configurations are tractable for arbitrary b ≥ 1.

II. THE MINIMAL MODEL: AVERAGE PROPERTIES
The exclusion interaction between particles is modeled by requiring that the two sites surrounding an occupied site are empty. Thus there is the blockade effect as each absorbed particle forbids future adsorption into the nearest-neighbor sites. Eventually, the system reaches a jammed state. For instance, is a jammed configuration with N = 9 occupied sites on the interval of length L = 19. Locally, jammed states vary from realization to realization. More global features of jammed states become deterministic in the thermodynamic limit, L → ∞; e.g., the fraction of short segments • • • is 1−3e −2 , while the fraction of the long segments • • • • is 3e −2 . Since we study fluctuations, so we must analyze finite systems. We consider open intervals that are relevant in the context of Rydberg atoms; the extension to rings is straightforward.
A. The average number of occupied sites Here we outline the computation of the average number A L = N of occupied sites on an open interval of length L. We employ an approach essentially developed by Flory [11], see also a textbook [29]; our analysis (Sec. III) of the cumulant generating function will proceed along similar lines.
For small L, one can compute A L = N by hand. For instance, N = 1 when L = 1 and 2; for L = 4, one gets N = 2. For L = 3 and L ≥ 5, the total number N of absorbed particles varies from realization to realization, e.g. N = 1 (probability 1/3) or N = 2 (probability 2/3) when L = 3. Thus To compute A L for arbitrary L, suppose k is the landing site of the first particle. The intervals of lengths k − 2 and L − k − 1 on the left and right of the first deposited particle are subsequently filled independently. This key feature makes the one-dimensional situation tractable. For our model, we arrive at the recurrence Using the generating function we convert the recurrence (3) into a differential equation Solving this linear inhomogeneous differential equation subject to the initial condition A(0) = 0 we obtain from which Using (7) one can extract the large L behavior: B. Behavior near the boundary In the L → ∞ limit the jamming density is When L 1, the probability that a site in the bulk is occupied is very close to (9). Near the boundary the densities are different.
Let us first compute the probability p L that the leftmost is occupied. For small L, one can compute p L by hand to yield etc. Generally, the probability p L satisfies the recurrence which is established similarly to the recurrence (3). Using the generating function we convert the recurrence (11) into a differential equation which is solved to yield Expanding P (x) we obtain In the L → ∞ limit, i.e. for the semi-infinite lattice, the left-most site is occupied with probability which substantially exceeds the probability (9) that a bulk site is occupied. The probability q L that the second site, viz. the site adjacent to the left-most site is occupied, is dual to the probability p L : for L ≥ 2. In particular The probability r L that the third site is occupied satisfies the recurrence Using the generating function we convert the recurrence (19) into a differential equation with P (x) given by (14). Solving (21) subject to R(0) = 0 we obtain from which In particular A longer calculation gives These results hint that with some positive rational A n , B n , C n , D n . Recalling that π ∞ = ρ jam , one gets

C. The total number of jammed states
The total number of jammed states J L satisfies an obvious recurrence The solution of this recurrence subject to the boundary conditions J 0 = J 1 = 1 and J 2 = 2 is encapsulated in the generating function In particular, the asymptotic behavior is where r is the real root of the cubic equation r 3 = r + 1 known as a plastic number. The Padovan sequence (26) is a cousin of the Fibonacci sequence, it arises in several applications, see e.g. [30,31].

III. THE MINIMAL MODEL: FULL COUNTING STATISTICS
One-dimensional RSA models are often tractable (see [1,2,29] for a review) and basic features like the jamming density have been probed analytically. For the minimal model with b = 1, we have shown (Sec. II) how to compute the jamming density, the behavior near the boundary, and the total number of jammed states. In this section we analyze fluctuations.

A. Full counting statistics
Consider an interval of length L. The RSA procedure brings the system into a jammed state. The total number N of absorbed particles fluctuates from realization to realization. Here we compute the cumulant generating function that encodes all the cumulants of N . Thus we want to compute the average where P (N, L) is the probability to have N particles in a jammed state. The standard relation then gives all the cumulants: the average N c = N , the variance N 2 c = N 2 − N 2 , etc. The function F (λ, L) ≡ e λN grows exponentially with L. The cumulant generating function encapsulates all cumulants: To determine F (λ, L) we proceed in the same way as in the derivation of the average and obtain a recurrence expressing that after the first deposition event, say at site k, the intervals on the left and the right are filled independently. (The one-dimensional nature of the problem is crucial for this property.) The recurrence (33) applies for all L ≥ 1 if we set We now introduce the generating function To recast (33) into an equation for the generating function we multiply (33) by Lx L−1 and sum over all L ≥ 1.
The left-hand side turns into The right-hand side becomes and using the boundary conditions (34) we simplify the double sum to and recast the recurrence (33) into a Riccati equation which turns out to be solvable. The solution of Eq. (36) subject to the initial condition Φ(λ, 0) = 1 reads where Λ ≡ e λ/2 . The generating function Φ(λ, x) has a simple pole at x = y = y(λ). Using (37) we find that the pole is implicitly determined by The cumulant generating function is plotted in Fig. 1. We already know the dominant exponential factor in F (λ, L) ∝ e LU (λ) . The exact result (37) allows one to find a pre-exponential factor. A straightforward calculation gives the residue of the simple pole of Φ(λ, x). We find when L 1. The cumulant generating function is implicitly defined through (38)- (39). Expanding U (λ) we recover already known value of the first cumulant: N = Lρ jam . In principle, one can determine an arbitrary cumulant. We haven't succeeded in finding an explicit general formula.
Here we merely list a few cumulants that are extracted by expanding U (λ) implicitly given by (38) The so-called Mandel Q parameter [32] indicates that the statistics is strongly sub-Poissonian in the present case. The ratios N n c / N of cumulants to the average are known as Fano factors [33]. Here are a few Fano factors

B. Extremal jammed states
Extremal jammed states are the states with the largest and the smallest number of occupied sites. Let us extract the probabilities of such states from the behavior of the cumulant generating function in the λ → ±∞ limits. An asymptotic analysis of (38) yields (see also Fig. 1) where we dropped the terms vanishing in the λ → ±∞ limits and denoted by u the positive root of the equation u tanh(u) = 1; numerically u = 1.19967864 . . .. Combining (40) and (44) we get To appreciate the first asymptotic in (45) we note that in the λ → −∞ limit the dominant contribution to the sum in Eq. (29) is provided by the jammed state with the smallest number of occupied sites. This jammed state has N min = (L+2)/3 particles explaining the dominant e λL/3 factor. The 3 −L/3 pre-factor gives the probability to end up in such jammed state.
In the λ → ∞ limit the dominant contribution to the sum (29) is provided by the jammed state with the largest number of occupied sites: We have N max = (L + 1)/2 and this explains the e λL/2 factor in (45). The u −L pre-factor is the probability to end up in the jammed state with the largest number of occupied sites. Thus the large deviation technique gives the asymptotic behaviors of the probability to reach the extremal jamming jammed states: The probabilities P (N min , L) and P (N max , L) can be also computed exactly as follows. Let us start with maximally dense jammed states (46b). The maximally dense state with N max = n particles may occur when L = 2n−1 or L = 2n. For concreteness, consider the first case. The probability µ n = P (n, 2n − 1) can be recurrently determined from The boundary conditions read Introducing the generating function we convert the recurrence (48) into a Riccati equation which is solved to yield Analyzing the divergence of the generating function when x → u 2 , we extract the large n asymptotic For maximally sparse jammed states (46a), say when L = 3n and N min = n, the probabilities ν n = P (n, 3n) satisfy the recurrence from which P (n, 3n) = 3 −n (55a) A similar but longer analysis gives two other series of probabilities of maximally sparse jammed states: P (n, 3n − 1) = 3n + 2 5 1 3 n−1 (55b) and P (n, 3n − 2) = 63n 2 + 45n + 8 350 Thus, in the realm of the RSA model, the defect-free arrays arise with exponentially small probabilities. The defect-free arrays (46a)-(46b) of Rb atoms excited to Rydberg states have been recently assembled experimentally by Bernien et al. [23].
One can verify that P (N, 1) = δ N,1 and P (N, 2) = δ N,1 . We now introduce the generating function This definition implies that Using (56)-(59) we obtain a differential equation for the generating function which is mathematically identical to Eq. (36). The solution of (60) subject to the boundary condition P(x, 0) = 1 is thus identical to (37) in different notation: Generally N lies within the bounds

IV. ARBITRARY BLOCKAGE RADIUS
In this section, we analyze the model with an arbitrary blockage radius and show that some characteristics remain analytically tractable. (In the context of ultra-cold Rydberg atoms, models with b > 1 are often relevant, see e.g. [34,35].)

A. The average number of occupied sites
The average number of occupied sites obeys This recurrence is applicable for all L ≥ 1 after setting A j = 0 for j ≤ 0. Using the generating function (4) we convert (63) into a differential equation Solving (64) subject to A(0) = 0 yields where we shortly write Using (65) we extract the density of occupied sites in the L → ∞ limit. For the minimal model we recover ρ 1 = (1 − e −2 )/2. The next jammed density ρ 2 also admits an explicit expression through standard special functions: is an error function. The following jammed densities are The maximally dense and maximally sparse jammed states provide obvious upper and lower bounds on the jammed density: These bounds hint that the limit exists and further suggest that 1 2 < C < 1. One can compute C using (67) and performing an asymptotic analysis. First we re-write (67) as We then write y = 1 − v/b and transform (71) into In the b → ∞ limit, we can use the asymptotic relation 1 − v b j → e −vj/b and replace summation over j by in- known as the Rényi's parking constant [12]. The efficiency of the coverage is measured by the ratio of the jamming density ρ b to the upper bound: The function F (λ, L) satisfies the recurrence which we recast into a differential equation for the generating function Φ = Φ(λ, x) defined by Eq. (35): This Riccati equation (74) appears solvable only for the minimal b = 1 model. It is still possible to determine the variance by analyzing ∂ λ Φ(λ, x) and ∂ 2 λ Φ(λ, x) at λ = 0. These functions satisfy equations which can be deduced from (74); solving these equation we determined the variance, see Appendix A.
We also note that the probability distribution P (N, L) satisfies a recurrence relation generalizing (56). Using the generating function (58) we recast the recurrence (75) into a Riccati equation which is mathematically identical to (74) and appears unsolvable when b = 2, 3, 4, . . ..

C. Extremal jammed states
The probabilities of the largest possible deviations, i.e. extremal jammed states, can be probed analytically using the same approach as in Sec. III B. We start with maximally dense jammed states and choose for concreteness such states with N max = n in the chains of length is an example of such maximally dense jammed state with N max = 6 and L = 16 for the model with the blockage radius is b = 2. The probabilities µ n = P (n, (b + 1)n − b) satisfy the recurrence and the boundary condition (49). The generating function (50) satisfies a Riccati equation Solving this Riccati equation subject to µ(0) = 1 yields Here I p is the modified Bessel function and we have used shorthand notation β = (1 + b) −1 and z = 2β √ bx. The denominator in (80) vanishes when x → y, where y = y(b) is found from This zero is simple and hence µ n ∼ [y(b)] −n . Re-writing in terms of the length of the chain, we arrive at The plot of u(b) is shown in Fig. 2 To extract more explicit results in the b → ∞ limit we recall that Since z = 2β by(b) 1 when b 1 limit, i.e. β 1, it suffices to keep the first three terms in (83). Using additionally Γ(β) β −1 we obtain in the leading order, from which we deduce This is asymptotically exact when b → ∞, but provides a reasonable approximation already for small b, see Fig. 2.
Maximally sparse jammed states are easier to analyze than maximally dense jammed states. For instance, for chains of length L = (2b + 1)n we have N min = n and the probabilities ν n = P (n, (2b + 1)n) to generate such states satisfy the recurrence from which ν n = (2b + 1) −n .

