[Paper Review] Discrete realization of group symmetric LOCC-detection of maximally entangled state

Masahito Hayashi|ArXiv.org|Oct 19, 2008

Quantum Information and Cryptography10 references3 citations

TL;DR

This paper proposes discrete, finite-outcome LOCC (one-way local operations and classical communication) measurements for detecting maximally entangled states in bipartite quantum systems, using symmetric informationally complete POVMs (SIC-POVMs) and mutually unbiased bases (MUBs) for the single-qudit case, and Clifford group-based constructions for the composite qudit case. The key contribution is a provably optimal discrete realization of group-symmetric, continuous-outcome tests, achieving exact equivalence to the ideal infinite-outcome measurement in both scenarios.

ABSTRACT

Group symmetric LOCC measurement for detecting maximally entangled state is considered. Usually, this type measurement has continuous-valued outcomes. However, any realizable measurement has finite-valued outcomes. This paper proposes discrete realizations of such a group symmetric LOCC measurement.

Motivation & Objective

• To resolve the gap between theoretically optimal, continuous-outcome group-symmetric LOCC measurements for detecting maximally entangled states and their practical realization with finite outcomes.
• To construct discrete, one-way LOCC measurement schemes that preserve group symmetry and achieve optimal detection fidelity.
• To extend the discrete realization from single-qudit to composite-qudit systems where each subsystem is a tensor product of two d-dimensional systems.
• To prove the optimality of the proposed discrete measurement schemes in terms of minimal number of outcomes and fidelity to the ideal measurement.

Proposed method

• Uses symmetric informationally complete POVMs (SIC-POVMs) as the local measurement on Alice’s system, with each rank-one projector corresponding to a pure state with uniform overlap 1/(d+1) for distinct states.
• Employs a two-step LOCC protocol: Alice performs a SIC-POVM and classically communicates the outcome; Bob performs a two-outcome projective measurement onto the conjugate state of Alice’s outcome.
• For the composite system case, constructs a discrete LOCC measurement using the unitary representation of the Clifford group acting on maximally entangled states, ensuring invariance under local unitary transformations.
• Derives the resulting positive operator-valued measure (POVM) as a sum over tensor products of conjugate state projectors, with uniform weights.
• Proves that the resulting measurement operator exactly matches the ideal group-invariant measurement operator $ T_{ ext{inv}}^{1,A\to B} $ and $ T_{ ext{inv}}^{2,A\to B} $, respectively.
• Establishes optimality by showing that any such discrete realization must have at least $ d^2 $ outcomes, which the proposed schemes achieve.

Experimental results

Research questions

• RQ1 Can the ideal, continuous-outcome group-symmetric LOCC measurement for detecting a maximally entangled state be discretized into a finite-outcome measurement while preserving group symmetry and optimality?
• RQ2 What is the minimal number of outcomes required for a discrete LOCC measurement to exactly realize the ideal group-invariant measurement for a single-qudit system?
• RQ3 How can the group-symmetric LOCC detection of maximally entangled states be generalized to composite systems composed of two-qudit subsystems?
• RQ4 Is there a finite, discrete LOCC measurement scheme based on finite groups (e.g., Clifford group) that achieves the same outcome as the ideal continuous measurement in the composite case?
• RQ5 Can the optimality of such discrete realizations be rigorously proven in terms of both outcome count and fidelity to the ideal measurement?

Key findings

• The SIC-POVM provides a discrete, one-way LOCC realization of the ideal group-symmetric measurement $ T_{ ext{inv}}^{1,A\to B} $, with exactly $ d^2 $ outcomes, matching the theoretical lower bound.
• The proposed SIC-POVM-based scheme achieves exact equality $ T(M_{ ext{sic}}) = T_{ ext{inv}}^{1,A\to B} $, proving its correctness and optimality in outcome count.
• For the composite system case with $ ext{dim}({f H}_{A_1}) = ext{dim}({f H}_{A_2}) = ext{dim}({f H}_{B_1}) = ext{dim}({f H}_{B_2}) = d $, a discrete LOCC scheme based on the Clifford group achieves the ideal measurement $ T_{ ext{inv}}^{2,A\to B} $.
• The Clifford-based scheme is proven to be optimal, as it achieves the minimal number of $ d^2 $ outcomes required for any discrete realization of $ T_{ ext{inv}}^{2,A\to B} $.
• The resulting measurement operators are invariant under local unitary transformations and exactly reproduce the form of the ideal group-invariant operators in both cases.
• The paper establishes that any discrete LOCC measurement achieving $ T(M) = T_{ ext{inv}}^{1,A\to B} $ must have at least $ d^2 $ outcomes, and the SIC-POVM construction meets this bound with equality.