text:
\section{Introduction}\label{sec:intro}
Quantum systems are extremely sensitive to noise arising from the interaction with their surroundings. This feature is at the very heart of the transition between the quantum and the classical description of the world and, at the same time, imposes limitations on the efficiency of quantum devices for quantum technologies. For this reason, a number of theoretical and experimental approaches focus on the modelization, characterization and reduction of noise induced by the environment, both by increasing the isolation of the system and by suitable engineering of the environment.

A key quantity in the dynamical description of open quantum systems is the dynamical map, a time-parametrized family of quantum channels \cite{Breuer}. Formally, the exact description of the dynamical map can be obtained starting from a microscopic Hamiltonian model of the system, the environment and their interaction. In many practical circumstances, however, one does not have accurate knowledge of the microscopic details of either the interaction or the environmental Hamiltonian. In addition, it is well known that different environments and couplings can lead to the same dynamical map. It is therefore clear that, both from an experimental and from a theoretical perspective, it is crucial to develop approaches able to i) discriminate between different dynamical maps; ii) identify the restrictions imposed on the form of the dynamical map from the specific choices of the environmental state, Hamiltonian and coupling, and iii) develop techniques to probe the environment without ideally any {\it a priori} assumption on its microscopic properties and on the way it is coupled to the system.

In this paper we address these issues by introducing a new family of channel fidelities which generalize the familiar (Uhlmann) fidelity of quantum states. Endowed with these quantifiers, we prove a powerful inequality between the fidelities of the two environmental states inducing the dynamics and the corresponding dynamical maps. More precisely, the introduced fidelity quantifies the difference between two dynamical maps by measuring how distinguishable they render a pair of initially identical (or more generally non-orthogonal) states, which are initially uncorrelated from their dynamics-inducing environments. This addresses point i). The derived inequality, on the other hand, allows us to answer the points ii) and iii) since, contrarily to the probing approaches existing in the literature \cite{recati2005, bruderer2006, johnson2011, dorner2013, mcendoo2013, haikka2013, haikka2014, Anil151, Anil152}, it does not rely on the knowledge of either the system-environment interaction Hamiltonian or the characteristic parameters of the environmental Hamiltonian. We prove the usefulness of our inequality by applying it to two different scenarios both highly relevant for the development of quantum technologies, namely quantum programming and quantum probing.

A programmable quantum processor, or gate array, is a device that implements different quantum channels on one system (the data register) depending on the state of another quantum system (the program register). The third element of this device is a fixed array of gates acting on both the data and the program register.
The virtue of this arrangement is its versatility: one can realize various maps simply by controlling the state of program register. The alteration of the programming state to perform the desired operation is called quantum programming.
Programmable quantum processors were first considered by Nielsen and Chuang in the late 90's  \cite{NielsenChuang}.  Remarkably, these authors showed that  a deterministic universal programmable processor cannot be realised. The reason for this impossibility is that any two states of the program register implementing a pair of inequivalent unitary transformations need to be orthogonal \cite{NielsenChuang,Hillery02, Hillery06,HeiTuk2015}. Therefore, the programming resources, bounded by the dimension of the program register, are insufficient to implement all unitaries, comprising an uncountable set.

The no-programming theorem above holds only if the programming is deterministic. Indeed, it was shown already in Ref.\,\cite{NielsenChuang} that it is possible to successfully implement any one-qubit unitary operation with $1/4$ probability, thus proving the existence of probabilistic universal quantum processors. This result was generalized in Refs.\,\cite{VidalarX, Vidal02}, where it was proven that the probability of success is $ p=1-\varepsilon$ and the error $\varepsilon$ can be made arbitrarily small; the results were later extend for qudits in Refs.\,\cite{Hillery01, Hillery04}. Probabilistic programming, however, tells little about programming states of general quantum transformations, described by completely positive and trace preserving (CPTP) maps. This is, however, important as all realistic implementations of quantum devices are subjected to environmental noise. In this more realistic scenario, known as approximate quantum programming, maps are near to, but not exactly, unitary.

Approximate quantum programming has been studied in Refs.\,\cite{VidalarX, Hillery02, Hillery06}. In particular, in Refs.\,\cite{Hillery02, Hillery06} relations between the overlap of pure programming states and the programmed channels are presented.
Our inequality, on the other hand, sets a novel upper bound for the programming states in terms of the programmed channels and is independent of the choice of the processor. This independence is important since, as we argued above, the microscopic description of the environmental noise affecting the processor may not be known. In addition, unlike in the previously proposed relations, our inequality holds for general (mixed) programming states. In other words, our bound is universally valid and enables us to tackle efficiently problems such as finding size bounds for quantum circuit designs able to implement the desired transformations in full generality.

The second application is in the framework of quantum probing. One particularly renowned quantum feature is summarized by the statement {\it no information without disturbance}. Namely, any measurement producing meaningful information generally transforms the state of the measured system. An observer is thus faced with a dilemma: how to ascertain and assign values to properties of the measured system since they may be, and typically are, altered due to the measurement? One way of obtaining some information on the system of interest without predisposing it to a direct measurement is to let it temporarily interact with a smaller ancillary quantum system and then perform a measurement on it: this technique is known as quantum probing. The system of interest in this framework acts as the environment of the quantum probe, and the induced probe dynamics carries information on the properties of the environment. If the coupling between the probe and the system of interest is sufficiently weak, measurements on the probe only lead to small perturbations on the system. In addition, measuring the full probe dynamics ideally allows one to extract even complete information on the system while leading to only minimal disturbance \cite{recati2005, bruderer2006, johnson2011, dorner2013, mcendoo2013, haikka2013, haikka2014}.

Typical quantum probing strategies assume the knowledge of the microscopic system-probe interaction as well as the system Hamiltonian. The novelty of our method is to use the quantum programming perspective in order to go beyond these existing probing approaches. Namely, by comparing the dynamics induced by an unknown environmental state with the one of a calibration state, some properties on the measured system can be extracted with none or minimal {\it a priori} assumptions.Indeed, we provide examples in which such a probing protocol can be used for parameter estimation without knowing anything about the system-probe coupling or the probe Hamiltonian.

The paper is structured as follows. In Sect.\,\ref{sec:prel} we introduce the mathematical methods and the notation used throughout this work and provide the main result of this paper: the inequality between the channel-fidelity of two state transformations and the fidelity of the corresponding two inducing states. In  Sect.\,\ref{sec:applic} we present four applications that capture the power of  this result from different perspectives. Finally, in Sect.\,\ref{sec:conc} we summarize our results and present conclusions.

\section{Inequality between inducing states and open system dynamics}\label{sec:prel}
We denote a complex separable Hilbert space by $\hi$
 and the trace class operators on $\hi$ by $\mc{T(H)}$.
A \emph{quantum state} is represented as a positive operator $\varrho \in \mc{T(H)}$ with $\tr{\varrho}=1$ and the set of quantum states of $\hi$ is denoted by $\mc{S(H)}$. We indicate with $\supp(\varrho)$ the support of the state $\varrho$ and say that $\varrho_1$ and $\varrho_2$ are orthogonal, i.e. $\varrho_1 \perp \varrho_2$, when the sets $\supp(\varrho_1)$ and $\supp(\varrho_2)$ are orthogonal. The extremal elements of $\mc{S(H)}$ are called \emph{pure} and such element can be written as $\varrho = |\varphi\rangle\langle\varphi|$ for some unit vector $| \varphi \rangle \in \hi$. The transformations of quantum states
 are represented by linear mappings $\mc E: \mc{T(H)} \rightarrow \mc{T(H')}$ that are completely positive and trace-preserving: such transformations are called {\it quantum channels}. In particular, a channel $\mc U$ is called a {\it unitary channel} whenever it is of the form $\mc U(\varrho) = U \varrho U^{\dagger}$ for some unitary operator $U$ on $\hi$ and for all $\varrho \in \mc{S(H)}$.

The dynamics of a closed quantum system with initial state $\varrho$ is described by a $t$-parametrized group of unitary operators $U^{(t)}$ as $\mc U^{(t)}(\varrho)= U^{(t)}\varrho U^{(-t)}$, $t\geq0$. In reality every physically realizable quantum system $S$, with Hilbert space $\hi_S$, interacts with some environment, $E$, with Hilbert space $\hi_E$, such that the pair $S + E$ can be considered a closed system, represented by $\hi_S\otimes\hi_E$. One commonly assumes that the system and the environment are initially uncorrelated, that is $\varrho_{S + E} = \varrho\otimes\xi$. The reduced dynamics of the system state can be calculated from the total system dynamics $\mc U^{(t)}(\varrho\otimes\xi)$ as a partial trace over $\hi_E$:
\begin{eqnarray}\label{eq:redchannel}
\mc E^{(t)} (\varrho) = \text{tr}_E[\mc U^{(t)}(\varrho\otimes\xi)]\,.
\end{eqnarray}
In particular, we say that the dynamics $\mc E^{(t)}(\varrho) $ is {\it induced} by the environmental state $\xi$.

The amount of information ascribed in the state $\varrho$ is measured in terms of entropy. In our investigation, we are focusing on a class of $\alpha$-R\'{e}nyi entropies $S_\alpha(\varrho) \doteq \frac{1}{1-\alpha} \ln \tr{\varrho^\alpha}$, $\alpha\in(0,1)\cup(1,\infty)$. The $\alpha$-R\'{e}nyi entropies generate a one-parameter family of divergences, that from a certain axiomatic point of view for classical random variables $X_1$ and $X_2$ takes the unique form
\begin{eqnarray}\label{eq:defrenyicl}
D_\alpha(X_1 || X_2) = \frac{1}{\alpha-1} \sum_{i=1}^n \ln\big[ p_1(x_i) ^\alpha p_2(x_i)^{1-\alpha} \big] \, ,
\end{eqnarray}
where the two probability measures $p_1$ and $p_2$ measure the probabilities for outcome $x_i$, $i=1,\ldots, n$, occurring in $X_1$ and $X_2$, respectively \cite{Renyi}.
In a quantum scenario, however, the uniqueness of the formulation of the divergences arising from the same set axioms is not guaranteed, and two divergent extensions of Eq.\,\eqref{eq:defrenyicl} to the quantum setting have been proposed in the literature
\begin{eqnarray}\label{eq:defrenyi0}
\widetilde{S}_\alpha(\varrho_1 || \varrho_2)\doteq \left\{ \begin{array}{ll}
\frac{1}{\alpha -1} \ln\left[ \tr{ \varrho_1^\alpha \varrho_2^{1-\alpha}} \right],& \varrho_1 \not\perp \varrho_2 \\
\infty, & \text{otherwise},
\end{array} \right.
\qquad
\end{eqnarray}
and
\begin{eqnarray}\label{eq:defrenyi}
&&S_\alpha \left( \varrho_1 || \varrho_2 \right) \doteq \nonumber \\
& & \left\{ \begin{array}{ll}
\frac{1}{\alpha-1} \ln \left[ \tr{\left(\varrho_2^{\frac{1-\alpha}{2\alpha}} \varrho_1 \, \varrho_2^{\frac{1-\alpha}{2\alpha}} \right)^\alpha} \right], & \text{when} \ \varrho_1\not\perp \varrho_2  \\
 \infty, & \text{otherwise,}
\end{array} \right.
\end{eqnarray}
defined for $\alpha \in (0,1)$ and a pair of quantum states $\varrho_1, \varrho_2 \in \sh$ \cite{Muller2013, Wilde2014}. In this study, we will be focusing for the latter definition $S_\alpha$ and call it the {\it quantum $\alpha$-R\'{e}nyi divergence}. It is, however, worth mentioning that all the results we will present in this Article could be also formulated in the context of $\widetilde{S}_\alpha$ and furthermore, since the $S_{\alpha}$ and $\widetilde{S}_\alpha$ coincide for commuting states $\big[\varrho_1, \varrho_2 \big] =0$ \cite{Muller2013}, most of these results are in fact equivalent for the two definitions.

Importantly, many of the most commonly used quantum (relative) entropies, such as the common (relative) entropy, the min- and the max-(relative)-entropies, can be derived from the quantum $\alpha$-R\'{e}nyi divergence as special cases; for the properties of $\alpha$-R\'{e}nyi divergence we refer the reader to \cite{Beigi2013, Datta2014, Datta2016, Carlen2016} and references therein. For our purposes, we recall here the following features
\begin{itemize}
\item[(S1)] $S_\alpha \big(\varrho_1||\varrho_2\big)\geq 0$, where the equality holds if and only if $ \varrho_1 = \varrho_2$,
\item[(S2)] $S_\alpha\big(\varrho_1 \otimes \xi_1 || \varrho_2 \otimes \xi_2\big) = S_\alpha\big(\varrho_1 || \varrho_2\big) + S_\alpha\big(\xi_1 || \xi_2\big)$,
\item[(S3)] $S_\alpha\big(\mc U(\varrho_1) || \mc U(\varrho_2)\big) = S_\alpha\big(\varrho_1 || \varrho_2\big)$
\end{itemize}
for all  $\alpha\in(0,1)$ and $\varrho_1,\varrho_2 \in \sh$,  $\xi_1,\xi_2 \in \se$ and for all unitary channels $\mc U$. Notice, that the order of inputs is important, since in general $S_{\alpha} \big(\varrho_1, \varrho_2\big) \neq S_{\alpha} \big(\varrho_2,\varrho_1\big)$. Additionally, for all $\alpha\in[1/2,1)$, the $\alpha$-R\'{e}nyi divergence satisfies the data processing inequality
\begin{itemize}
\item[(S4)] $S_\alpha\big(\mc E(\varrho_1) || \mc E(\varrho_2)\big) \leq S_\alpha\big(\varrho_1 || \varrho_2\big)$,
\end{itemize}
for all  $\varrho_1, \varrho_2 \in \sh$ and for arbitrary quantum channels $\mc E:\trh \rightarrow \mc T(\hi_S')$.

From Eq.\,\eqref{eq:redchannel} it is clear that altering the inducing state $\xi\in\se$, while keeping the coupling interaction fixed, may lead to different channels. A question then arises: How do the two different inducing states and the corresponding induced channels relate to each other? The following proposition provides insight into this question.
\begin{proposition}
Suppose that two (different) channels $\mc E_1$ and $\mc E_2:\trh \rightarrow \trh$ are induced by states $\xi_1$ and $\xi_2\in \se$, respectively, from some fixed (unitary) coupling $\mc U:\mc T(\hi_S\otimes\hi_E)\rightarrow\mc T(\hi_S\otimes\hi_E)$; see Eq.\,\eqref{eq:redchannel}. Then for all $\alpha\in[1/2,1)$ and $\varrho_1,\varrho_2 \in \sh$
\begin{eqnarray}\label{eq:mainineq1}
S_\alpha \big(\mc E_1(\varrho_1)||\mc E_2(\varrho_2)\big) - S_\alpha\big(\varrho_1||\varrho_2\big) \leq  S_\alpha\big(\xi_1||\xi_2\big).
\end{eqnarray}
\end{proposition}
\begin{proof}
The claim follows from the properties (S1)-(S4), namely
  \begin{eqnarray}
&&   S_\alpha \big( \varrho_1 || \varrho_2 \big) + S_\alpha \big( \xi_1 || \xi_2 \big) \nonumber \\
&=&  S_\alpha \big( \mc U(\varrho_1\otimes\xi_1) || \mc U(\varrho_2\otimes\xi_2) \big) \nonumber \\
   &\geq &    S_\alpha \big( \text{tr}_{E} \left[ \mc U(\varrho_1\otimes\xi_1) \right] || \text{tr}_{E} \left[ \mc U(\varrho_2\otimes\xi_2)\right] \big) \nonumber \\
   &=&   S_\alpha \big(\mc E_1(\varrho_1)||\mc E_2(\varrho_2)\big) \, .
  \end{eqnarray}
\end{proof}
We will use a short hand notation
\begin{eqnarray}
&&F_{\alpha} \big(\varrho_1, \varrho_2\big) \doteq \tr{\left(\varrho_2^{\frac{1-\alpha}{2\alpha}} \varrho_1 \, \varrho_2^{\frac{1-\alpha}{2\alpha}} \right)^\alpha},
\end{eqnarray} since it will often appear in the following, and call the quantity $F_{\alpha}$ the {\it $\alpha$-fidelity of states} \footnote{In one of the founding papers  \cite{Wilde2014}, this quantity goes by the name ``sandwiched quasi-relative entropy''.}. In the particular case $\alpha=1/2$, $F_{1/2}$ corresponds to the usual (Uhlmann) fidelity of states. Using this notation and the monotonicity of the logarithm function the Ineq.\,(\ref{eq:mainineq1}) implies
\begin{eqnarray}\label{eq:mainineq2}
F_{\alpha} \big(\varrho_1, \varrho_2\big) F_{\alpha} \big(\xi_1, \xi_2\big) \leq F_{\alpha} \big(\mc E_1(\varrho_1), \mc E_2(\varrho_2)\big),
\end{eqnarray}
for $\alpha\in[1/2,1)$, which can be seen as a generalization of the data processing inequality $F_{\alpha} \big(\varrho_1, \varrho_2\big) \leq F_{\alpha} \big(\mc E(\varrho_1), \mc E(\varrho_2)\big)$, for $\alpha\in[1/2,1)$.  Motivated by this inequality, for $\alpha\in(0,1)$, we define
\begin{eqnarray}\label{eq:chanafid} \mc F_{\alpha}(\mc E_1, \mc E_2) \doteq \inf_{\varrho_1,\varrho_2\in\sh} \frac{ F_{\alpha} \big(\mc E_1(\varrho_1), \mc E_2(\varrho_2)\big) }{ F_{\alpha} \big(\varrho_1, \varrho_2 \big) },
\end{eqnarray} and call it the {\it $\alpha$-fidelity of channels} $\mc E_1$ and $\mc E_2$. We justify the terminology, since $\mc F_{\alpha}$ shares many of the basic properties of fidelity-measures as we shall verify next.
\begin{proposition}
For all $\alpha\in(0,1)$ and channels $\mc E_i:\mc T(\hi_S)\to\mc T(\hi_S)$, $i=1,2$, $\alpha$-fidelity of channels $\mc F_{\alpha}(\mc E_1, \mc E_2)$ has the following properties:
\begin{itemize}
\item[(F1)] $\mc F_{\alpha}(\mc E_1, \mc E_2)\in [0,1]$,
\item[(F2)] $\mc F_{\alpha}(\mc E_1, \mc E_2)=\mc F_{\alpha}(\mc U \circ \mc E_1 \circ \mc V , \mc U\circ \mc E_2 \circ \mc V)$, for all unitary channels $\mc U$ and $\mc V$.
\end{itemize}
Additionally, for $\alpha\in[1/2,1)$,
\begin{itemize}
\item[(F3)] $\mc F_{\alpha}(\mc E_1, \mc E_2)=1$ iff $\mc E_1=\mc E_2$,
\item[(F4)] $\mc F_{\alpha}(\mc E_1, \mc E_2) \leq \mc F_{\alpha}(\mc E \circ \mc E_1,\mc E \circ \mc E_2)$,  for all channels $\mc E:\trh\rightarrow\mc T(\hi_S')$,
\item[(F5)] $\mc F_{\alpha}(\mc E_1, \mc E_2) \leq \mc F_{\alpha}(\mc E_1 \circ \mc E,\mc E_2 \circ \mc E)$, for all channels $\mc E:\mc T(\hi_S')\rightarrow\trh$.
\end{itemize}
\end{proposition}
\begin{proof}
Clearly $\mc F_{\alpha}(\mc E_1, \mc E_2)$ is non-negative by definition. Since $F_\alpha(\varrho_1,\varrho_2)\leq 1$ for all $\varrho_1, \varrho_2 \in \sh$, as verified from (S1) of $\alpha$-R\'{e}nyi divergence
, $\mc F_\alpha\big(\mc E_1, \mc E_2\big) \leq \inf_{\varrho_1, \varrho_2} F_{\alpha}(\varrho_1,\varrho_2)^{-1} =1$ proving (F1). Using the properties (S3) and (S4) of $\alpha$-R\'{e}nyi divergence, the monotonicity of the logarithm function and the fact $\sh \simeq \mc U \big( \sh \big)$ we conclude (F2) and (F4). Since $F_{\alpha} \big(\varrho_1,\varrho_2\big) \leq F_{\alpha} \big(\mc E(\varrho_1),\mc E(\varrho_2)\big)$, for all channels $\mc E$ when $\alpha\in[1/2,1)$, we confirm $\mc F_{\alpha}(\mc E, \mc E) = 1$. Assume on the other hand that $\mc F_{\alpha}(\mc E_1, \mc E_2) = 1$, but make a counter assumption $\mc E_1 \neq \mc E_2$. Then there exists a state $\varrho\in \sh$ such that $\mc E_1 (\varrho) \neq \mc E_2(\varrho)$ and therefore by (S1) $F_{\alpha} \big(\mc E_1(\varrho), \mc E_2(\varrho)\big)<1$. Since  $1=\mc F_{\alpha}(\mc E_1, \mc E_2) \leq F_{\alpha} \big(\mc E_1(\varrho), \mc E_2(\varrho)\big)<1$ this leads to a contradiction and we get (F3). Lastly we prove (F5). From the properties above we get the inequality
\begin{align}
&\frac{F_{\alpha} \big((\mc E_1 \circ \mc E)(\varrho_1),(\mc E_2 \circ \mc E)(\varrho_2)\big) }{F_{\alpha} \big(\mc E(\varrho_1),\mc E(\varrho_2)\big)} \leq  \nonumber \\
&\frac{F_{\alpha} \big((\mc E_1 \circ \mc E)(\varrho_1),(\mc E_2 \circ \mc E)(\varrho_2)\big) }{F_{\alpha} \big(\mc \varrho_1,\varrho_2\big)}\,,
\end{align}
which together with $\mc E (\mc S(\hi_S')) \subset \mc S(\hi_S)$ implies that
\begin{eqnarray}
\mc F_{\alpha}(\mc E_1, \mc E_2) &\leq &\inf_{\mc E(\varrho_1), \mc E(\varrho_2)} \frac{F_{\alpha} \big(\mc E_1( \mc E(\varrho_1) ),\mc E_2(\mc E (\varrho_2) )\big) }{F_{\alpha} \big(\mc E(\varrho_1),\mc E(\varrho_2)\big)} \nonumber \\
&= & \inf_{\varrho_1, \varrho_2} \frac{F_{\alpha} \big((\mc E_1 \circ \mc E)(\varrho_1),(\mc E_2 \circ \mc E)(\varrho_2)\big) }{F_{\alpha} \big(\mc E(\varrho_1),\mc E(\varrho_2)\big)} \nonumber \\
 &\leq & \inf_{\varrho_1,\varrho_2} \frac{F_{\alpha} \big((\mc E_1 \circ \mc E)(\varrho_1),(\mc E_2 \circ \mc E)(\varrho_2)\big) }{F_{\alpha} \big(\mc \varrho_1,\varrho_2\big)} \nonumber \\
 &=&  \mc F_{\alpha}(\mc E_1 \circ \mc E,\mc E_2 \circ \mc E).
\end{eqnarray}
\end{proof}

Different fidelity measures of pairs of quantum channels have been formulated including the process fidelity measuring the distinguishability of the corresponding Choi-states \cite{Raginsky2001}, the minimax fidelity \cite{Raginsky2005} with operational connection to the single-shot discrimination of channels done with the so-called process POVMs \cite{Mario2008, Mario2009} and the plethora of gate fidelities considered in \cite{Nielsen2005, NielsenInfo}. All of these channel fidelities have a common feature: they measure the distinguishability of pair of quantum channels in different operational scenarios.
The $\alpha$-fidelity of channels we propose does not, however, admit operational connection to the above channel discrimination tasks due to a peculiar property the other channel fidelities do not share. Namely, as we will see shortly, it vanishes for any pair of different unitary channels. This unique feature makes our fidelity particularly appealing in quantum programming.

We note that the double infima in Eq.\,\eqref{eq:chanafid} can make $\mc F_{\alpha}$ difficult to evaluate. It is, however, readily verified that $\mc F_{\alpha}\big(\mc E_1, \mc E_2\big)\leq \inf_{\varrho} F_{\alpha} \big(\mc E_1(\varrho), \mc E_2(\varrho) \big)$, where the latter quantity also sets an upper bound for the inducing states and it is much easier to handle. Furthermore, since it has been shown in \cite{Mosonyi2014} that $F_{\alpha}$ is jointly concave for $\alpha\in[1/2,1)$, that is in particular for an arbitrary $\varrho = \sum_i \lambda_i |i\rangle \langle i|$ it satisfies $F_{\alpha} \big(\mc E_1(\sum_i \lambda_i |i\rangle\langle i|), \mc E_2(\sum_i \lambda_i |i\rangle\langle i|) \big) \geq \sum_i \lambda_i F_{\alpha} \big(\mc E_1(|i\rangle\langle i|), \mc E_2(|i\rangle\langle i|) \big)$ for $\alpha\in[1/2,1)$, it can be concluded that it is enough to consider the infimum over the set of pure states. In the special case $\alpha=1/2$ the quantity $F_{\text{min}} (\mc E_1, \mc E_2) \doteq \inf_{\varrho} F_{1/2} \big(\mc E_1(\varrho), \mc E_2(\varrho) \big)$ is known as the {\it minimal gate fidelity} \cite{Nielsen2005, NielsenInfo}. We leave it as an open problem, whether also in $\mc F_{\alpha}$ it is sufficient to evaluate the infima over the set pure states or not.

Above an ordering $\mc F_{1/2} \leq F_{\text{min}}$ was pointed out, and one may wish to reveal such relations between other fidelities also, {\it e.g.} between $\alpha$-fidelity and the commonly used process fidelity defined for channels $\mc E_i: \mc T(\hi_S) \to \mc T(\hi_S), $ $i=1,2,$ via $F_{\text{proc} } (\mc E_1, \mc E_2) \doteq F_{1/2} ( \mc E_1 \otimes \mc I (\Omega), \mc E_2 \otimes \mc I (\Omega) )$, where $\mc I: \mc T(\hi_S) \to \mc T(\hi_S)$ is the identity channel and $\Omega = \frac{1}{\dim (\hi_S)} \sum_{n,m=1}^{\dim (\hi_S)} |n n\rangle \langle m m|$ is the maximally entangled state in $\hi_S \otimes \hi_S$ with respect to an orthonormal basis $\{ \ket n,\, n=1,\ldots \dim (\hi_S) \}$. Such an ordering cannot, however, be established. Namely, it can be confirmed that for the one-qubit channels $\mc E_1=\mc I$ and $\mc E_2(\varrho) = 1/3 \sum_{i=1}^3 \sigma_i \varrho \sigma_i^\dagger$, where $\sigma_i$, $i=1,2,3,$ are the Pauli matrices, the process fidelity vanishes, $F_{\text{proc} } (\mc E_1, \mc E_2) = 0$, or more generally $F_{\alpha} \big( \mc E_1 \otimes  \mc I, \mc E_2 \otimes \mc I(\Omega) \big)=0$ for all $\alpha \in (0,1)$. On the other hand, by writing $\varrho_{\vec{m}} = \frac{1}{2} ( \id + \vec m \cdot \vec \sigma),$ where $\vec m \cdot \vec \sigma = \sum_{i=1}^3 m_i \sigma_i$ and $\vec{m}\in \R^3$ has $|| \vec m || \leq 1$, we see that $\mc E_2(\varrho_{\vec{m}}) =  \frac{2}{3} \cdot (\frac 12 \id) + \frac 13 \cdot \varrho_{-\vec m}$ and therefore
\begin{eqnarray}
\mc F_{\alpha} (\mc E_1, \mc E_2) &=& \inf_{\varrho_{\vec n}, \varrho_{\vec m}} \frac{F_{\alpha} ( \varrho_{\vec n}, \frac{2}{3} \cdot (\frac 12 \id) + \frac 13 \cdot \varrho_{-\vec m})}{F_{\alpha}( \varrho_{\vec n}, \varrho_{\vec m})} \nonumber \\
&\geq & \frac 23 \inf_{\varrho_{\vec n}} F_{\alpha} ( \varrho_{\vec n}, \frac 12 \id )  + \frac 13 \inf_{\varrho_{\vec n}, \varrho_{\vec m}}  \frac {F_{\alpha}(\varrho_{\vec n}, \varrho_{-\vec m})}{F_{\alpha}(\varrho_{\vec n}, \varrho_{\vec m})} \nonumber \\
&=& \frac 23 (\frac 12)^{1-\alpha} \neq 0 \, ,
\end{eqnarray}
for $\alpha \in [1/2, 1)$, where the estimation follows from the joint concavity of the $\alpha$-fidelity \cite{Mosonyi2014}. Therefore, there exist channels $\mc E_1$ and $\mc E_2$ for which $F_{\text{proc}} (\mc E_1, \mc E_2) < \mc F_{\alpha} (\mc E_1, \mc E_2)$ for all $\alpha \in [1/2, 1)$. However, in Prop.\,\ref{prop:programming} we will show that $\mc F_{\alpha} (\mc U_1, \mc U_2) =0$ for any pair of unitary channels $\mc U_1 \neq \mc U_2$, which does not hold for $F_{\text{proc} }$ in general \cite{Raginsky2001}. In conclusion, there is no ordering between $\mc F_{\alpha}$ and $F_{\text{proc}}$. Furthermore, since the process fidelity and the minimax fidelity \cite{Raginsky2005} $F_{\text{mm}}(\mc E_1, \mc E_2) \doteq \inf_{\varrho \in \mc S(\hi_S)} \tr{|\sqrt{ \mc E_1 \otimes \mc I (\Omega)} \, \id \otimes \varrho \, \sqrt{ \mc E_2 \otimes \mc I (\Omega)}|}$ satisfy $F_{\text{mm}} \leq F_{\text{proc}}$, and since there exist unequal unitary channels $\mc U_i$, $i=1,2$, such that $F_{\text{mm}}(\mc U_1, \mc U_2) \neq 0$, it may be concluded that the channel fidelities $F_{\text{mm}}$ and $\mc F_{\alpha}$ are not related either. As a corollary, we can prove that the $\alpha$-fidelity of channels is not (sub)multiplicative for separable channels.
\begin{proposition}
$\mc F_\alpha$ does not satisfy the multiplicativity property
\begin{eqnarray}
\mc F_{\alpha} \big( \mc E_1 , \mc E_2 \big) \,  \mc F_{\alpha} \big(  \mc C_1, \mc C_2 \big)  = \mc F_{\alpha} \big( \mc E_1 \otimes  \mc C_1, \mc E_2 \otimes \mc C_2 \big) \, ,
\end{eqnarray} for all channels $\mc E_i: \mc T(\hi_S) \to \mc T(\hi_S)$, $\mc C_i: \mc T(\hi'_S) \to \mc T(\hi'_S)$, $i=1,2$.
\end{proposition}
\begin{proof}
It is easily seen, that $\mc F_{\alpha} ( \mc E_1 , \mc E_2) \, \mc F_{\alpha} (  \mc C_1, \mc C_2 )  \geq \mc F_{\alpha} ( \mc E_1 \otimes  \mc C_1, \mc E_2 \otimes \mc C_2 )$, since $\mc S(\hi_S) \otimes \mc S(\hi'_S) \subset \mc S (\hi_S \otimes \hi'_S)$. The other direction, however, leads to a contradiction. Namely, assume that $\mc F_{\alpha} ( \mc E_1 , \mc E_2) \mc F_{\alpha} (  \mc C_1, \mc C_2 )  \leq \mc F_{\alpha} ( \mc E_1 \otimes  \mc C_1, \mc E_2 \otimes \mc C_2 )$ holds for all channels $\mc E_i: \mc T(\hi_S) \to \mc T(\hi_S)$, $\mc C_i: \mc T(\hi'_S) \to \mc T(\hi'_S)$, $i=1,2$. Then in particular
\begin{eqnarray}
\mc F_{\alpha} \big( \mc E_1 , \mc E_2\big) &\leq & \mc F_{1/2} \big( \mc E_1 \otimes \mc I , \mc E_2 \otimes \mc I \big) \nonumber \\
&\leq & F_{\alpha}\big( \mc E_1 \otimes \mc I (\Omega), \mc E_2 \otimes \mc I(\Omega) \big) \, ,
\end{eqnarray}
which we have noted before does not hold in general.
\end{proof}
\noindent We observe, that similar calculations to those above can be used to verify that $F_{\text{min}}$ is not multiplicative either.

So far we have only considered static quantum channels, however, the calculations above hold as well for dynamical maps. The emerging inequality
\begin{eqnarray}\label{eq:mainineq}
F_{\alpha}\big(\xi_1,\xi_2\big) \leq \mc F_{\alpha}\big(\mc E_1^{(t)}, \mc E_2^{(t)}\big), \,\, \alpha\in[1/2,1), \,\, t\ge 0,
\end{eqnarray} has the following twofold interpretation. Firstly, if the initial environmental states $\xi_i$, $i=1,2$, induce dynamics $\mc E^{(t)}_i$, respectively, then $\mc F_{\alpha}\big(\mc E^{(t)}_1, \mc E^{(t)}_2\big)$ cannot decrease below $F_{\alpha}\big(\xi_1,\xi_2\big)$ for any $t\ge 0$. This sets limitations on what dynamics are compatible with the inducing environmental states. On the other hand, since $\inf_{t\ge 0} \mc F_{\alpha}\big(\mc E_1^{(t)}, \mc E_2^{(t)}\big)$ majorises $F_{\alpha}\big(\xi_1,\xi_2\big)$, it is possible to gain some information about the environment by measuring the open system dynamics, {\it i.e.} by probing. Remarkably, these two strategies work even if the environment and the interaction were not specified. The next section is devoted to examples that demonstrate these properties.

\section{Applications of the inequality}\label{sec:applic}
In the following we will present three applications of Ineq.\,\eqref{eq:mainineq2} for quantum programming and quantum probing purposes.
Figure \ref{fig:progprob} illustrates the quantum probing and quantum programming protocols.
Despite their apparent similarity, the main objectives in the two protocols are complementary:
in quantum programming the aim is to induce different channels for the data register in a controlled way,
whereas in quantum probing the goal is to extract information of the system of interest which induced the dynamics.

\begin{figure}
\includegraphics[width=0.45\textwidth]{progprob.eps}
\caption{{\bf Quantum programming and probing.} (a) In a quantum programming scheme one varies the state $\xi_i$ of the program register ($\hi_E$) coupled to the data register ($\hi_S$).
By varying $\xi_i$ we can induce multiple different channels $\mc E_i$ on the data register. In this scenario emphasis in interest is on $\hi_S$.
(b) In a quantum probing scheme, on the other hand, the system of interest is $\hi_E$ which is coupled to the probe $\hi_S$.
Different initial states, $\xi_i$, of the system lead to different dynamics, $\mc E_i$, of the probe.
By measuring the probe dynamics $\mc E_i$ we can obtain some information of the corresponding inducing state $\xi_i$.}
\label{fig:progprob}
\end{figure}

\subsection{Quantum programming}
The altering of the inducing state $\xi\in\se$ in Eq.\,\eqref{eq:redchannel} is an action known as {\it quantum programming} and accordingly, the pair $\langle \hi_E, \mc U \rangle$ formed by the program space $\hi_E$ and the (unitary) coupling $\mc U$ between $\hi_S$ and $\hi_E$ is called a {\it programmable processor}.

From the inequality
\begin{eqnarray}\label{eq:progineq}
F_{\alpha}\big(\xi_1,\xi_2\big) \leq \mc F_{\alpha}\big(\mc E_1, \mc E_2\big), \qquad \alpha \in [1/2,1),
\end{eqnarray}
we get a limit for the fidelity of the inducing states in terms of the induced channels. We recall once again that a sharp equality between the inducing states and the induced channels has been introduced in \cite{Buzek2006}. This equality depends on the programmable processor. In Ineq.\,\eqref{eq:progineq}, on the contrary, no such dependence is present.

In the case $\alpha=1/2$, Ineq.\,\eqref{eq:progineq} has a particularly nice interpretation, {\it viz.} $F_{1/2}(\xi_1,\xi_2)$ measures the distinguishability of the states $\xi_i$, $i=1,2$. Also $\mc F_{1/2}$ has a similar interpretation: $\mc F_{1/2}(\mc E_1,\mc E_2)=\inf_{\varrho_1,\varrho_2} \frac{F_{1/2}\left(\mc E_1(\varrho_1),\mc E_2(\varrho_2) \right)}{F_{1/2}(\varrho_1,\varrho_2)}$ quantifies the mutual {\it distinguishability of the channels} $\mc E_1,\mc E_2$ by measuring how distinguishable these channels can render a pair of initially non-orthogonal states $\varrho_1$ and $\varrho_2$. The inequality then naturally stems from the fact that the distinguishabilities should not increase under physical processes, such as programming. We connect Ineq.\,\eqref{eq:progineq} to
earlier known programming results \cite{NielsenChuang, Hillery02, Hillery06, HeiTuk2015} by proving that unitary channels are, in this sense, the most distinguishable.

