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Chinese Physics Letters

Fidelity Tradeoff in Estimation of Partial Entanglement State with Local Operation and Classic Communication

ShengLi Zhang, Kun Wang, Jiansheng Guo, Jianhong Shi

Published Jul 1, 2015• 1 citation
Publisher

Abstract

In quantum mechanics, there is no measurement process that could gain some information of an unknown quantum state without causing any disturbance. A tradeoff bound between the amount of information gain and the concomitant disturbance in the measurement process of a bipartite entangled state is actually ingrained. Such a bound is fundamental and closely connected with the entangled degree b. In this work, the bound for estimation of a partial entangled state with a local strategy is investigated. It is shown that, with local operation with classical communication, a monotonic change in the picture will be spotted. This is due to the fact that the partial entanglement gradually becomes two individual qubits and, consequently, the optimal operation boils down to local operations. A quantum circuit which achieves the optimal tradeoff is also obtained.

DOI
10.1088/0256-307x/32/7/070302

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In quantum mechanics, there is no measurement process that could gain some information of an unknown quantum state without causing any disturbance. A tradeoff bound between the amount of information gain and the concomitant disturbance in the measurement process of a bipartite entangled state is actually ingrained. Such a bound is fundamental and closely connected with the entangled degree b. In this work, the bound for estimation of a partial entangled state with a local strategy is investigated. It is shown that, with local operation with classical communication, a monotonic change in the picture will be spotted. This is due to the fact that the partial entanglement gradually becomes two individual qubits and, consequently, the optimal operation boils down to local operations. A quantum circuit which achieves the optimal tradeoff is also obtained.
We consider the optimal measurement procedures for estimating an unknown maximally entangled state with local operations and classical communications (LOCC). By the ``optimal,'' we mean that there is no other LOCC measurements that can provide a higher estimation fidelity. In fact, we first use the positive partial transpose constraint to obtain an upper bound of the fidelity; then, we show there exists a kind of LOCC measurement that can attain such a bound. The estimation fidelity of the optimal LOCC estimation procedures is D+2(D+1)D2\frac{D+2}{(D+1){D}^{2}}(D+1)D2D+2​ with DDD denoting the dimension of the relevant quantum system.
We solve the problem of the optimal cloning of pure entangled two-qubit states with a fixed degree of entanglement using local operations and classical communication. We show that, amazingly, classical communication between the parties can improve the fidelity of local cloning if and only if the initial entanglement is higher than a certain critical value. It is completely useless for weakly entangled states. We also show that bound entangled states with positive partial transpose are not useful as a resource to improve the best local cloning fidelity.
We give a direct derivation for the information–disturbance tradeoff in estimating a maximally entangled state, which was first obtained by Sacchi (2006 Phys. Rev. Lett. 96 220502) in terms of the covariant positive operator valued measurement (POVM) and Jamiołkowski's isomorphism. We find that, the Cauchy–Schwarz inequality, which is one of the most powerful tools in deriving the tradeoff for a single-particle pure state still plays a key role in the case of the maximal entanglement estimation. Our result shows that the inequality becomes equality when the optimal tradeoff is achieved. Moreover, we demonstrate that such a tradeoff is physically achievable with a quantum circuit that only involves single- and two-particle logic gates and single-particle measurements.