V. CONCLUSIONS
We have studied a class of one-dimensional random sequential adsorption (RSA) models parametrized by a positive integer b ≥ 1, the number of sites on the left and the right of an occupied site which must be empty. This class of models is tractable. For instance, we have determined the average number of occupied sites, the average densities near the boundaries, and the extreme probabilities to fall into maximally dense or minimally dense jammed states. We have mostly focused on large deviations and derived a Riccati equation for the cumulant generating function. This Riccati equation is solvable only for the minimal model, b = 1. When b ≥ 2, it is still possible to extract lower cumulants via perturbation analysis. In Appendix A we determined the variance. The complexity of such calculations quickly increases with the order of the cumulant, the computation of the variance is already rather involved. The probabilities of maximally sparse jammed states exhibit a simple dependence on b; in the most clean situation of chains of length L = (2b + 1)n when N min = n, the probabilities are (2b + 1) −n . The probabilities of maximally dense jammed states are more challenging: to extract the asymptotic behavior one must solve a Riccati equation for the generating function encoding these probabilities, this is possible for arbitrary b.
The RSA models are occasionally tractable on quasione-dimensional structures, e.g., on ladders. In particular, the RSA model with b = 1 has been studied, and the jamming density has been computed [36,37]. The derivation [36,37] of the jamming density is significantly more convoluted than in the purely one-dimensional setting, but perhaps the extreme probabilities would be easier to derive. The maximal density is again ρ max = 1 2 , while the minimal density on the ladder, ρ min = 1 4 , is smaller than in the one-dimensional setting.
The RSA models are often analytically tractable on lattices without loops, e.g. on the Bethe lattices [1] and some classes of infinite trees [38]. Large tree-like graphs, i.e. graphs with few loops, can be also tractable. For instance, the RSA model on the sparse Erdős-Rényi random graphs with b = 1 has been studied, see [39][40][41][42]. It would be interesting to determine the full counting statistics of the occupation number for this RSA model. which follows from F (0, L) ≡ 1, see (29), and the definition (36) of the generating function Φ(λ, x).
Differentiating Eq. (74) twice over λ, setting λ = 0 and using (A3) we find that B(x) obeys where we have used the shorthand notation Integrating (A4) and using B(0) = 0 we obtain We extract the first two cumulants by analyzing the singular behaviors of A(x) and B(x) near x = 1. Using Eq. (64) we find that A(x) has a pole of degree two Similarly using (A5)-(A7a) one finds that B(x) has a pole of degree three Extracting the amplitudes from the exact solution (A6) is not straightforward. One may try to insert (A7b) into (A4), with the right-hand side computed with the help of (A7a). This gives B 3 = 2ρ 2 b , but does not fix B 2 . To determine B 2 we combine (A6) and (A7b) to find To appreciate that the singular behaviors of A(x) and B(x) near x = 1 give the cumulants we start by recalling that the function Φ(λ, x) has a simple pole at y(λ), so when x → y(λ), with Y (0) = 1, y(0) = 1 (A11) as it follows e.g. from Eq. (A3). The average and the variance are found from the cumulant generating function U (λ) = − ln y(λ) to yield where (· · · ) = d(··· ) dλ λ=0 . Differentiating Eq. (A10) over λ, setting λ = 0 and using (A11) we find Comparing (A7a) and (A13a) we conclude that and thus re-derive that ρ b is indeed the jammed density. Comparing (A7b) and (A13b) and using (A14) we get Collecting these results we arrive at the final formula for the Mandel Q parameter. The jammed density ρ b admits an integral representation, Eq. (67), while J b given by (A9) requires taking the limit of an integral. (The naive replacement, x → 1, in (A9) gives erroneous integral representation.) It is still possible to extract very accurate results, e.g. one finds Q 2 ≈ −0.9518 indicating that the statistics of the occupation number is also strongly sub-Poissonian when b = 2. One can similarly compute higher cumulants. For instance, to determine the third cumulant one deduces the governing equation for ∂ 3 λ Φ(λ, x) λ=0 from (74), solves it and from the singular behavior at x = 1 extracts y ; the third cumulant is then 3y y − 2(y ) 3 − y .


