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Conclusive Local State Marking: More Nonlocality With No Entanglement

Samrat Sen

Abstract

Nonlocality exhibited by ensembles of composite quantum states, wherein local operations and classical communication (LOCC) yield suboptimal discrimination probabilities compared to global strategies, is one of the striking nonclassical features of quantum theory. A variant of this phenomenon arises in conclusive local state discrimination, where the correct state must be identified with zero error, albeit allowing for inconclusive outcomes. More recently, the notion of local state marking has been introduced, with the focus shifted to correctly identifying the permutation of a subset of states randomly chosen from a given set of multipartite states under LOCC. In this work, we unify these two approaches by introducing the task of conclusive local state marking, which reveals a finer hierarchy of nonlocality in multipartite quantum state ensembles. Notably, we demonstrate that certain ensembles of product states can exhibit a stronger form of nonlocality than entangled ensembles traditionally considered highly nonlocal.

Conclusive Local State Marking: More Nonlocality With No Entanglement

Abstract

Nonlocality exhibited by ensembles of composite quantum states, wherein local operations and classical communication (LOCC) yield suboptimal discrimination probabilities compared to global strategies, is one of the striking nonclassical features of quantum theory. A variant of this phenomenon arises in conclusive local state discrimination, where the correct state must be identified with zero error, albeit allowing for inconclusive outcomes. More recently, the notion of local state marking has been introduced, with the focus shifted to correctly identifying the permutation of a subset of states randomly chosen from a given set of multipartite states under LOCC. In this work, we unify these two approaches by introducing the task of conclusive local state marking, which reveals a finer hierarchy of nonlocality in multipartite quantum state ensembles. Notably, we demonstrate that certain ensembles of product states can exhibit a stronger form of nonlocality than entangled ensembles traditionally considered highly nonlocal.

Paper Structure

This paper contains 6 sections, 18 theorems, 51 equations, 2 figures.

Table of Contents

  1. Proof of Proposition \ref{['propNOTPOSSIBLE']}
  2. Proof of Proposition \ref{['propLI']}
  3. Proof of Theorem \ref{['theo1']}
  4. Proof of Theorem \ref{['theo2']}
  5. The ensemble of states $\mathcal{S}^{D}$ does not allow $2$-CLSM.
  6. Converse of theorem \ref{['theo1']}

Key Result

Lemma 1

[Chefles CheflesGlobal] A set of $N$ quantum states can be conclusively distinguished if and only if the states are linearly independent.

Figures (2)

  • Figure 1: (Color online) In the bipartite scenario, the task of $2$-CLSM is demonstrated for a set containing three bipartite states: $Z := \{\ket{\Psi_1}, \ket{\Psi_2}, \ket{\Psi_3}\}$. Two states, chosen randomly from the set, are distributed between spatially separated Alice and Bob without revealing their identities. The possible input pairs are: $\{\ket{\Psi_1\Psi_2}, \ket{\Psi_1\Psi_3}, \ket{\Psi_2\Psi_1}, \ket{\Psi_2\Psi_3}, \ket{\Psi_3\Psi_1}, \ket{\Psi_3\Psi_2}\}$. They must conclusively identify the indices of the two states (No. 1 and No. 2) using LOCC.
  • Figure 2: (Color online) A strict hierarchy between the tasks of CLSD and $2$-CLSM is shown. Theorem \ref{['theo4']} ( also Proposition \ref{['propositionUPB']}) is represented by the symbol $\textcolor{green!50!black}{\bigstar}$, which denotes a binary ensemble of states whose CLSD is not possible but for which $2$-CLSM, i.e. CLSM is possible. In contrast, Proposition \ref{['propositionBell']} corresponds to the symbol $\textcolor{red}{$\blacktriangle$}$, representing a set of states that is not locally conclusively distinguishable, and for which even $2$-CLSM is not possible.

Theorems & Definitions (32)

  • Lemma 1
  • Definition 1
  • Lemma 2
  • Definition 2
  • Lemma 3
  • Proposition 1
  • proof
  • Definition 3
  • Definition 4
  • Proposition 2
  • ...and 22 more