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Journal of Physics A Mathematical and Theoretical

Quantum circuit implementation of the optimal information–disturbance tradeoff of maximally entangled states

ShengLi Zhang, XuBo Zou, Ke Li, ChenHui Jin, Guang‐Can Guo

Published Jan 4, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/41/3/035309

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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.
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.
Information-theoretic definitions for the noise associated with a quantum measurement and the corresponding disturbance to the state of the system have recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401 (2014)]. These definitions are invariant under relabeling of measurement outcomes, and lend themselves readily to the formulation of state-independent uncertainty relations both for the joint estimate of observables (noise-noise relations) and the noise-disturbance tradeoff. Here we derive such relations for incompatible qubit observables, which we prove to be tight in the case of joint estimates, and present progress towards fully characterizing the noise-disturbance tradeoff. In doing so, we show that the set of obtainable noise-noise values for such observables is convex, whereas the conjectured form for the set of obtainable noise-disturbance values is not. Furthermore, projective measurements are not optimal with respect to the joint-measurement noise or noise-disturbance tradeoffs. Interestingly, it seems that four-outcome measurements are needed in the former case, whereas three-outcome measurements are optimal in the latter.
We show that the global balance of information dynamics for general quantum measurements given in [F. Buscemi, M. