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Entanglement in single-shot quantum channel discrimination

Daniel Puzzuoli

Published Aug 30, 2018• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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 εεε-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.
Quantum channel discrimination is a fundamental task in quantum information processing. In the one-shot regime, discrimination between two candidate channels is characterized by the diamond norm. Beyond this basic setting, however, many scenarios in distributed quantum information processing remain unresolved, motivating notions of distinguishability that capture the power of the available resources. In this work, we formulate a theory of testers for bipartite channel discrimination, leading to the concept of the entanglement cost of bipartite channel discrimination: the minimum Schmidt rank kkk of a shared maximally entangled state required for local protocols to achieve the globally optimal success probability. We introduce kkk-injectable testers as a tester-based description of entanglement-assisted local discrimination and, in particular, study the class of kkk-injectable positive-partial-transpose (PPT) testers, which constitutes a numerically tractable relaxation of the practically relevant class of LOCC testers. For every kkk, we derive a semidefinite program (SDP) for the optimal success probability, which in turn yields an efficiently computable one-shot PPT entanglement cost. To render these optimization problems numerically feasible, we prove a symmetry-reduction principle for covariant channel pairs, thereby reducing the effective dimension of the associated SDPs. Finally, by dualizing the SDP, we derive bounds on the composite channel-discrimination problem and illustrate our framework with proof-of-principle examples based on the depolarizing channel, the depolarized SWAP channel, and the Werner--Holevo channels.
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.
Abstract Distinguishing physical processes is one of the fundamental problems in quantum physics. Although distinguishability of quantum preparations and quantum channels have been studied considerably, distinguishability of quantum measurements remains largely unexplored. We investigate the problem of single-shot discrimination of quantum measurements using two strategies, one based on single quantum systems and the other one based on entangled quantum systems. First, we formally define both scenarios. We then construct sets of measurements (including non-projective) in arbitrary finite dimensions that are perfectly distinguishable within the second scenario using quantum entanglement, while not in the one based on single quantum systems. Furthermore, we show that any advantage in measurement discrimination tasks over single systems is a demonstration of Einstein–Podolsky–Rosen ‘quantum steering’. Alongside, we prove that all pure two-qubit entangled states provide an advantage in a measurement discrimination task over one-qubit systems.
We study a broad class of quantum process discrimination problems that can handle many optimization strategies such as the Bayes, Neyman-Pearson, and unambiguous strategies, where each process can consist of multiple time steps and can have an internal memory. Given a collection of candidate processes, our task is to find a discrimination strategy, which may be adaptive and/or entanglement assisted, that maximizes a given objective function subject to given constraints. Our problem can be formulated as a convex problem. Its Lagrange dual problem with no duality gap and necessary and sufficient conditions for an optimal solution are derived. We also show that if a problem has a certain symmetry and at least one optimal solution exists, then there also exists an optimal solution with the same type of symmetry. A minimax strategy for a process discrimination problem is also discussed. As applications of our results, we provide some problems in which an adaptive strategy is not necessary for optimal discrimination. We also present an example of single-shot channel discrimination for which an analytical solution can be obtained.