\begin{proposition}\label{prop:programming}
Let $\mc U_1$ and $\mc U_2$ be two unitary channels. If $\mc U_1\neq \mc U_2$ then $\mc F_{\alpha}(\mc U_1, \mc U_2)=0$ for all $\alpha\in (0,1)$. In particular, the inducing states of different unitary channels must be orthogonal regardless of the choice of the processor $\langle \hi_E, \mc U\rangle$.
\end{proposition}
\begin{proof}
We first prove the proposition for $\alpha=1/2$. Using the properties of fidelity of pure states we have $F_{1/2}\big(\mc U_1 ( |\varphi_1 \rangle \langle \varphi_1| ), \mc U_2 ( |\varphi_2 \rangle \langle \varphi_2| ) \big) = | \langle \varphi_1 |  U_1^{\dagger} U_2 \varphi_2 \rangle |$. It suffices to show that for any operator $U \neq c \cdot \id_{\hi_S}$, $|c| =1$, there exists non-orthogonal unit vectors $|\varphi_1\rangle,|\varphi_2 \rangle \in \hi_S$ such that $\langle  \varphi_1 | U \varphi_2 \rangle =0$. Make a counter-assumption that $ \langle  \varphi_1 | U \varphi_2 \rangle \neq 0$ whenever $\langle \varphi_1 | \varphi_2 \rangle \neq 0$. One can express $|\varphi_2\rangle = \alpha |\varphi_1\rangle + \beta |\eta\rangle$, where $\langle \varphi_1 | \eta\rangle =0$ and $|\alpha|^2 + |\beta|^2 =1$. Fix $|\varphi_1\rangle$ such that $|\langle \varphi_1| U \varphi_1\rangle| \neq 1$. We can then choose a unit vector $|\eta\rangle = \big(U^{\dagger} \varphi_1 - \langle \varphi_1| U^{\dagger} \varphi_1 \rangle \varphi_1 \big)/\sqrt{1-|\langle \varphi_1| U \varphi_1 \rangle |^2}$ orthogonal to $\varphi_1$. With the above choices we have $0\neq \langle  \varphi_1 | U \varphi_2 \rangle = \alpha \langle \varphi_1 | U \varphi_1 \rangle + \beta \sqrt{1-|\langle \varphi_1| U \varphi_1 \rangle|^2}$ for all $\alpha$ and $\beta$. Since $\alpha = \sqrt{1-|\langle \varphi_1| U \varphi_1 \rangle|^2}$ and $\beta = - \langle \varphi_1 | U \varphi_1\rangle$ contradict this while satisfying $|\alpha|^2 + |\beta|^2 =1$, the counter-assumption is falsified. Going the other way around, for any unit vector $|\varphi_1\rangle$ such that $|\langle \varphi_1| U \varphi_1\rangle| \neq 1$ we find that the unit vector
\begin{align}\label{eq:varphi2}\nonumber
|\varphi_2\rangle &= \sqrt{1-|\langle \varphi_1 | U \varphi_1 \rangle |^2} \, |\varphi_1\rangle \\
&-\frac{\langle \varphi_1 | U \varphi_1 \rangle}{\sqrt{1-|\langle \varphi_1 | U \varphi_1 \rangle |^2}} \big( U^{\dagger} |\varphi_1\rangle - \langle \varphi_1 | U^{\dagger} \varphi_1 \rangle |\varphi_1\rangle\big)
\end{align} which satisfies $\langle \varphi_1 | U \varphi_2 \rangle =0$ and $\langle \varphi_1 | \varphi_2 \rangle \neq 0$. Fixing $U = U_1^{\dagger} U_2$ proves that $\mc F_{1/2} \big( \mc U_1, \mc U_2 \big) = 0$. For general $\alpha\in(0,1)$ the claim follows from the above considerations when noticing that $F_{\alpha}\big(\varrho_1,\varrho_2)=0$ if and only if $\varrho_1 \perp \varrho_2$; see Eq.\eqref{eq:defrenyi}.

\end{proof}

It should be stressed that for any pair of quantum channels $\mc E_i$, $i=1,2$ acting on a same Hilbert space one can find a processor $\langle \hi_E, \mc U\rangle$ from which the two channels can be induced. To see this, let us recall that any channel $\mc E:\trh \rightarrow \trh$ admits a Stinespring dilation of the form $\mc E(\varrho) = \text{tr}_{\hi_S} \left[ G \, \varrho \otimes |\eta \rangle \langle \eta | \, G^{\dagger}\right]$, where $G$ is a unitary on $\hi_S\otimes \hi_S$ and $|\eta\rangle\in\hi_S$ is some fixed unit vector. We can therefore assume without loss of generality that $\mc E_i(\varrho) = \text{tr}_{\hi_S}\left[ G_i \, \varrho \otimes |\eta \rangle \langle \eta | \, G_i^{\dagger}\right]$ for unitaries $G_i$, $i=1,2$. Consider any pair of orthogonal unit vectors $|\phi_1\rangle$ and $|\phi_2\rangle$, $\langle\phi_1| \phi_2\rangle =0$ in a two dimensional Hilbert space $\C^2$. Then it is easily verified that $U = G_1 \otimes |\eta\rangle\langle \eta| \otimes |\phi_1\rangle\langle\phi_1|+G_2 \otimes |\eta\rangle\langle \eta| \otimes |\phi_2\rangle\langle\phi_2|+\id_{\hi_S} \otimes \big( \id_{\hi_S}- |\eta\rangle\langle \eta|\big) \otimes \id_{\C^2}$ defines a unitary operator on $\hi_S \otimes \big( \hi_S \otimes \C^2\big)$. It follows that, by defining $\hi_E = \hi_S\otimes \C^2$ and $\mc U(\cdot) = U \cdot U^{\dagger}$, the vectors $|\eta \otimes \phi_i\rangle$ program the two channels $\mc E_i$, $i=1,2$, via processor $\langle \hi_E, \mc U\rangle$. As a result, any pair of quantum channels, even identical ones, can be programmed with orthogonal programming states. On the other hand, we showed above that for identical quantum channels the channel $\alpha$-fidelity reaches its maximum value 1. Importantly, this example alludes that is impossible to generally talk about the tightness of Ineq.\,\eqref{eq:progineq}.

In connection to the above, there are two complementary questions related to the tightness of Ineq.\,\eqref{eq:progineq} that we would like to address. Firstly, one may wonder if for any pair of programming states $\xi_i\in \mc S(\hi_E)$ there exists a processor $\langle \hi_E, \mc U\rangle$ realizing channels $\mc E_i$, $i=1,2$, respectively, that would satisfy $F_{\alpha} (\xi_1, \xi_2) = \mc F_{\alpha}\big(\mc E_1, \mc E_2 \big)$. The answer is affirmative, namely any quantum state $\xi\in \mc S(\hi_E)$ can be considered as the preparation channel $\mc E_\xi: \mc T(\hi_E) \to \C\{\xi\}$, $\mc E_\xi (\varrho) = \xi,$ for all $\varrho\in \mc S(\hi_E)$. Moreover, any such channel may be programmed with state $\xi\in \mc S(\hi_E)$ by using the processor $\langle \hi_E, \mc U_{\text{SWAP}} \rangle$, where $\mc U_{\text{SWAP}}(T_1 \otimes T_2) = T_2 \otimes T_1$, for all $T_i \in \mc T(\hi_E)$, that is $\mc E_\xi(\varrho) = \text{tr}_{\hi_E}\left[ \mc U_{\text{SWAP}}( \varrho \otimes \xi)\right]$, for all $\varrho\in \mc S(\hi_E)$. It is evident that in this case $\mc F_{\alpha} \big( \mc E_{\xi_1}, \mc E_{\xi_2} \big) = F_{\alpha} \big(\xi_1,\xi_2 \big)$, for all $\alpha\in(0,1)$. It is natural to ask if the contrary also holds, that is, if for any channels $\mc E_i$ one can find some processor and programming states $\xi_i$, $i=1,2$, respectively, such that $F_{\alpha} (\xi_1, \xi_2) = \mc F_{\alpha}\big(\mc E_1, \mc E_2 \big)$? It turns out that the answer in this case is negative. To confirm this, we recall that the programming states $\xi_i\in \mc S(\hi_E)$, $i=1,2,$ resulting into a unitary and any other different extremal channel are necessarily orthogonal to each other, that is, in particular, $F_{1/2} (\xi_1, \xi_2) =0$; see Ref.\,\cite[Prop.\,8]{HeiTuk2015}. It is, however, possible find a unitary channel $\mc U$ and an extremal channel $\mc E\neq \mc U$ with $\mc F_{1/2}\big(\mc U, \mc E) \neq 0$. Indeed, let us choose $\mc U$ as the one-qubit identity channel and for any $\varrho_{\vec{m}} \in \mc S(\C^2)$ define $\mc E(\varrho_{\vec{m}}) = \frac{1}{2}(\id + m_2 \, \sigma_2)$. The extremality of $\mc E$ can be verified by using results of Ref.\,\cite{Ruskai2002}, and furthermore using inequalities found in Ref.\,\cite{Miszczak09} one can shown that
\begin{eqnarray}
& &\mc F_{1/2}\big(\mc U, \mc E) \geq \nonumber \\
& &  \inf_{\substack{ ||\vec n||\leq 1, \\ ||\vec m||\leq 1\phantom{,}}} \frac{\tr{\varrho_{\vec n} \, \mc E(\varrho_{\vec{m}} )}}{\tr{\varrho_{\vec n} \, \varrho_{\vec m} } + \sqrt{(1- \tr{\varrho_{\vec n}^2}) (1- \tr{\varrho_{\vec m}^2})}} = 1/2\,. \nonumber \\
\end{eqnarray} We conclude that Ineq.\,\eqref{eq:progineq} fails to be tight in general.

\begin{figure}[t!]
\hspace{-1cm}\includegraphics[width=0.45\textwidth]{programming.eps}
\caption{{\bf Dimension of programmable processor in approximate programming.} We consider the dimension of a processor capable of implementing the four noisy unitary channels $\mc E_i: \varrho \mapsto (1-\varepsilon) \sigma_i \varrho \sigma_i^\dagger + \varepsilon \frac 12 \id,$ $i=0,\ldots, 3$, in terms of the noise parameter $\varepsilon\in[0,1]$. It may be confirmed that the process fidelity satisfies $F_{\text{proc}}\big(\sigma_i \cdot \sigma_i^\dagger, \mc E_i \big) \geq 1-\varepsilon$ for all $i= 0, \ldots, 3$. Accordingly, an analysis done in Ref.\,\cite{Hillery06} implies that any processor implementing $\mc E_i$, $i=0, \ldots, 3$, is at least four dimensional for $0 \leq \varepsilon < \left[3 (13 + 2 \sqrt{42}) \right]^{-1}$, at least three dimensional for $\left[3 (13 + 2 \sqrt{42}) \right]^{-1} \leq \varepsilon < \left[2 (9 + 4 \sqrt{5} )\right]^{-1}$ and at least two dimensional for $\left[2 (9 + 4 \sqrt{5}) \right]^{-1} \leq \varepsilon< 1$ excluding the dimensions within the red area. However, our approach results to tighter limits which we have denoted in blue: the processor is necessarily at least four dimensional for $0\leq \varepsilon < \frac13 (3 - \sqrt 6)$, at least three dimensional for $\frac13 (3 - \sqrt 6) \leq \varepsilon < \frac12 (2 - \sqrt 2)$ and at least two dimensional for $\frac12 (2 - \sqrt 2) \leq \varepsilon< 1$. }
\label{fig:apprprog}
\end{figure}
Nevertheless, the channel $\alpha$-fidelity $\mc F_\alpha$ is a genuinely important figure of merit in quantum programming, as we will see in the following. According to the previous proposition, programming $N$ different unitary channels requires $\dim(\hi_E) \geq N$. The processor's dimension may, however, be lowered if some error is accepted. Let us denote $d=\dim(\hi_S)$. Suppose we wish to program the approximate unitary channels $\mc E_{U_i}(\varrho) = (1-\varepsilon) \mc U_j(\varrho) + \varepsilon \frac{1}{d} \id_{\hi_S}$ with programming vectors $|\phi_i\rangle$, respectively, where $U_i$, $i=1,\ldots, N$ are unequal unitaries and $\varepsilon \in [0,1]$ describes a fixed error rate. From Ref.\,\cite{Miszczak09} we find that $F_{1/2}\big(\mc E_{U_j}(\varrho_1),\mc E_{U_k}(\varrho_2)\big) \leq \tr{\mc E_{U_j}(\varrho_1)\,\mc E_{U_k}(\varrho_2)} + \sqrt{\big(1-\tr{\mc E_{U_j}(\varrho_1)^2}\big)\big(1-\tr{\mc E_{U_k}(\varrho_2)^2}\big)}.$ Choosing $\varrho_i = |\varphi_i\rangle \langle \varphi_i|$, $i=1,2$ simplifies this inequality yielding $\frac{F_{1/2}\big(\mc E_{U_j}(\varrho_1),\mc E_{U_k}(\varrho_2)\big)}{F_{1/2}(\varrho_1, \varrho_2)} \leq \frac{(1-\varepsilon)^2 |\langle U_j \varphi_1| U_k \varphi_2 \rangle|^2}{|\langle \varphi_1| \varphi_2 \rangle|}+\frac{(2-\varepsilon)\varepsilon}{|\langle \varphi_1| \varphi_2 \rangle|}.$
In particular, choosing $\varphi_2$ as in Eq.\,\eqref{eq:varphi2} for $U=U_j^{\dagger}U_k$ implies
\begin{eqnarray}\label{eq:approbound}
F_{1/2} (\phi_j,\phi_k) &\leq & \inf_{|\varphi_1\rangle} \frac{(2-\varepsilon)\varepsilon}{\sqrt{1-|\langle U_j \varphi_1| U_k \varphi_1 \rangle|^2}} \nonumber \\
& =& \frac{(2-\varepsilon)\varepsilon}{\sqrt{1-\inf_\varrho F_{1/2}\big(\mc U_j(\varrho),\mc U_k(\varrho)\big) }} \doteq g_{jk}(\varepsilon), \nonumber \\
\end{eqnarray}
when $j\neq k$. We may use this bound to solve for the largest set of linearly independent programming vectors using the following result: if $|\phi_j\rangle,\, j=1,\ldots,K$, are unit vectors and $F_{1/2}(\phi_j, \phi_k) \leq \frac{1}{K-1}$ whenever $j\neq k$, then the vectors $|\phi_j\rangle, \, j=1,\ldots,K$, are linearly independent \cite{Hillery02}. Let us denote with $K_\varepsilon$ the largest integer such that $K_\varepsilon < \max_{j\neq k \in\{1,\ldots N\}}\big(1/g_{jk}(\varepsilon)+1\big)$. The previous result then implies that any set of vectors whose size is less than or equal to $K_\varepsilon$ is linearly independent. Therefore, if $N \leq K_\varepsilon$, then all of the programming vectors are linearly independent and $\dim(\hi_E) \geq N$. On the other hand if $N> K_\varepsilon$, then $\dim(\hi_E) \geq K_\varepsilon$ \cite{Hillery06}.

As an example, let us consider approximate programming of qubit unitaries $\sigma_i$, $i=0,\ldots,3$, where $\sigma_0 = \id_{\C^2}$. For all pairs the quantity $\inf_\varrho F_{1/2}\big(\sigma_j \varrho \sigma_j,\sigma_k \varrho \sigma_k\big)$ vanishes. Therefore, according to Eq.\,\eqref{eq:approbound} the processor in approximate programming of the above unitaries is at least four dimensional for $0\leq \varepsilon < \frac13 (3 - \sqrt 6)$, at least three dimensional for $\frac13 (3 - \sqrt 6) \leq \varepsilon < \frac12 (2 - \sqrt 2)$ and at least two dimensional for $\frac12 (2 - \sqrt 2) \leq \varepsilon< 1$. In Fig.\,\ref{fig:apprprog} these bounds have been compared to those previously solved in Ref.\,\cite{Hillery06}. It is noteworthy that our approach gives tighter limits for the amount of noise necessary to be present in order to approximately implement the four unitary channels $\sigma_i$, $i=0,\ldots,3$.

\subsection{Ruling out incompatible environmental properties} Let us consider a general thermal environment $\hi_E$ and two initial states of the environment in different temperatures $T_1$ and $T_2$. The environmental states therefore attain the Gibbs form $\xi(\beta_i) \doteq \expo{- \beta_i \Ho_{E}} / Z(\beta_i, \Ho_E)$, $i=1,2$, where $\beta_i = (k_B T_i)^{-1}$, where $\Ho_{E}$ is a Hamiltonian of the environment and $Z(\beta_i,\Ho_{E}) \doteq \tr{ \expo{ -\beta_i \Ho_{E}}}$ are the corresponding partition functions for $i=1,2$.

	Our goal is to exclude some properties of environment inscribed in the spectrum of the Hamiltonian $\Ho_E$ that are incompatible with Ineq.\,\eqref{eq:mainineq}. Since for thermal states $\left[\xi(\beta_1),\xi(\beta_2)\right]=0$, we easily solve
\begin{widetext}
\begin{equation}
S_\alpha\big(\xi(\beta_1)||\xi(\beta_2)\big) = \frac{1}{\alpha -1} \big( \ln\left[ Z(\alpha \beta_1 + (1-\alpha) \beta_2,\Ho_{E}) \right]
- \alpha \ln\left[ Z( \beta_1,\Ho_{E}) \right] -  (1-\alpha) \ln\left[ Z( \beta_2,\Ho_{E} ) \right] \big).
\end{equation}
In particular, for all $\alpha \in [\frac{1}{2}, 1)$ Ineq.\,(\ref{eq:mainineq}) implies the following inequality
\begin{equation}
\ln\left[ Z(\alpha \beta_1 + (1-\alpha) \beta_2,\Ho_{E}) \right]- \alpha \ln\left[ Z( \beta_1,\Ho_{E}) \right]
-  (1-\alpha) \ln\left[ Z( \beta_2,\Ho_{E} ) \right
\leq  \inf_{t\geq 0} \ln\left[\mc F_{\alpha} \big(\mc E_1^{(t)},\mc E_2^{(t)} \big)\right].
\end{equation}
In fact, due to the commutativity of the thermal states, we can expand this for $\alpha \in (0, 1)$ such that
\begin{equation}\label{eq:mainineq3}
\ln\left[ Z(\alpha \beta_1 + (1-\alpha) \beta_2,\Ho_{E}) \right]- \alpha \ln\left[ Z( \beta_1,\Ho_{E}) \right]
-(1-\alpha) \ln\left[ Z( \beta_2,\Ho_{E} ) \right]  \leq  \inf_{t\geq 0} \left\{ \begin{array}{cc}
\ln[\mc F_{\alpha} \big(\mc E_2^{(t)},\mc E_1^{(t)}\big) ],  \text{for} \  \alpha \in (0, \frac{1}{2}) \\
\ln[\mc F_{\alpha} \big(\mc E_1^{(t)},\mc E_2^{(t)}\big)],  \text{for} \ \alpha \in [\frac{1}{2}, 1)
\end{array} \right. .
\end{equation}
\end{widetext}

Let us consider the implications of the above inequalities. Assume that it is possible to prepare the environment in two different known temperatures and
perform the full-process tomography of the two induced dynamics.
This enables us to determine the values of $\mc F_{\alpha}$ in the right-hand side of Ineq.\,(\ref{eq:mainineq3}). On the other hand, any hypothesis about the Hamiltonian of the environment determines the left-hand side of the inequality. If the inequality is not satisfied, the hypothesized Hamiltonian of the system can be ruled out as incompatible with the induced pair of dynamics. We highlight that in order to do so neither the system-environment coupling nor the Hamiltonian of the system needs to be specified.

As a demonstration of the power of this method, let us consider a specific example of a qubit system coupled to an environment consisting of a single harmonic oscillator: such a situation could occur, for instance, when a two-level atom is passing through an optical cavity including only a single quantized mode. Assume that the mechanism of interaction is unknown as well as the oscillator frequency which we nevertheless wish to determine. Mathematically, we only know that the Hamiltonian of the environment is of the form $\Ho_E=\omega \big(b^{\dagger} b + \frac{1}{2}\id\big)$. Choosing different values for $\omega$ one plots the dashed coloured curves in Fig.\,\ref{fig:hamiltonians} given by the left-hand side of Ineq.\,\eqref{eq:mainineq3}. For a concrete example of the (exactly solvable) reduced dynamics giving the right-hand-side of Ineq.\,(\ref{eq:mainineq3}) and the solid black curve in Fig.\,\ref{fig:hamiltonians}, we consider the interaction model of the form
\begin{eqnarray}
 \Ho	&=& \Ho_S + \Ho_E + \Ho_I\nonumber \\
	&=& \frac{\omega_0}{2}\sigma_3 + \omega \big(b^{\dagger} b + \frac{1}{2}\id\big) + \sigma_3\otimes \big(g b^\dagger + g^* b\big)\,
\end{eqnarray}
analysed in \cite{Breuer}. One can see, that the inequality forbids frequency values greater than 3.1 times the real frequency $\omega$.
For details we refer the reader to Appendix \ref{app:b}.
\begin{figure}[t!]
\includegraphics[width=0.45\textwidth]{discrimination.eps}
\caption{ {\bf Excluding the frequency values of a harmonic oscillator.} (In this example we have fixed $\hbar=1=k_B$.) We prepare the environment in two temperatures and measure the induced dynamics, which gives us right-hand-side of Ineq.\,(\ref{eq:mainineq3}) (black line). The three dashed coloured lines correspond to the left-hand-side of Ineq.\,\eqref{eq:mainineq3} with different choices for $\omega_k$, $k=1,2,3$, in units of the actual frequency $\omega$. Knowing the two temperatures of the environment ($T_1/\omega=0.25$) and ($T_2/\omega =0.75$) we notice that some frequencies ($\omega_1/\omega=3$) are compatible with Ineq.\,(\ref{eq:mainineq3}) while others ($\omega_2/\omega=3.1$ and $\omega_3/\omega=3.25$) violate it.
In fact, we see that Ineq.\,(\ref{eq:mainineq3}) is violated for all frequencies larger than $\omega_2/\omega=3.1$, and thus $\omega_2/\omega=3.1$ is the crossover between the compatible and incompatible frequencies.
As the probe we use a qubit and the induced dynamics result from an exactly solvable model described in \cite{Breuer}. The optimal initial state(s) of the qubit are $\varrho = |+\rangle \langle +|$ with $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ being the eigenstate of Pauli operator $\sigma_1$; for details see Appendix B.}\label{fig:hamiltonians}
\end{figure}

\subsection{Quantum thermometry} Let us consider the situation of the previous example from a different perspective. Suppose now that a thermal environment described by a single harmonic oscillator with Hamiltonian $\Ho_E=\omega \big(b^{\dagger} b + \frac{1}{2}\id\big)$ is initially prepared at calibration temperature $T_0 = 0$, that is $\xi(\beta_0=\infty) = |0 \rangle \langle 0|$. Let us assume we are tasked with probing another temperature $T$ of the environment. The $\alpha$-R\'{e}nyi divergence between the states $\xi(\beta_0)$ and $\xi(\beta=1/(k_B T) )$ now reads $S_\alpha \big(\xi(\beta_0)|| \xi(\beta)\big) = \frac{1-\alpha}{\alpha - 1} \big(-\frac{\beta \hbar \omega}{2} - \ln[ Z(\beta, \Ho_{E}) ] \big) $ and therefore
after some simple algebra we solve from Ineq.\eqref{eq:mainineq1} a lower bound for the temperature
\begin{align}\nonumber
&\frac{k_B T}{\hbar \omega}  \geq \\ \label{eq:thermoineq1}
&-1 \times
\left\{ \begin{array}{ll}
\left[\ln\left[1-\inf_{t\geq 0} \mc F_{\alpha} \big(\mc E_T^{(t)}, \mc E_0^{(t)} \big)^{\frac{1}{\alpha}}\right]\right]^{-1}, &  \alpha \in (0, \frac{1}{2}) \\
\left[\ln\left[1- \inf_{t\geq 0} \mc F_{\alpha} \big(\mc E_0^{(t)} ,\mc E_T^{(t)} \big)^{\frac{1}{1-\alpha}}\right]\right]^{-1}, &   \alpha \in [\frac{1}{2}, 1)
\end{array} \right.
\end{align}
for all $T\geq 0$. It can be confirmed, that with temperatures lower than a limiting temperature $(k_B T)/ (\hbar \omega) \lesssim 1.03$ the term $-\ln[ Z(\beta, \Ho_{E}) ] = \frac{\beta \hbar \omega}{2} + \ln[1-e^{-\beta \hbar \omega}]$ is non-negative, from which using Ineq.\,\eqref{eq:mainineq1} we may also obtain an upper bound for the temperature
\begin{align} \nonumber
&\frac{k_B T}{\hbar \omega} \leq   \\ \label{eq:thermoineq2}
& -1/2 \times
\left\{ \begin{array}{ll}
\left[\ln\left[\inf_{t\geq 0} \mc F_{\alpha} \big(\mc E_T^{(t)}, \mc E_0^{(t)} \big)^{\frac{1}{\alpha}}\right]\right]^{-1}, & \alpha \in (0, \frac{1}{2}) \\
\left[\ln\left[\inf_{t\geq 0} \mc F_{\alpha} \big(\mc E_0^{(t)} ,\mc E_T^{(t)} \big)^{\frac{1}{1-\alpha}}\right]\right]^{-1}, &  \alpha \in [\frac{1}{2}, 1)
\end{array} \right. \qquad
\end{align}

Since there is no dependency on the parameter $\alpha$ on the left-hand-side of the above inequalities, we optimize over this parameter. The quantities on the right-hand-side of the inequalities above can be solved (numerically) after measuring the dynamics; these observations allow us to estimate the unknown temperature $T$.

The inequalities (\ref{eq:thermoineq1}) and (\ref{eq:thermoineq2}) may, unfortunately, be difficult to evaluate in general due to double optimization with respect to states and therefore a more accessible form is desirable. To achieve this, we use the fact that $\mc F_{\alpha} \big(\mc E_1, \mc E_2\big) \leq \inf_\varrho F_{\alpha}\big(\mc E_1(\varrho), \mc E_2(\varrho)\big)$, for all $\alpha\in(0,1)$. Since $S_{\alpha}(\cdot || \cdot)$ is monotonically increasing in $\alpha$ \cite{Muller2013}, that is, $S_{\alpha}\big(\varrho_1 || \varrho_2\big)\leq S_{\alpha'}\big(\varrho_1 || \varrho_2\big)$ whenever $\alpha \leq \alpha'$, and since the logarithm function is monotonically increasing, we conclude that $F_{\alpha} ( \varrho_1, \varrho_2)^{\frac{1}{1-\alpha}}$ monotonically decreases in $\alpha$. On the other hand, since for all positive semidefinite linear operators $A,B$ and numbers $q\geq 0, r \geq 1$ the Araki-Lieb-Thirring inequality \cite{Lieb1976, Araki1990} states that $\tr{\big(B^{1/2} A B^{1/2}\big)^{rq}} \leq \tr{\big(B^{r/2} A^r B^{r/2}\big)^q}$ holds and since $F_{\alpha}(\varrho_1,\varrho_2)$ may be recast in the form \footnote{This follows from the fact that the operators $X X^\dagger$ and $X^\dagger X$ have the same non-zero eigenvalues, and therefore $\tr{(X X^\dagger)^\alpha} =\tr{(X^\dagger X)^\alpha}$.} $F_{\alpha}(\varrho_1,\varrho_2) = \tr{\big( \varrho_1^{1/2} \varrho_2^{(1-\alpha)/\alpha} \varrho_1^{1/2} \big)^{\alpha}}$, we have
\begin{eqnarray}\nonumber
F_{\alpha}(\varrho_1,\varrho_2)&=&\tr{\big( \varrho_1^{1/2} \varrho_2^{(1-\alpha)/\alpha} \varrho_1^{1/2} \big)^{\frac{\alpha}{1-\alpha}(1-\alpha)}} \\ \nonumber
&\leq& \tr{\big( \varrho_1^{\alpha/2(1-\alpha)} \varrho_2 \varrho_1^{\alpha/2(1-\alpha)} \big)^{1-\alpha}}  \\
&=& F_{1-\alpha}(\varrho_2,\varrho_1)
\end{eqnarray} and therefore $F_{\alpha}(\varrho_1,\varrho_2)^{1/(1-\alpha)} \leq F_{1-\alpha}(\varrho_2,\varrho_1)^{1/(1-\alpha)}$ whenever $\alpha\geq 1/2$ ($\Leftrightarrow r=\frac{\alpha}{1-\alpha} \geq 1$). To summarize
 \begin{eqnarray} \label{eq:thermoineq3}
\frac{k_B T}{\hbar \omega}  &\geq & - \ln\left[1-\inf_{\varrho,t\geq 0}\lim_{\alpha \nearrow 1}   F_{\alpha} \big(\mc E_0^{(t)}(\varrho), \mc E_T^{(t)}(\varrho) \big)^{\frac{1}{1-\alpha}}\right]^{-1} \nonumber \\
&=& - \ln\left[1- e^{-\sup_{\varrho,t\geq 0} S_1\big(\mc E_0^{(t)}(\varrho)||\mc E_T^{(t)}(\varrho)\big)}\right]^{-1}\,
 \end{eqnarray}
for $T\geq 0$, where $\lim_{\alpha \nearrow 1} S_\alpha (\varrho_1||\varrho_2 \big) = S_1\big(\varrho_1||\varrho_2 \big) \doteq \tr{ \varrho_1\big(\ln\varrho_1 - \ln\varrho_2 \big)}$ is the Kullback-Leibler divergence \cite{Muller2013}. Similarly, for the upper bound of temperature one derives
\begin{eqnarray} \label{eq:thermoineq4}
\frac{k_B T}{\hbar \omega}  &\leq & -1/2  \ln\left[\inf_{\varrho,t\geq 0} \lim_{\alpha \nearrow 1}  F_{\alpha} \big(\mc E_0^{(t)}(\varrho), \mc E_T^{(t)}(\varrho) \big)^{\frac{1}{1-\alpha}}\right]^{-1} \nonumber \\
&=&\left( 2 \sup_{\varrho,t\geq 0} S_1\big(\mc E_0^{(t)}(\varrho)||\mc E_T^{(t)}(\varrho)\big) \right)^{-1}\,,
\end{eqnarray}
when $0 \leq (k_B T)/ (\hbar \omega) \lesssim 1.03.$
\begin{figure}[t!]
\includegraphics[width=0.45\textwidth]{thermometry3.eps}
\caption{ {\bf Estimation of temperature.} We assume that the environment is prepared in two different temperatures ($(k_B T_0)/(\hbar \omega) = 0$ and $(k_B T)/(\hbar \omega)$) and measure the induced dynamics. The green line represents the actual temperature $(k_B T)/(\hbar \omega)$. From Ineq.\,\eqref{eq:thermoineq3} we solve a lower bound of $(k_B T)/(\hbar \omega)$, plotted above as the blue dashed curve. The red dotted curve corresponds to the upper bound of the temperature valid whenever $(k_B T)/(\hbar \omega) \lesssim 1.03$ given by Ineq.\,\eqref{eq:thermoineq4}. The system that we use for probing the temperature is a qubit and the dynamics are due to Jaynes-Cummings model in the resonant case (see Ref.\,\cite{Smirne2010}). The initial state of the qubit that we have used for in the plot is $\varrho=|+\rangle \langle + |$, since numerics suggest that it leads to the optimal bounds. } \label{fig:thermo}
\end{figure}

In Fig.\,\ref{fig:thermo} we have plotted $(k_B T)/(\hbar \omega)$ and the optimal bounds given by the right-hand-sides of Ineqs.\,(\ref{eq:thermoineq3}) and (\ref{eq:thermoineq4}) for the resonant Jaynes-Cummings model, which describes a two-level atom interacting with a single quantized mode of radiation in an optical cavity \cite{Breuer, Smirne2010}; for details we refer the reader to Appendix C. In the figure, we have chosen the initial state of the qubit probe as $\varrho=|+\rangle \langle + |$, since numerics suggest that it leads to the optimal bounds.

A standard procedure in (quantum) thermometry is to bring the thermometer into contact with a bath, let it thermalize and then read out the temperature. In the quantum scenario, we say that a qubit probe has {\it thermalized} with the bath if any initial state $\varrho$ of the probe evolves in the long time limit $t\to \infty$ into the equilibrium Gibbs state
$\varrho \to \varrho_{\text{eq}} \doteq  \big(1-p(T) \big) \, \ket 0 \bra 0 + p(T) \, \ket 1 \bra 1$, where $p(T) = \big[ 1 +  \text{exp}(\hbar \omega/k_B T ) \big]^{-1}$.
Quantum thermometry protocols  \cite{Brunelli2012,Higgins2013,Jevtic2015,Johnson2016} and optimality of initial probe states \cite{Correa2015} have been studied for both fully and partially thermalizing probes. To our best knowledge all pre-existing protocols are based on knowing the exact form of the coupling. Also the optimal initial probe states have been solved in the partially thermalized probes only for fixed system-probe couplings.

In the case of fully thermalized probes, calculating the $\alpha$-fidelity becomes particularly easy: $\lim_{t\to \infty} \mc F_{\alpha} \big(\mc E_0^{(t)}, \mc E_T^{(t)} \big) = \big(1-p(T) \big)^{1-\alpha}$. Accordingly, our protocol sets the limits
\begin{eqnarray}
- \big[ \ln[ p(T) ] \big]^{-1} \leq \frac{k_B T}{\hbar \omega} \leq - 1/2 \big[ \ln[ 1- p(T) ] \big]^{-1},
\end{eqnarray}
with the upper bound valid whenever $0 \leq (k_B T)/ (\hbar \omega) \lesssim 1.03.$ It is noteworthy, that due to the special form of the state $\varrho_{\text{eq}}$ the temperature $T$ could be solved analytically from the state occupations, which shows that our protocol is the not the optimal one for thermometry under the assumption of thermalization. However, there is no {\it a priori} need to require thermalization of the probe with the bath in our protocol. This is an important advantage, since thermalization itself in a fully quantum scenario may not be a well-defined concept due to the ever present quantum fluctuations and the long time-scales needed compared to the survival time of the quantum properties of the bath.

We wish to emphasize that knowing the specific system environment coupling is not required in the above two parameter estimation protocols {\bf B} and {\bf C}: to the best of our knowledge, such probing approaches independent of the description of the coupling have never been proposed before. While the precisions of the frequency estimation and thermometer are clearly not optimal, these examples serve as proofs-of-principle demonstrations of the ability to extract information on certain physical quantities by means of our approach, making only minimal assumptions on the microscopic details of the system to be probed.
From a broader point of view, the two protocols {\bf B} and {\bf C} could be applied to estimate, not only the frequency and temperature, but also other physical quantities of a system of interest. The task to probe some of such quantities could be unattainable ''conventionally'', since the description of the coupling model between the system and the probe is not either accurately known or is borderline totally missing. An ultimate example of such case is the quantum gravity, since no commonly agreed model of gravitational interaction in quantum regime exists.

\subsection{Bounds for the Loschmidt Echo}

In our last application we change the perspective altogether and consider a different implication of Ineq.\,\eqref{eq:mainineq2}. We show how from the $\alpha$-fidelity of the initial environmental states we can infer certain properties of the induced dynamics, such as information back-flow (non-Markovianity). Importantly, back-flow of information can be used to protect and restore vital quantum properties of the system subjected to detrimental noise \cite{Rivas2014, Breuer2016, Hinarejos2016}.

As an example we concentrate on one-qubit dephasing channels arising from system environment interaction. Although the method of this subsection holds generally for such channels, we demonstrate the power of the protocol by focusing on a specific situation with known solutions only for certain initial states of environment: a qubit transversely coupled to its environment, which is an Ising spin chain in a transverse field.
The total evolution of such a composite system is governed by the Hamiltonian $\Ho(\lambda,\delta) = - J \,\Sigma_j\big( \sigma_3^{(j)} \otimes \sigma_3^{(j+1)} + \lambda \, \sigma_1^{(j)} + \delta \, |e\rangle\langle e| \otimes \sigma_1^{(j)} \big)$. Here the parameters $J$, $\lambda$ and $\delta$ characterize the strength of the nearest neighbour interactions in the Ising chain, its coupling to a transverse field and the coupling between system and environment, respectively \cite{Quan2006, Haikka2012}. It is known, that the initial environmental state $|\phi\rangle$ induces dynamics corresponding to a pure dephasing channel $
\mc E^{(t)}_\phi(\varrho) = p_\phi(t) \varrho + \big(1-p_\phi(t)\big) \sigma_3 \varrho \sigma_3$ with the probability $p_\phi(t)$ given by
\begin{eqnarray}
p_\phi(t)=\frac{1}{2}\big(1+\sqrt{L_\phi(\lambda,t)}\big),
\end{eqnarray}
where $L_\phi(\lambda,t) = |\langle \phi | e^{i \Ho(\lambda,0) t} e^{-i \Ho(\lambda+\delta,0) t} \phi \rangle|^2$ is the Loschmidt echo corresponding to the state $|\phi\rangle$. Despite the apparent simplicity, finding an analytical expression of $L_\phi(\lambda,t)$ is a difficult task for a general initial state $|\phi\rangle$. In fact to our best knowledge, the only known analytical solution exists for the ground state $|\phi_0\rangle$ of the Hamiltonian $\Ho(\lambda,0)$ for which one obtains
\begin{eqnarray}
L_{\phi_0}(\lambda,t)= \Pi_{k>0} \big(1-\sin^2(2 \alpha_k) \sin^2(\varepsilon_k t) \big),
\end{eqnarray}
where $\alpha_k$ are the Bogoliubov angles and $\varepsilon_k$ the single quasiparticle excitation energies of the system with the qubit in the excited state $|e\rangle$ \cite{Quan2006, Haikka2012}.
\begin{figure*}[t!]
\begin{minipage}[t]{0.9\textwidth}
\includegraphics[width=0.9\textwidth]{loschmidt.eps}
\end{minipage}
\caption{{\bf Bounding the Loschmidt echo.} (a) The dynamics of the Loschmidt echo induced by the environmental ground state $\phi_0$. The parameter values of the Hamiltonian $\Ho(\lambda)$ are set as $J = 1,\, \delta = 0.1$ and $N = 4000$. The blue dashed line corresponds to the choice $\lambda = 0.01$, the green solid line corresponds to the critical value $\lambda = 0.9$ and the red dotted line corresponds to the value $\lambda = 1.8$ (b) - (g) The dynamics of the upper and lower bound of the unknown Loschmidt echo are represented by the curves in lighter and darker hue of the corresponding color choices of the parameter $\lambda$, respectively. The black solid line indicates the level of the first local minimum of the upper bound. In (b) and (f) the lower bounds cross the black line after the local minimum of the upper bound, which implies revivals in the Loschmidt echo and thus information back-flow.
(b) $\lambda = 0.01$, $F_{1/2}(\phi_0, \phi) = 0.98$.
(c) $\lambda = 0.01$, $F_{1/2}(\phi_0, \phi) = 0.966675$.
(d) $\lambda = 0.9$, $F_{1/2}(\phi_0, \phi) = 0.98$.
(e) $\lambda = 0.9$, $F_{1/2}(\phi_0, \phi) = 0.95$.
(f) $\lambda = 1.8$, $F_{1/2}(\phi_0, \phi) = 0.999$.
(g) $\lambda = 1.8$, $F_{1/2}(\phi_0, \phi) = 0.99761$.
} \label{fig:loschmidt}
\end{figure*}

Using Ineq.\,\eqref{eq:mainineq2} allows us to give an estimate for the Loschmidt echo $L_\phi(\lambda,t)$ that is valid for any state $|\phi\rangle$. Indeed, fixing an initial state $\varrho = |+\rangle\langle +|$ of the qubit, $|+\rangle = \frac{1}{\sqrt{2}}( |0\rangle + |1\rangle)$ being the eigenstate of $\sigma_1$, we can solve
\begin{eqnarray}\label{eq:lossi}
\mc E^{(t)}_\phi(\varrho) &= &\frac{1}{2} \left(
\begin{array}{cc} 1 & \sqrt{L_\phi(\lambda,t)} \\
\sqrt{L_\phi(\lambda,t)} & 1
\end{array}
\right).
\end{eqnarray}
It can be concluded that
\begin{align} \nonumber
F_{1/2}\big( \phi_0, \phi\big)&\leq  F_{1/2}\big(\mc E^{(t)}_{\phi_0}(\varrho), \mc E^{(t)}_\phi(\varrho)\big)\nonumber\\ \nonumber
&= \frac{1}{2} \sqrt{\big(1-\sqrt{L_{\phi_0}(\lambda,t)}\big)\big(1-\sqrt{L_\phi(\lambda,t)}\big)}\nonumber\\ \label{eq:loschmidt}
&+\frac{1}{2} \sqrt{\big(1+\sqrt{L_{\phi_0}(\lambda,t)}\big)\big(1+\sqrt{L_\phi(\lambda,t)}\big)},
\end{align}
from which an estimate for $L_\phi(\lambda,t)$ in terms of $F_{1/2}\big( \phi_0, \phi\big)$ may be solved. In Fig.\,\ref{fig:loschmidt} we have presented the results for three different values of the strength of the transverse field $\lambda$ fixing the parameters $\delta = 0.1$, $J = 1$ and the number of spins $N = 4000$. In particular, we see that for those $\lambda$ resulting to
revivals in Loschmidt echo
in the ground state $|\phi_0\rangle$, after a certain limiting value of $F_{\text{lim}}\leq F_{1/2}\big( \phi_0, \phi\big)$  such revivals are guaranteed
for the state $|\phi\rangle$ also. The revivals of Loschmidt echo are linked to revivals of coherences via Eq.\,\eqref{eq:lossi} and hence indicate back-flow of information from environment to system.

It should be pointed out, that the upper and lower bounds of the Loschmidt echo $L_\phi$ could be made tighter by considering general $\alpha$-fidelities in the above Ineq.\,\eqref{eq:loschmidt}. However, since solving $L_\phi$ from Ineq.\,\eqref{eq:loschmidt} in such a general situation can only be done numerically, we have left this examination as a topic for future investigation.

\section{Conclusions and Outlook} \label{sec:conc}

In this paper we have introduced a novel family of $\alpha$-fidelities of quantum channels. We also have derived an inequality between the $\alpha$-fidelity of the initial states of the environment and the $\alpha$-fidelity of the corresponding induced dynamics: $F_\alpha(\xi_1,\xi_2) \leq \mc F_\alpha\big(\mc E_1^{(t)}, \mc E_2^{(t)}\big)$, $\alpha\in[1/2,1)$. From a more practical viewpoint, the inequality was then considered in the context of four different applications: {\bf A)} quantum programming, {\bf B)} discrimination of environmental properties, {\bf C)} quantum thermometry and in {\bf D)} deriving bounds for the Loschmidt echo of a general (pure) initial state of the spin-chain.

In a nutshell, the general procedure of estimating an unknown $\xi_2$
requires 1) the possibility to prepare the environment in a known ''calibration'' state $\xi_1$ and 2) the possibility to
compare the probe dynamic $\mc E_1^{(t)}$ and  $\mc E_2^{(t)}$ induced by these states. To apply our framework to quantum probing experimentally, as proposed in examples {\bf B)}, {\bf C)} and {\bf D)}, the experimenter needs first to perform quantum process tomography and then numerical optimization for all times, which may be extremely difficult. However, the starting point underlying behind our results is the Ineq.\,\eqref{eq:mainineq2}, which shows that non-trivial bounds for $F_{\alpha}(\xi_1, \xi_2)$ may be found
for plethora of choices of initial probe states $\varrho_1$ and $\varrho_2$; the notion of $\alpha$-fidelity of dynamics was developed as the best achievable resolution of this inequality. In particular, choosing $\varrho_i= \varrho,$ $i=1,2$, simplifies the Ineq.\,\eqref{eq:mainineq2} to
\begin{eqnarray}
F_{\alpha}(\xi_1, \xi_2) \leq F_{\alpha}\big(\mc E_1^{(t)} (\varrho) , \mc E_2^{(t)} (\varrho) \big) \leq 1 \, ,
\end{eqnarray}
for all $\varrho \in \mc S(\hi_S)$ and $\alpha \in [1/2, 1)$ and any given time $t\in [0,\infty)$. Although the resulting upper bound $F_{\alpha}\big(\mc E_1 (\varrho) , \mc E_2 (\varrho) \big)$ may not be the optimal one, any choice of the initial state of $\varrho \in \mc S(\hi_S)$ allows one to extract some information of system of interest via the protocols described above. In addition, the full process tomography is now reduced from the full channel tomography to mere state tomography of the evolved states of the probe $\mc E_1 (\varrho)$ and $\mc E_2 (\varrho)$.

In our theoretical framework the introduced fidelities are based on the definition of the $\alpha$-R\'enyi divergence, given in Eq.\,\eqref{eq:defrenyi}, mainly because of its connection to the Uhlmann fidelity for $\alpha = 1/2$. However as we mentioned before, there exists another quantum divergence often encountered in the literature, namely $\widetilde{S}_{\alpha}$ defined in Eq.\eqref{eq:defrenyi0}. It is known that also $\widetilde{S}_\alpha$ satisfies both the properties (S1)-(S3) and the data processing inequality (S4) for all $\alpha \in (0,1)$ \cite{Muller2013}. Hence, also for $\widetilde{F}_\alpha(\varrho_1, \varrho_2 ) \doteq \tr{\varrho_1^\alpha \varrho_2^{1-\alpha}}$, we have $\widetilde{F}_\alpha(\varrho_1,\varrho_2) \widetilde{F}_\alpha(\xi_1,\xi_2) \leq \widetilde{F}_\alpha\big(\mc E_1^{(t)}(\varrho_1), \mc E_2^{(t)}(\varrho_2)\big)$, $\alpha\in(0,1)$. Furthermore, for commuting states $[\varrho_1, \varrho_2]=0$ the quantities $S_\alpha$ and $\widetilde{S}_\alpha$ coincide \cite{Muller2013}.  Although it is known that $\widetilde{F}_\alpha \leq F_\alpha$, since $ S_\alpha \leq \widetilde{S}_\alpha $ \cite{Wilde2014, Datta2014}, an interesting question to investigate in the future is which of the two definitions $\widetilde{F}_\alpha$ or $F_\alpha$ leads to a tighter inequality between the inducing states and induced dynamics -- or whether such optimality even holds in general. It is noteworthy, that the results shown in Figs.\,\ref{fig:hamiltonians}, \ref{fig:thermo} and \ref{fig:loschmidt} are independent of the choice of  $\widetilde{F}_\alpha$ or $F_\alpha$ due to the facts that the time-evolved final states of the probe are commuting in all the three cases as well as the environment states in applications {\bf B)}, {\bf C)}, and since $\widetilde{F}_\alpha$ and $F_\alpha$ coincide when comparing pure states in {\bf D)}.

Due to the generality of our formalism, our results can be used in different fields of physics: from quantum information theory to solid state physics, from particle physics to cosmology, from quantum gravity to atomic and molecular physics, and from quantum optics to quantum thermodynamics. Therefore our approach has the potential to pave the way to new fundamental theoretical and experimental discoveries in numerous physical scenarios.

\section*{Acknowledgements}
The authors are grateful to M\'ario Ziman for his comments on the manuscript.
S.M was supported by the EU Collaborative project QuProCS (Grant Agreement 641277) and the Academy of Finland (Project no. 287750).
M.T. and H.L. acknowledge financial support from the University of Turku
Graduate School (UTUGS) and M.T. additionally acknowledges the hospitality of Mathematical Physics Group in Toru\'n.
G.S. was partially supported by the National Science Centre (project 2015/17/B/ST2/02026), the Magnus Ehrnrooth Foundation and wishes to acknowledge the hospitality of Turku Quantum Technology Group.

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text:
Q: With an odd number of terms, the Taylor polynomial of cosine is greater than cosine, but with an even number of terms it's less than cosine If you use an odd number of terms,
$$f_{4n}(x) = 1 - \frac{x^2}{2!} + ... + \frac{x^{4n}}{(4n)!},$$
this Taylor polynomial $f_{4n}(x) \geq \cos(x)$. With an even number of terms,
$$f_{4n-2}(x) = 1 - \frac{x^2}{2!} + ... - \frac{x^{4n-2}}{(4n-2)!}$$
the Taylor polynomial $f_{4n-2}(x) \leq \cos(x).$ How do we prove this? The trouble is that the power series for $\cos$ is infinite, and there are infinitely-many negative terms and infinitely-many positive terms.

A: Suppose that you want to prove that$$(\forall x\in\Bbb R):\cos(x)\geqslant1-\frac{x^2}{2!}.$$This is the same thing as proving that$$(\forall x\in\Bbb R):\cos(x)-1+\frac{x^2}{2!}\geqslant0.$$Let $f(x)=\cos(x)-1+\frac{x^2}{2!}$. Then $f(0)=0$ and $f'(x)=-\sin(x)+x$. If it can be proved that $f'(x)\leqslant0$ when $x<0$ and that $f'(x)\geqslant0$ when $x>0$, we're done. But$$f''(x)=-\cos(x)+1\geqslant0.$$So, since $f'(0)=0$, and since $f'$ is increasing, we do indeed have that $f'(x)\leqslant0$ when $x<0$ and that $f'(x)\geqslant0$ when $x>0$. So, $(1)$ is indeed true.
Now, suppose that you want to prove that$$(\forall x\in\Bbb R):\cos(x)\leqslant1-\frac{x^2}{2!}+\frac{x^4}{4!}.$$Define$$g(x)=\cos(x)-1+\frac{x^2}{2!}-\frac{x^4}{4!}.$$ Then $g(0)=0$ and $g'(x)=-\sin(x)+x-\frac{x^3}{3!}$. If it can be proved that $g'(x)\geqslant0$ when $x<0$ and that $g'(x)\leqslant0$ when $x>0$, we're done. But $g''(x)=-\cos(x)+1-\frac{x^2}{2!}$, and we have already seen that then we always have $g''(x)\leqslant0$. So, since $g'(0)=0$, it is indeed true that $g'(x)\geqslant0$ when $x<0$ and that $g'(x)\leqslant0$ when $x>0$.
Can you take it from here?

A: Using $\sin' = \cos$ and $\cos' = -\sin$ get:
$$\sin(x) = \int_0^x \cos(t) dt$$
and
$$\cos(x) = 1-\int_0^x \sin(t) dt$$
From these, by induction, you can show that
the successive sin and cos polynomials
are alternatively above and below
the corresponding functions.
Note:
This does not originate with me.
I saw this in Dorre's
"100 Great Problems in Elementary Mathematics"
which I highly recommend.

A: Alternate approach: use the Liebniz rule:
If $\langle a_n\rangle$ is a strictly decreasing sequence of positive terms, with limit $0$, then the alternating series
$\displaystyle \sum_{n=1}^\infty (-1)^{(n-1)}a_n$ converges.
Further, if $S$ denotes it sum and $s_n$ is its $n$-th partial sum then
$$0 < (-1)^n (S-s_n) < a_{n+1}.$$
This means that there is no reason to focus on the trigonometric association here.  For any conditionally convergent alternating series, that is as described at the start of this answer, the partial sums will always oscillate from less than $S$ to greater than $S$ and back.  Further, this oscillation will always get strictly closer and closer to $S$.
Of course, for the present problem, for any fixed value of $(x)$, you will have to wait a finite number of terms, before the absolute value of the terms start strictly decreasing.

