content_image/444424.jpg content_image/444425.jpg
# Separability Criterion for One-Sided Gaussian Channels 

Jason Hoelscher-Obermaier and Peter van Loock ∗ Optical Quantum Information Theory Group, Max Planck Institute for the Science of Light and Institute of Theoretical Physics I, Universit¨ at Erlangen-N¨ urnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany 

We show that the following nontrivial necessary precondition for an entanglement evolution equa- tion for pure Gaussian states under one-sided Gaussian channels holds. Suppose a Gaussian quantum channel acts on one mode of a pure entangled multi-mode Gaussian input state. Then, for a fixed channel, either all output states are entangled or none of them are. In other words, if the input state is Gaussian, pure and entangled, the separability after a one-sided Gaussian quantum channel does not depend on the input state, but only on the channel. Furthermore, a simple linear-algebraic separability criterion allows to decide whether a given channel destroys the entanglement of pure entangled input states or leaves them entangled. 

A large fraction of the known quantum information protocols (QIPs) — including such important protocols as quantum key distribution [1] and measurement-based quantum computation [2] — relies on entangled states. Unfortunately, entanglement is a quite fragile resource and unwanted interactions with the environment tend to destroy it rapidly. In order to assess the feasibility of many entanglement-based QIPs, it is therefore important to know how entanglement evolves in time. 

A quantum system in state ρ which interacts with an environment in state ρ$_{E}$ is in general subject to a non- unitary time evolution T 

where U is the time evolution of both the system and its environment. T is a completely positive, trace-preserving map from the set of density operators on a certain Hilbert space H into itself, a so-called quantum channel . The standard way to calculate the evolution of entan- 

glement under a given quantum channel is to directly calculate the evolution of the quantum state in question according to (1) and then calculate the entanglement of the output state T ( ρ ). In particular, the calculation has to be performed separately for every input state. In the general case of arbitrary quantum channels and arbitrary (pure as well as mixed) input states, this is the only vi- able way. 

Fortunately, we do not have to consider the most gen- eral quantum channel since typical noise effects can be described as local quantum channels , that is, they can be considered as acting uncorrelated on the different parts of the quantum system. It is therefore practically rele- vant to characterize the evolution of entanglement under local quantum channels [13]. As a special case of local quantum channels, this paper considers one-sided quan- tum channels which act only on a single qubit or a single mode of an entangled state. 

<img src='content_image/444424.jpg'>

## MOTIVATION 

## QUBIT ENTANGLEMENT EVOLUTION EQUATION 

In the qubit case, the restriction to one-sided quantum channels is known to simplify the calculation of the en- tanglement evolution drastically. More specifically, con- sider an arbitrary one-sided quantum channel $ which acts on one qubit of an initially pure two-qubit state | χ 〉 (hence the output state is ( 1 ⊗ $)[ | χ 〉 〈 χ | ]) and consider the evolution of the concurrence C under this channel [14]. Konrad et al. [3] have shown that, for all pure two- qubit input states | χ 〉 , the concurrence of the output state is given by 

<img src='content_image/444425.jpg'>

where | Φ 〉 is one of the Bell states. Hence, up to a pref- actor given by the entanglement of the input state, the entanglement (as quantified by the concurrence) evolves always exactly like the entanglement of the Bell states. 

In particular, this implies that a given one-sided quan- tum channel will either disentangle all initially entan- gled pure two-qubit states or leave all of them entan- gled. For if the initial state | χ 〉 is entangled, we have C {| χ 〉 〈 χ |} ̸ = 0 and therefore, according to (2), the en- tanglement of the output state is zero if and only if C { ( 1 ⊗ $)[ | Φ 〉 〈 Φ | ] } = 0 — independent of the state | χ 〉 . 

## GAUSSIAN ENTANGLEMENT EVOLUTION 

Equation (2) is relevant because qubits are ubiquitous in quantum information. But in many quantum-optical implementations of QIPs, continuous variables are pre- ferred over qubits because of practical advantages [15]. In continuous variable quantum information processing, Gaussian states are widely used and typical errors are described by Gaussian quantum channels. Therefore, an evolution equation for the entanglement of Gaus- sian states under one-sided Gaussian channels similar to (2) would be highly relevant. But, an equation of the 

