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arXiv:2504.14499 · Physical Review A

Limitation of maximally entangled probes for single-shot distinguishability of unitaries

Satyaki Manna, Anandamay Das Bhowmik, Debashis Saha

Published Sep 26, 2025• 3 citations
arXivHTMLPDF

Abstract

There have been many instances where the maximally entangled state as a probe acts better than the product and the nonmaximally entangled states in the task of distinguishing quantum channels. We provide a proof that for the single-shot discrimination of two unitary channels, entangled and product states are operationally equivalent. However, we identify pairs of unitaries that are perfectly distinguishable using a nonmaximally entangled state, but not with a maximally entangled one. This contrast becomes more pronounced when the number of unitaries exceeds two. In every $\text{dimension}\phantom{\rule{0.16em}{0ex}}\ensuremath{\geqslant}3$, we show that there exists a class of unitaries that is indistinguishable under maximally entangled probes, yet perfectly distinguishable using product or nonmaximally entangled inputs. Another interesting set of unitaries in every $\text{dimension}\phantom{\rule{0.16em}{0ex}}\ensuremath{\geqslant}3$ has been presented where only a nonmaximally entangled state acts as a successful probe, while product states and maximally entangled states cannot.

DOI
10.1103/bsxv-q9x7

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There have been many instances where the maximally entangled state as a probe acts better than the product and the nonmaximally entangled states in the task of distinguishing quantum channels. We provide a proof that for the single-shot discrimination of two unitary channels, entangled and product states are operationally equivalent. However, we identify pairs of unitaries that are perfectly distinguishable using a nonmaximally entangled state, but not with a maximally entangled one. This contrast becomes more pronounced when the number of unitaries exceeds two. In every dimension\ensuremath⩾3\text{dimension}\phantom{\rule{0.16em}{0ex}}\ensuremath{\geqslant}3dimension\ensuremath⩾3, we show that there exists a class of unitaries that is indistinguishable under maximally entangled probes, yet perfectly distinguishable using product or nonmaximally entangled inputs. Another interesting set of unitaries in every dimension\ensuremath⩾3\text{dimension}\phantom{\rule{0.16em}{0ex}}\ensuremath{\geqslant}3dimension\ensuremath⩾3 has been presented where only a nonmaximally entangled state acts as a successful probe, while product states and maximally entangled states cannot.
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.
The notion of antidistinguishability captures the possibility of ruling out certain alternatives in a quantum experiment without identifying the actual outcome. Although extensively studied for quantum states, the antidistinguishability of quantum channels remains largely unexplored. In this work, we investigate the single-shot antidistinguishability of unitary operations. We analyse two scenarios: antidistinguishability with single-system probes and with entangled probes. For sets of three unitaries, we first prove that all maximally entangled states are equivalent in their performance as probe. In the qubit case, we further establish that maximally entangled probes are always sufficient: if a set of three qubit unitaries is antidistinguishable with either a single-system or non-maximally entangled probe, then it is also antidistinguishable with a maximally entangled one. However, in higher dimension, this equivalence fails. In \textit{dimension 3}, there exists a set of unitaries that are antidistinguishable with non-maximally entangled probe or single-system probe but not with maximally entangled probe. We also establish that union of two antidistinguishable sets of three qubit unitaries also forms a set of antidistinguishable unitaries. Lastly, we provide methods to construct antidistinguishable unitaries from non-antidistinguishable ones.
We investigate the task of local marking for locally implementable unitary operations. In this setting, multipartite quantum unitary channels, chosen randomly from a known set, are distributed among spatially separated parties without revealing their identities. The objective is to correctly identify (mark) the applied process using only local operations supplemented with classical communication (LOCC). While local distinguishability implies local marking, local marking does not guarantee either local or even global distinguishability of a set of unitaries. Thus the task of marking is not equivalent to the task of discrimination. We demonstrate a stronger manifestation of nonlocality without entanglement by constructing a set of globally distinguishable tripartite product unitaries that cannot be locally marked. In contrast to state marking, we find that marking a subset of product unitaries does not imply the ability to mark a larger subset. Finally, we explore the hierarchy of probes-entangled and product-in the context of local marking with respect to the standard discrimination scenario.
Among the surprising features of quantum measurements, the problem of distinguishing and antidistinguishing general quantum measurements is fundamentally appealing. Unlike classical systems, quantum theory offers entangled states and peculiar state update rule of the postmeasurement state, which gives rise to four distinct scenarios for distinguishing (and antidistinguishing) quantum measurements: (i) probing single systems and without access to the postmeasurement states, (ii) probing entangled systems and without access to the postmeasurement states, (iii) probing single systems with access to the postmeasurement states, and (iv) probing entangled systems with access to the postmeasurement states. In these scenarios, we consider the probability of distinguishing (and antidistinguishing) quantum measurements sampled from a given set in the single-shot regime. For some scenarios, we provide the analytical expressions of distinguishability (and antidistinguishability) for qubit projective measurements. We show that the distinguishability of any pair of qubit projective measurements in scenario (iii) is always greater than its values in scenario (ii). Interestingly, we find certain pairs of qubit nonprojective measurements for which the optimal distinguishability in scenario (ii) is achieved using a nonmaximally entangled state. It turns out that, for any set of measurements, distinguishability (and antidistinguishability) in scenario (i) is always less than or equal to in any other scenario, while it reaches its highest possible value in scenario (iv). We establish that the relations form a strict hierarchy, and there is no hierarchical relation between scenarios (ii) and (iii). In particular, we introduce different variants of the well-known ``trine'' qubit measurement to construct pairs (and triples) of qubit quantum measurements such that they are perfectly distinguishable (and antidistinguishable) in scenario (ii) but not in scenario (iii) and vice versa. Additionally, we present qubit measurements that are perfectly distinguishable (and antidistinguishable) in scenario (iv) but not in any other scenario.
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.