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Dr. Zhang's primary research goals are directed toward understanding customer engagement in the hospitality and tourism industries with a special focus on technology usage and adoption (such as virtual reality, social media, mobile apps, and online communities, etc.). Dr. Zhang's research dissertation explores the service innovation strategies that engage Generation Y consumers, an emerging and powerful cohort in the market, to co-create through mobile technologies. Her research articles are published in service management and mainstream hospitality administration journals. Additionally, Dr. Zhang has been serving as a reviewer for several refereed journals, such as International Journal of Contemporary Hospitality Management, Journal of Service Management, and Journal of Hospitality and Tourism Management. Dr. Zhang serves on editorial advisory board for the Journal of Interdisciplinary Business-Economics Advancement (IIBA Journal). Dr. Zhang has also been serving as a reviewer for referred international conferences, such as 2017 Graduate Conference Hospitality and Tourism.
Dr. Zhang joined the Rosen College of Hospitality Management as an Assistant Professor in August 2016. In her current position, she teaches face-to-face course of Hospitality Information Systems and mixed mode course of Event Technology.
Dr. Zhang received her Ph.D. degree and Master's degree in Hospitality Management from the Ohio State University. She received her Master's degree in Translation and Interpretation from Beijing Foreign Studies University, China. She also received her Bachelor of Science degree in Marketing with emphasis on Hospitality and Tourism Management from Beijing International Studies University, China.
Before joining UCF's Rosen College of Hospitality Management, Dr. Zhang held multiple teaching positions, including teaching assistant, instructor, and mentor. She has taught both undergraduate and graduate courses in hospitality and tourism management and marketing. In addition to that, Dr. Zhang advised honors undergraduate student for honors thesis that received $10000 grant support from Pelotonia Fellowship Program. Dr. Zhang also held an instructor position at Beijing International Studies University (BISU). During her teaching at BISU, she joined faculty trainee program offered by Marriott Group. This allowed her to work in multiple function departments including Front Desk, Food & Beverage, Executive Lounge, and Sales & Marketing. This unique work experience gave her a deeper understanding of hotel management across various departments and thus enhanced her ability to teach on these topics.


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Vince McMahon: 10 Reasons Why WWE's Chairman Is the Best Heel of All Time
Mike Chiari@mikechiariTwitter LogoFeatured ColumnistOctober 19, 2012
Over the course of wrestling history, there have been countless great heels, but none of them have measured up to Vince McMahon. The WWE chairman helped revolutionize the business during the Attitude Era, and because of that, he has earned the title of best heel of all time.
McMahon may not have wrestled often, although he wasn't shy about mixing it up, but as an authority figure, he was among the most hated men in wrestling history. It is very difficult to pinpoint one aspect of McMahon's character that made him such a fantastic heel, rather it was a combination of many things that helped him excel.
As much as people like to give credit to "Stone Cold" Steve Austin for the company's explosion in the late 1990s, McMahon was an equally big part of it. Without McMahon, the WWE, as we know it, wouldn't exist today, and without his contributions during the Attitude Era, perhaps, there wouldn't even be a WWE, period.
Here are the top 10 reasons why Mr. McMahon is unquestionably the greatest heel in the long-and-decorated history of professional wrestling.
Feud with "Stone Cold" Steve Austin
Every great hero needs a great villain, and McMahon had no issue when it came to playing the yin to "Stone Cold" Steve Austin's yang.
I could probably name a thousand great professional wrestling feuds, but I don't believe that any of them ever reached the level that Austin vs. McMahon did. It surpassed everything that came before it and I don't think that anything will ever top it.
Austin was wildly popular during the Attitude Era, and for good reason, but Mr. McMahon's devilish ways certainly helped when it came to Austin getting over with the fans. Austin feuded with all the top stars of the time period. However, his rivalry with McMahon was constant and never really subsided over the course of the Attitude Era.
Top faces can very easily expose the weaknesses of those that they feud with, but McMahon was always at the same level as Austin in terms of entertainment value. Austin will always be considered a bigger star than Vince since he was an actual wrestler, but McMahon more than held his own and was paramount in the rise of Austin 3:16 and the WWE for a while during the Attitude Era.
Montreal Screwjob
The only thing more helpful in getting a heel over than a dastardly storyline act is a dastardly real-life act. Almost everything that happens in professional wrestling is scripted, but there are rare occasions where the original plans are ignored.
Perhaps, the prime example of that is the Montreal Screwjob, which took place at Survivor Series in 1997 and involved McMahon screwing Bret Hart out of the WWE Championship.
Hart was about to move on to WCW, and McMahon wanted him to drop the title to Shawn Michaels at Survivor Series, but Hart declined. Hart's issues with Michaels back then have been well-documented, and he had no interest in putting him over.
In the interest of protecting himself from Hart taking the title with him to WCW a la Madusa, McMahon organized a ruse that involved him calling for the bell when Michaels had Hart in the sharpshooter.
Hart realized immediately that he had been tricked, and he proceeded to spit in McMahon's face and then later assaulted him in the locker room. This was essentially the birth of the Mr. McMahon character, and it was a big reason for his success. McMahon and Hart ultimately made up years later, but many consider the Montreal Screwjob to be the start of the Attitude Era.
Jerk Boss Persona
Wrestling fans tend to react well to things that they can relate to in their real lives, and the Mr. McMahon character certainly delivered in that way. I'm sure that many of us have had terrible bosses in the past, and that is precisely what McMahon represented. He had a ton of power, and he abused for the sole purpose of making people's lives miserable.
Professional wrestling has always been filled with larger-than-life characters, many of whom didn't translate to the real world. Everything about Mr. McMahon seemed real, though, and the fact that he could get himself over as a heel in that manner is what made him so great. Nobody likes a corrupt authority figure, and that made McMahon the perfect bad guy during that time period.
As mentioned before, McMahon being the evil boss made Steve Austin an even more sympathetic character as far as the viewers were concerned. Austin was essentially the common man who was doing something that many wished they could do: getting back at his boss.
McMahon represented something that most people have experienced and that everyone hated.
Got Nuclear Heat Despite Not Being a Regular Wrestler
Most of the top heels in professional wrestling history have naturally been in-ring competitors, but there are a select few who have gotten over in other manners.
Managers like Bobby "The Brain" Heenan and Jimmy Hart come to mind, but McMahon trumps both of them in that regard. McMahon had his fair share of matches against Steve Austin and others, but he largely became the hottest heel in wrestling based solely on his character.
Heels usually get heat based on what they do in the squared circle, but essentially, all of McMahon's heat was the result of his promos and his abuse of power. In fact, since McMahon wasn't an actual wrestler, he was even more hated when he did step into the ring because fans wanted to see Austin, The Rock, Mick Foley or whoever else beat him to a pulp.
There have been wrestlers who have been able to get over with the crowd with only their wrestling skills over the years, but McMahon didn't have that luxury. He had to nail it with his character, or else, he was essentially useless.
If McMahon failed to get the crowd to hate him as a person, then he would have fallen flat, but McMahon elicits a hugely negative response, despite his limitations as a wrestler.
Put Over Every Top Face
As much as many current wrestling fans like to see heels go over, the tried-and-true method for success in the business is ultimately putting over the top faces.
McMahon has always subscribed to this theory, and it rang true during his run as the top heel in the business. While McMahon often got his licks in against Austin and pretty much every other top star, he would almost always come out with the short end of the stick.
Look no further than the Chairman's WrestleMania record for proof as he is 0-5 overall with losses to his son, Shane, Hulk Hogan, Shawn Michaels, Donald Trump in a hair vs. hair match involving Umaga and Bobby Lashley and Bret Hart.
McMahon may have a big ego, but he has never let pride get in the way of making the right creative decision with regard to himself. He knows that putting over guys like Austin, Hogan and Michaels is good business, and that is what he has always done.
It is the job of a great heel to be selfish from a storyline perspective, but they must also be selfless when it comes to booking. McMahon was never worried about embarrassing himself, and he was fully committed to sacrificing himself for the greater good of the WWE.
His Real-Life Role Made It Believable
One thing that often prevents a heel from getting over is the fact that their gimmick simply isn't believable.
There are many such examples over the years, but that was never an issue for McMahon. As soon as the Mr. McMahon character was created, it was made very clear that he was basically the head honcho of the WWE and that he could do pretty much anything he wanted.
Many authority figures over the years have been lame since it was obvious that they didn't have any real power.
It dates back to Jack Tunney as WWE president, and it continues to this day with A.J. Lee and Booker T acting as general managers for Raw and SmackDown, respectively. All of them have storyline power, but it is well-known that the real power belongs to McMahon.
That is why McMahon was so easy to buy into his role as the corrupt owner of WWE. He truly was the main man, and it wasn't a stretch to believe that he was actually a terrible person who let the power go to his head. Realism is a tough thing to achieve in wrestling, but McMahon's heel character had it from the start.
Trish Stratus Storyline
As if it wasn't already apparent that Mr. McMahon was a terrible person, it was further solidified in 2001 when he began cheating on his wife, Linda, with Trish Stratus. This led to McMahon asking for a divorce, which caused Linda to enter a catatonic state. During this time, McMahon would wheel Linda around and often make out with Trish in front of her.
McMahon's evil ways didn't end there, however, as he eventually turned on Trish as well and told her that she was a toy that he had grown tired of. He continued to abuse his power by forcing Trish to get on her hands and knees and bark like a dog while in her underwear. Not only was McMahon a jerk to guys like Steve Austin, now he was demeaning women as well.
The hallmark of a truly great heel is finding ways to make people hate you even more than they already do, and McMahon always managed to one-up himself somehow. Many consider McMahon's storyline with Trish and Linda to be offensive to this day, and that goes to show you that he knew exactly what he was doing.
Perfect Heel Theme Music
I believe that fitting theme music has always been an underrated aspect of a great character in professional wrestling, but McMahon's music was definitely perfect for what he was trying to portray.
McMahon often seemed like an impossible hurdle for Steve Austin and all the other top faces during the Attitude Era, and his "No Chance In Hell" theme song blaring over the loudspeakers further emphasized his supreme power.
Theme music can set the tone for a great match or promo, and it was always obvious what McMahon was setting out to do when he walked down the ramp with his signature, cocky strut.
He was looking to screw someone over at every turn. When you really think about it, there haven't been many more perfect combination of character and theme song over the course of wrestling history.
McMahon's theme song allowed him to get immediate heat as soon as the first chord was played, and I believe it was a big part of why he was so successful as a heel. Some may consider a theme song to be an auxiliary factor with no real bearing on a person's character, but it's tough to argue against the impact that "No Chance In Hell" made on McMahon.
Higher-Power Storyline
It may have been one of the Attitude Era's lamer storylines, but the higher-power angle definitely helped McMahon get over as a heel.
After a long-running feud with Steve Austin, McMahon's Corporation stable began to get involved with Undertaker's Ministry of Darkness in 1999. Taker kidnapped McMahon's daughter, Stephanie, on a couple occasions, and it eventually led to McMahon turning face.
McMahon went at it with Undertaker for a couple months before it was finally revealed who was directing Undertaker and his Ministry. He revealed himself to be the ring leader, and while it disappointed a lot of fans, it definitely made him seem even more dastardly.
Not only did he put his own daughter in harm's way, but he pulled the wool over the fans' eyes as they were sympathetic toward him for a time.
McMahon is known as one of the most deceptive villains in WWE history, and the higher-power storyline is probably one of his signature moments in that regard. There aren't many heels who could have made something out of a disappointing conclusion to an angle like that, but McMahon managed to make it work.
Transitioned to Top Heel Role Seamlessly
One of the most impressive things about McMahon's instant heat as a heel was the fact that he served as a well-liked commentator for many years prior to becoming Mr. McMahon.
Most fans didn't really have any idea that he was running the show, and they just thought that he was a random announcer. McMahon went from being a fairly anonymous person behind the announce table to the most hated person in WWE with the flip of a switch essentially.
It usually takes months or even years to develop an effective heel persona, but McMahon did it quicker than anyone with the possible exception of Hulk Hogan becoming Hollywood Hogan. It was easier for Hogan, though, because he was the biggest face in the industry, so doing one heel-like thing naturally made him the top bad guy.
McMahon wasn't really even on the radar prior to his turn, but the Montreal Screwjob helped launch him and the WWE to new heights. It takes somebody very special to elicit such an overwhelming response from the crowd so quickly, but McMahon was able to, and that is why he is the best heel of all time.
Follow @MikeChiari on Twitter and listen to him on Ring Rust Radio.

redpajama_set_name: RedPajamaCommonCrawl


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