Temporal Link Prediction Using Node Centrality and Time Series

Link prediction is an important task in the area of complex networks. Some networks can be better modeled by temporal networks where the patterns of link appearance and disappearance varying with time. However, most of the previous link prediction researches ignore the temporal behaviors of links. The temporal link prediction needs to predict future links via a known network, considering the temporal relationship of node pairs. We propose a method combining the node centrality with time series. We distinguish the contributions of common neighbors to link generation by their centralities. Compared with benchmark approaches in several temporal networks, the proposed method can improve the accuracy of temporal link prediction efficiently.

timestamps. The final link prediction scores are computed according to such model. In [16], the time series for each node pairs is calculated by the frequency of link occurrence between adjacent snapshot. Then, the likelihood of links are estimated based on the time series model. In [17], a collection of time-series to model are adopted to produce similarity scores over time.
Several methods try to utilize the time-evolving patterns of temporal networks. O'Madadhain et al. [18] regarded the historic events and node attributes as the input of a logistic regression classifier and decided the future links according to the output. In [19], the local probabilistic model is extended to include time awareness.
We present a node centrality-based temporal link prediction approach by the extendence of Salton. The main contributions of this paper are as follows: 1) We assess the influence of common neighbors in temporal network by the eigenvector centrality of nodes. In order to describe the influence of the previous timestamps on the later ones, we use time series model to capture the evolution of node centrality over time. Then, the future node centralities are produced. 2) We regard the centrality of common neighbors as their contribution to the connection likelihood. We extend the definition of global neighbors from multilayer networks to temporal networks. 3) We execute experiments on several real temporal networks. The results show that the performance of the proposed approach outperforms other benchmark methods.

A. Similarity-Based Methods
A lot of similarity-based link prediction methods are depicted in the literature. We present several approaches briefly. For a node x , let () x  be the neighborhood set of x and x k be the degree of x .
Common Neighbors (CN) [4] is one of the basic methods for link prediction. The similarity of node x and y is measured by the number of their common neighbors. The definition is written as: For Adamic-Adar (AA) [5], the node pairs that share fewer common nodes are weighted more heavily. The AA score is defined as: Temporal Link Prediction Using Node Centrality and Time Series For node pairs without direct connection, Jaccard Index [6] produces higher score value when they have more common neighbors in proportion to their total number of neighbors. It is defined as: Preferential Attachment (PA) [7] gives the score according to the degrees of node neighbors, and it is defined as: x PA y S k k  = (4) Analogous to AA, Resource-Allocation (RA) [8] assigns more weights to the nodes with lower-degree neighbors. The RA score is defined as follows: Salton index [9] is based on cosine similarity, and it is defined as follows:

B. Eigenvector Centrality
In complex network, eigenvector centrality can measure the influential nodes. The eigenvector centrality is related to the number of node neighbors and the influence of each neighbor. The eigenvector centrality of node x can be calculated as follows: where  is a constant and () x  is the neighbor set of node x .
For the adjacent matrix A , the eigenvector c associated with the largest eigenvalue of A is obtained as follows: The centrality of node x is represented by the x th element of eigenvector c .

