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Physical Review A

Optimal estimation of an unknown maximally entangled state with local operations and classical communication

ShengLi Zhang, Xu‐Bo Zou, Ke Li, Chenhui Jin, Guang‐Can Guo

Published Oct 18, 2007• 3 citations
Publisher

Abstract

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 $\frac{D+2}{(D+1){D}^{2}}$ with $D$ denoting the dimension of the relevant quantum system.

DOI
10.1103/physreva.76.042323

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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 study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of k≤dk \leq dk≤d orthogonal maximally entangled states in $\complex^{d}\otimes\complex^{d}$ that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012)] gives an affirmative answer for the case k=dk = dk=d. We give, for the first time, a proof that such sets of states indeed exist even in the case k<dk < dk<d. Our result is constructive and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT). The proof uses the characterization of the PPT-distinguishability problem as a semidefinite program.
The problem of optimally estimating an unknown unitary quantum operation with the aid of entanglement is addressed. The idea is to prepare an entangled pair, apply the unknown unitary to one of the two parts, and then measure the joint output state. This measurement could be an entangled one or it could be separable (e.g., measurements which can be implemented with local operations and classical communication or LOCC). A comparison is made between these possibilities and it is shown that by using nonseparable measurements one can improve the accuracy of the estimation by a factor of 2(d+1)/d2(d+1)/d2(d+1)/d where d is the dimension of the Hilbert space on which U acts.
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 show that there exist sets of three mutually orthogonal ddd-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of ddd. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal ddd-dimensional maximally entangled states can be perfectly distinguished using measurements with a positive partial transpose (PPT) and can be distinguished with one-way LOCC with high probability. These results circle around the question of whether there exist three maximally entangled states which cannot be distinguished using the full power of LOCC; we discuss possible approaches to answer this question.
We consider a single copy of a mixed state of two qubits and prove tight lower and upper bounds for its fidelity in terms of its concurrence or negativity. Next we derive the optimal filtering operations and the optimal trace-preserving local operations assisted by classical communication (LOCC) such as to maximize the fidelity of a given copy of a mixed state of two qubits. The optimal LOCC operations turn out to be very simple and implementable by 1-LOCC protocols, and always yield a fidelity larger then 1/2 if the original state was entangled.