In this work, we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.
In this work, we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.
The topic of causality has recently gained traction quantum information research. This work examines the problem of single-shot discrimination between process matrices which are an universal method defining a causal structure. We provide an exact expression for the optimal probability of correct distinction. In addition, we present an alternative way to achieve this expression by using the convex cone structure theory. We also express the discrimination task as semidefinite programming. Due to that, we have created the SDP calculating the distance between process matrices and we quantify it in terms of the trace norm. As a valuable by-product, the program finds an optimal realization of the discrimination task. We also find two classes of process matrices which can be distinguished perfectly. Our main result, however, is a consideration of the discrimination task for process matrices corresponding to quantum combs. We study which strategy, adaptive or non-signalling, should be used during the discrimination task. We proved that no matter which strategy you choose, the probability of distinguishing two process matrices being a quantum comb is the same.
We present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case of minimum error discrimination, we focus on discrimination of measurements with the assistance of entanglement. We provide an alternative proof of the fact that all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. Moreover, we analytically find the minimal number of queries needed for perfect discrimination. We also show that in this scenario querying the measurements in parallel gives the optimal strategy, and hence any possible adaptive methods do not offer any advantage over the parallel scheme. In the unambiguous discrimination scenario, we give the general expressions for the optimal discrimination probabilities with and without the assistance of entanglement. Finally, we show that typical pairs of Haar-random von Neumann measurements can be perfectly distinguished with only two queries.
Entanglement is known to be a powerful resource that improves performance in various quantum information and computational tasks. A standard example of such a phenomenon is the possibility of perfectly discriminating all four Pauli operations in a single shot via the superdense coding protocol. While entanglement is often a powerful resource for quantum channel discrimination, this is not necessarily the case. In this work, we identify scenarios in which the maximally entangled state is a bad choice of input state and, more generally, show that excessive entanglement can reduce channel discriminability dramatically. To do so, we present an explicit pair of unitary channels which are perfectly discriminable without entanglement, but for which any strategy with maximally entangled input states is ε-close to a blind uniform guessing strategy. To develop a systematic approach, we introduce the concepts of Maximal Entanglement Worst Case (MEWC) and Maximal Entanglement Best Case (MEBC) pairs of channels, and present conditions for a pair of channels to be MEWC or MEBC. With these conditions, we show that the optimal input states for discriminating MEWC pairs of channels are necessarily separable, and provide non-trivial examples of measurement channels for which entanglement necessarily reduces the maximum probability of discrimination.
Among the intriguing features of quantum theory, the problem of distinguishing quantum channels is of fundamental interest. In this paper, we focus on the single-shot discrimination of two noisy quantum channels using two distinct classes of probes: single-system (product) probes and entangled probes. Our aim is to identify optimal probing state for specific discrimination tasks and to analyze the necessity and role of entanglement in enhancing channel distinguishability. We show that maximally entangled probes are optimal for discriminating two qubit depolarizing channels, with any nonzero entanglement providing an advantage over single-system probes. In contrast, for dephasing channels in arbitrary dimensions, we prove that single-system probe can be optimal and that entanglement offers no improvement, even when the dephasing unitary is generalized. For qubit amplitude-damping channels, we identify distinct noise-dependent regimes in which either single-system probe outperforms maximally entangled probes and vice-versa. Moreover, we demonstrate that non-maximally entangled probes can act as the optimum probe if the noise parameters restricted to certain values in this task. We also present examples of noisy unitary channels for which discrimination is possible using non-maximally entangled probe, while both single-systems and maximally entangled probes fail. We introduce another class of noisy unitary channels for which perfect discrimination is achievable with a single system, while maximally entangled probes are insufficient. Finally, we show that two erasure channels can be optimally discriminated using any pure single-system probe, with no advantage gained from entanglement.
Single-shot quantum channel discrimination is the fundamental task of determining, given only a single use, which of two known quantum channels is acting on a system. In this thesis we investigate the well-known phenomenon that entanglement to an auxiliary system can provide an advantage in this task. In particular, we consider the questions: (1) How much entanglement is in general necessary to achieve an optimal discrimination strategy? (2) What is the maximal advantage provided by entanglement? \n \nGiven a linear map Ψ : L(Cn) → L(Cm), its multiplicity maps are defined as the family of linear maps Ψ ⊗ 1L(Ck) : L(Cn ⊗ Ck) → L(Cm ⊗ Ck), where 1L(Ck) is the identity on L(Ck). Due to the Holevo-Helstrom theorem, the optimal performance using an auxiliary system of dimension k is quantified in terms of the norm of Ψ⊗1L(Ck), where Ψ is a linear map that depends on the parameters of the discrimination problem. Hence, the advantage provided by entanglement is represented in the growth of the norm of Ψ ⊗ 1L(Ck ) with k, a classic phenomenon in the theory of operator algebras. \n \nWe formalize question (1) by investigating, relative to the input and output dimensions of the channels to be discriminated, how large of an auxiliary system is necessary to achieve an optimal strategy. Mathematically, this is connected to when the norm of Ψ ⊗ 1L(Ck) stops growing with k. It is well- known that an auxiliary system of dimension equal to the input is always sufficient to achieve an optimal strategy, and that this is sometimes necessary when the output dimension is at least as large as the input. We prove that, even when the output dimension is arbitrarily small compared to the input, it is still sometimes necessary to use an auxiliary system as large as the input to achieve an optimal strategy. \n \nFor question (2), we investigate, with respect to a fixed input dimension, how large the gap between the optimal performances with and without entanglement can be. Mathematically, this is quantified by the rate of growth of the norm of Ψ ⊗ 1L(Ck ) in k. It is known that matrix transposition has the fastest possible growth, and we prove that it is essentially the unique linear map with this property. We use this to prove that a discrimination problem defined in terms of the Werner-Holevo channels is essentially the unique game satisfying a norm relation that states that the game can be won with certainty using entanglement, but is hard to win without entanglement. \n \nAlong the way, we prove characterizations of the structure of maximal entanglement, as measured according to the entanglement negativity, as well as relative to a large class of entanglement measures. We also give various characterizations of complete trace-norm isometries, and reversible quantum channels.
We find optimality conditions for testers in discrimination of quantum channels. These conditions are obtained using semidefinite programming and are similar to optimality conditions for discrimination of quantum states. We get a simple condition for existence of an optimal tester with any given input state with maximal Schmidt rank, in particular with a maximally entangled input state. In the case when maximally entangled state is not optimal, an upper bound on the optimal success probability is obtained. The results for discrimination of two channels are applied to covariant channels, qubit channels, unitary channels, and simple projective measurements.