C. Time Series
The Auto-Regressive Integrated Moving Average (ARIMA) includes autoregressive and moving average models. It is a combination model for time series data. The ARIMA model is defined as follows: 1 1 Here, t X represents the estimated variable at t th snapshot,

III. THE TSALTON APPROACH
Many activities are closely related to more influential nodes. Node centrality is regarded as the weight that represents the contribution of nodes to link generation. Therefore, the key issue is how to distinguish the nodes in important positions in the future when predicting links. Our proposed approach is referred to as the extension of Salton index [9] for temporal link prediction (TSalton).

A. The Proposed Model
In the TSalton method, we utilize the eigenvector centrality of each node to distinguish their roles to the likelihood of links. We use the ARIMA model to capture the changes of node centrality with time as follows: where t C represents the estimated node centrality at t th snapshot. We calculates the eigenvector centrality at each timestamp. Then, we forecast the future node centrality by ARIMA. Inspired by the node-centrality-based method in [12], we proposed the TSalton which is eigenvector-centrality-based Salton. The proposed model is read: We extend the neighbors to temporal neighbors according to the definition of neighbors in multilayer network [20]. The global neighbors of node represents the unique neighbors over time: where xy e is the connections between node x and y , t E is the link set at the t th snapshot.

B. The Proposed Algorithm
The TSalton algorithm consists of three main components: node centrality calculation, centrality-based score construction and temporal link prediction. Framework of TSalton algorithm is presented as follows:

A. Evaluation Metrics
We adopt two standard metrics, AUC [21] and precision [22] to estimate the accuracy of temporal link prediction algorithms.
The AUC score is defined as The precision is defined as The precision value is the ratio of relevant links to the number of selected links. If we predict L links to be connected and among them Lr links are correct. The temporal networks applied in this paper includes Workplace [23], EnronEmail [24], Manufacture [25] and CollegeMessage [24]. Table I reports the important statistics of the real-world network.

C. Evaluation of Link Prediction
In these experiments, we compare the temporal link prediction approaches on real-world networks. The comparative approaches are similarity-based methods in Section II. Experiments have been performed adopting the networks in Table I The AUC results of these methods are given in Table II and the related ROC (Receiver operator characteristic) curves are depicted in Fig. 1. We can see that the TSalton outperforms all the other approaches. The similarity-based methods ignore the temporal information of network and only use local topological information, so they perform poorly on temporal link prediction.
Meanwhile, TSalton takes the temporal information into account, and the time series model can reflect the influential nodes more accurately. Let's take the example of EnronEmail. TSalton achieves the highest AUC 0.9134 and is about 20% higher than CN which is in second place.
We list the precision results in Table III. TSalton yields higher precision than similarity-based methods. As for EnronEmail, TSalton is about 0.4% higher than PA in second place. This result shows the advantages of time influence in link prediction. In general, we can see that TSalton performs the best among the baseline approaches in term of AUC and precision on real networks. This results verify that TSalton is suitable for temporal link prediction. The possible reasons are from two aspects :(1) TSalton employs the time series model of eigenvector centrality which can represent the influential nodes more accurately in temporal networks. (2) TSalton considers the contributions of each neighbor to link generation via Salton scores to adjust such contribution.

V. CONCLUSION
The paper proposes a novel algorithm named TSalton. It International Journal of Future Computer and Communication, Vol. 9, No. 3, September 2020 can obtain the topology information, which considers the influential nodes over time. The TSalton scores the likelihood of links by taking the eigenvector centralities of common neighbors into account. We have considered link prediction methods in terms of AUC and precision. Experiments in real networks show that, TSalton performs better than the methods without using temporal information. As a future direction, we aim to consider other information on each snapshot included in the model, such as the context and the community.

CONFLICT OF INTEREST
The authors declare no conflict of interest.

AUTHOR CONTRIBUTIONS
Ting Zhang conducted the research and wrote the paper. Kun Zhang analyzed the data. All authors had approved the final version.