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Minimal-error quantum state discrimination versus robustness of entanglement:More indistinguishability with less entanglement

Debarupa Saha, Kornikar Sen, Chirag Srivastava, Ujjwal Sen

TL;DR

This work establishes universal bounds connecting minimum-error quantum state discrimination performance to the robustness of entanglement, via the closest separable ensemble and the ensemble-wide random robustness. It proves an upper bound on the discrimination success that scales with the maximal random robustness and a lower bound that depends on the ensemble's minimum random robustness, applicable to arbitrary multipartite systems and discrimination strategies. A key finding is the phenomenon 'More indistinguishability with less entanglement': for two equally entangled states, the entangled pair is easier to distinguish than the corresponding closest separable pair under global measurements, with numerical evidence suggesting a threshold on minimum entanglement for two-qubit ensembles to activate this effect. The results provide a rigorous, dimension-free framework linking entanglement structure to information-processing tasks and offer practical insight into when entanglement is advantageous for state discrimination, supported by Haar-uniform two-qubit numerical studies.

Abstract

We relate the the distinguishability of quantum states with their robustness of the entanglement, where the robustness of any resource quantifies how tolerant it is to noise. In particular, we identify upper and lower bounds on the probability of discriminating the states, appearing in an arbitrary multiparty ensemble, in terms of their robustness of entanglement and the probability of discriminating states of the closest separable ensemble. These bounds hold true, irrespective of the dimension of the constituent systems the number of parties involved, the size of the ensemble, and whether the measurement strategies are local or global. Additional lower bounds on the same quantity is determined by considering two special cases of two-state multiparty ensembles, either having equal entanglement or at least one of them being separable. The case of equal entanglement reveals that it is always easier to discriminate the entangled states than the ones in the corresponding closest separable ensemble, a phenomenon which we refer as "More indistinguishability with less entanglement". Furthermore, we numerically explore how tight the bounds are by examining the global discrimination probability of states selected from Haar-uniformly generated ensembles of two two-qubit states. We find that for two-element ensembles of unequal entanglements, the minimum of the two entanglements must possess a threshold value for the ensemble to exhibit "More indistinguishability with less entanglement".

Minimal-error quantum state discrimination versus robustness of entanglement:More indistinguishability with less entanglement

TL;DR

This work establishes universal bounds connecting minimum-error quantum state discrimination performance to the robustness of entanglement, via the closest separable ensemble and the ensemble-wide random robustness. It proves an upper bound on the discrimination success that scales with the maximal random robustness and a lower bound that depends on the ensemble's minimum random robustness, applicable to arbitrary multipartite systems and discrimination strategies. A key finding is the phenomenon 'More indistinguishability with less entanglement': for two equally entangled states, the entangled pair is easier to distinguish than the corresponding closest separable pair under global measurements, with numerical evidence suggesting a threshold on minimum entanglement for two-qubit ensembles to activate this effect. The results provide a rigorous, dimension-free framework linking entanglement structure to information-processing tasks and offer practical insight into when entanglement is advantageous for state discrimination, supported by Haar-uniform two-qubit numerical studies.

Abstract

We relate the the distinguishability of quantum states with their robustness of the entanglement, where the robustness of any resource quantifies how tolerant it is to noise. In particular, we identify upper and lower bounds on the probability of discriminating the states, appearing in an arbitrary multiparty ensemble, in terms of their robustness of entanglement and the probability of discriminating states of the closest separable ensemble. These bounds hold true, irrespective of the dimension of the constituent systems the number of parties involved, the size of the ensemble, and whether the measurement strategies are local or global. Additional lower bounds on the same quantity is determined by considering two special cases of two-state multiparty ensembles, either having equal entanglement or at least one of them being separable. The case of equal entanglement reveals that it is always easier to discriminate the entangled states than the ones in the corresponding closest separable ensemble, a phenomenon which we refer as "More indistinguishability with less entanglement". Furthermore, we numerically explore how tight the bounds are by examining the global discrimination probability of states selected from Haar-uniformly generated ensembles of two two-qubit states. We find that for two-element ensembles of unequal entanglements, the minimum of the two entanglements must possess a threshold value for the ensemble to exhibit "More indistinguishability with less entanglement".
Paper Structure (15 sections, 6 theorems, 39 equations, 2 figures)

This paper contains 15 sections, 6 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Robustness of entanglement
  4. Minimum error state discrimination
  5. Global discrimination between two states
  6. Bounds on probability of success
  7. Upper bound
  8. Lower bound
  9. Example
  10. The smallest non-trivial ensembles: Ensembles of size two
  11. When the two states have equal random robustness of $n$-entanglement
  12. When one of the states is separable
  13. Analysis on the tightness of the upper bound
  14. Conclusion And Discussion
  15. Acknowledgement

Key Result

Theorem 1

The probability, $P_X$, of discriminating the states chosen from any fixed bipartite ensemble $\eta_2:=\{p_{b},\rho_{b}\}$, using any particular discrimination scheme, $X$, is upper bounded as where $\epsilon_{\eta_2}$ is the closest ensemble of fully separable states to ${\eta_2}$ and $\mathbb{R}^M_1(\eta_2)$ is the maximum RRE of the ensemble $\eta_2$.

Figures (2)

  • Figure 1: Analysis of the tightness of the upper bound on the probability of successful global discrimination of states. We plot $\gamma(i)$ along the vertical axis with respect to $\mathbb{R}(i)$ presented in the horizontal axis for ensembles, $\eta^2_2$, consisting of Haar-uniformly generated states. In the left panel, the green, blue, yellow, and red points represent ensembles consisting of two rank-4, rank-3, rank-2, and rank-1 states, respectively. The green, blue, yellow, and red points of the right panel represent ensembles having one pure product state and one rank-4, rank-3, rank-2, and rank-1 state, respectively. There are 1000 points for each color in both of the panels, i.e., 1000 ensembles of each type, specified by the rank of the enclosed states. In both of the panels, the black line presents $\gamma=1/({1+\mathbb{R}})$ curve. All the axes are dimensionless.
  • Figure 2: Behavior of the difference between the probability of globally discriminating states of an ensemble and the states of the closest ensemble of separable states to the former. The plot shows the nature of $\delta P (r)$ (presented along the vertical axis) against fixed minimum RRE, $r$, (presented along the horizontal axis) with red points. The orange dashed line parallel to the vertical axis represents $r=r_c=0.073$ line indicating the value of $r$ above which $\delta P (r)$ becomes zero. Both the horizontal and vertical axes are dimensionless.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof