[Paper Review] Trade-off between Performance and Reversibility of Entanglement Concentration for Pure Entangled State

Wataru Kumagai, Masahito Hayashi|arXiv (Cornell University)|May 27, 2013

Quantum Information and Cryptography4 citations

TL;DR

This paper establishes that entanglement concentration for bipartite pure states is fundamentally irreversible in the asymptotic limit, demonstrating a trade-off between concentration performance and reversibility. It proves that the minimal concentration-recovery error (MCRE) converges to 1 for any non-maximally entangled state, implying that perfect recovery after compression is impossible, even with large numbers of copies.

ABSTRACT

In quantum information theory, it is widely believed that entanglement concentration for bipartite pure states is asymptotically reversible. In order to examine this, we give a precise formulation of the problem, and show a trade-off relation between performance and reversibility, which implies the irreversibility of entanglement concentration. Then, we regard entanglement concentration as entangled state compression in an entanglement storage with lower dimension. Because of the irreversibility of entanglement concentration, an initial state can not be completely recovered after the compression process and a loss inevitably arises in the process. We numerically calculate this loss and also derive for it a highly accurate analytical approximation.

Motivation & Objective

• To rigorously examine whether entanglement concentration is asymptotically reversible, challenging the widely held belief in reversibility.
• To quantify the inherent trade-off between the performance of entanglement concentration and the feasibility of recovering the original state.
• To define and analyze the minimal concentration-recovery error (MCRE) as a measure of irreversibility in LOCC-based entanglement compression.
• To derive analytical approximations and numerical evidence for the irreversibility of entanglement concentration in non-asymptotic and asymptotic regimes.
• To establish that perfect entanglement compression and recovery via LOCC is fundamentally impossible for non-maximally entangled states.

Proposed method

• Formalizes entanglement concentration and recovery as LOCC operations, defining two error metrics: concentration error and minimum recovery error.
• Introduces the minimal concentration-recovery error (MCRE) as the sum of both errors, minimized jointly over the concentration operation and target EPR state copies.
• Uses the fidelity-based error metric $ e^{ ext{C}}_n(m,C| ho) = 1 - F^2(C( ho^{igotimes n}), ilde{ ho}^{igotimes m}) $ and $ e^{ ext{R}}_n(C| ho) = ext{min}_{D:LOCC} ig[1 - F^2( ho^{igotimes n}, D igcirc C( ho^{igotimes n}))ig] $.
• Applies the central limit theorem to the von Neumann entropy and variance of the state to derive asymptotic approximations of the MCRE using the Gaussian cumulative distribution function $ G(x) $.
• Proves that $ ilde{K}(b,b'| ho) = Gig( rac{b - S_ ho b'}{ u_ ho} ig) $, enabling asymptotic analysis of error rates under finite-size corrections.
• Derives the key result: $ ext{MCRE} o 1 $ as $ n o ty $ for non-maximally entangled states, with a refined asymptotic form depending on the entropy and variance of the initial state.

Experimental results

Research questions

• RQ1 Is entanglement concentration for bipartite pure states asymptotically reversible under LOCC operations?
• RQ2 What is the fundamental trade-off between the performance of entanglement concentration and the success of state recovery?
• RQ3 Can the minimal concentration-recovery error (MCRE) be made arbitrarily small in the asymptotic limit?
• RQ4 How does the irreversibility of entanglement concentration depend on the initial state’s entropy and variance?
• RQ5 What is the optimal asymptotic rate of entanglement concentration and dilution when irreversibility is taken into account?

Key findings

• The minimal concentration-recovery error (MCRE) converges to 1 in the asymptotic limit for any non-maximally entangled bipartite pure state, proving irreversibility.
• For non-maximally entangled states, the concentration error can be made arbitrarily small, but this forces the recovery error to approach 1, making perfect recovery impossible.
• The MCRE is bounded below by $ 2Gig( rac{S_ ho b'}{2 u_ ho} ig) $ when the number of copies is $ n + b' u_ hoig/ ext{const} + o( u_ ho) $, with $ b' < 0 $, showing a non-trivial trade-off.
• When the number of copies is $ n + b' u_ ho + o( u_ ho) $ with $ b' o 0 $, the MCRE converges to 1, confirming that even small deviations from the optimal rate lead to irreversibility.
• The asymptotic error rate is analytically approximated using the Gaussian cumulative distribution function $ G(x) $, with the error depending on the initial state’s entropy $ S_ ho $ and variance $ V_ ho $.
• The paper derives a precise asymptotic expansion for the minimal number of EPR pairs $ N_n( ho| ho) $ needed to achieve a given error $ ho $, showing $ N_n( ho| ho) = n + b_{ ho, ho} u_ ho + o( u_ ho) $ with $ b_{ ho, ho} = -2 u_ ho S_ ho^{-1} G^{-1}(1 - ho/2) $.