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arXiv:2506.02575

Classical pair of states as optimal pair for quantum distinguishability quantifiers

Bassano Vacchini, Andrea Smirne, Nina Megier

Published Jun 3, 2025• 0 citations
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Abstract

The capability to quantitatively distinguish quantum states is of great importance for a variety of tasks, and has recently played an important role in the study of quantum reduced dynamics and their characterization in terms of memory effects. A crucial property of quantum distinguishability quantifiers considered in the latter framework is the contractivity under the action of completely positive trace-preserving maps. We show that this requirement warrants that the pairs on which these quantifiers attain their maximal value are pairs of orthogonal, and in this sense classical, states.

DOI
10.48550/arxiv.2506.02575

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The capability to quantitatively distinguish quantum states is of great importance for a variety of tasks, and has recently played an important role in the study of quantum reduced dynamics and their characterization in terms of memory effects. A crucial property of quantum distinguishability quantifiers considered in the latter framework is the contractivity under the action of completely positive trace-preserving maps. We show that this requirement warrants that the pairs on which these quantifiers attain their maximal value are pairs of orthogonal, and in this sense classical, states.
The distinguishability between two quantum states can be defined in terms of their trace distance. The operational meaning of this definition involves a maximization over measurement projectors. Here we introduce an alternative definition of distinguishability which, instead of projectors, is based on maximization over normalized states (density matrices). It is shown that this procedure leads to a distance (between two states) that in contrast to the usual approach based on a 1-norm, is based on an infinite-norm. Properties such as convexity, monotonicity, and invariance under unitary transformations are fulfilled. Equivalent operational implementations based on maximization over classical probabilities and hypothesis-testing scenarios are also established. When considering the action of completely positive transformations, contractivity is only granted for unital maps. This feature allows us to introduce a measure of the quantumness of nonunital maps that can be written in terms of the proposed distinguishability measure and corresponds to the maximal possible deviation from contractivity. Particular examples sustain the main results and conclusions.
We study a recently proposed measure for the quantification of quantum non-Markovianity in the dynamics of open systems which is based on the exchange of information between the open system and its environment. This measure relates the degree of memory effects to certain optimal initial state pairs featuring a maximal flow of information from the environment back to the open system. We rigorously prove that the states of these optimal pairs must lie on the boundary of the space of physical states and that they must be orthogonal. This implies that quantum memory effects are maximal for states which are initially distinguishable with certainty, having a maximal information content. Moreover, we construct an explicit example which demonstrates that optimal quantum states need not be pure states.
It is a fundamental property of quantum mechanics that information is lost as a result of performing measurements. Indeed, with every quantum measurement one can associate a number -- its POVM norm constant -- that quantifies how much the distinguishability of quantum states degrades in the worst case as a result of the measurement. This raises the obvious question which measurements preserve the most information in these sense of having the largest norm constant. While a number of near-optimal schemes have been found (e.g. the uniform POVM, or complex projective 4-designs), they all seem to be difficult to implement in practice. Here, we analyze the distinguishability of quantum states under measurements that are orbits of the Clifford group. The Clifford group plays an important role e.g. in quantum error correction, and its elements are considered simple to implement. We find that the POVM norm constants of Clifford orbits depend on the effective rank of the states that should be distinguished, as well as on a quantitative measure of the "degree of localization in phase space" of the vectors in the orbit. The most important Clifford orbit is formed by the set of stabilizer states. Our main result implies that stabilizer measurements are essentially optimal for distinguishing pure quantum states. As an auxiliary result, we use the methods developed here to prove new entropic uncertainty relations for stabilizer measurements. This paper is based on a very recent analysis of the representation theory of tensor powers of the Clifford group.
In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this Rapid Communication, I propose a framework for the distinguishability by one-way local operations and classical communication (1-LOCC) of sets of orthogonal bipartite states in a ${d}_{A}\ensuremath{\bigotimes}{d}_{B}$ bipartite system, where dA,dB{d}_{A},{d}_{B}dA​,dB​ are the dimensions of both subsytems, labeled as AAA and BBB. I show that if the ithi\mathrm{th}ith party (where i=A,Bi=A,Bi=A,B) can initiate a 1-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 1-LOCC protocol lies in a subspace of ${d}_{i}\ifmmode\times\else\texttimes\fi{}{d}_{i}$ Hermitian matrices, denoted by ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$, and that the method to extract this information (of the existence of this 1-LOCC protocol) from ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ depends on the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. In this way one can give sweeping results for the 1-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. Thus I propose that the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ gives the common underlying property based on which sweeping results for the 1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
The quantum channels with memory, known as non-Markovian channels, are of crucial importance for a realistic description of a variety of physical systems, and pave ways for new methods of decoherence control by manipulating the properties of an environment such as its frequency spectrum. In this work, the reduced dynamics of a coin in a discrete-time quantum walk is characterized as a non-Markovian quantum channel. A general formalism is sketched to extract the Kraus operators for a ttt-step quantum walk. Non-Markovianity, in the sense of P indivisibility of the reduced coin dynamics, is inferred from the nonmonotonous behavior of distinguishability of two orthogonal states subjected to it. Furthermore, we study various quantum information-theoretic quantities of a qubit under the action of this channel, putting in perspective the role such channels can play in various quantum information processing tasks.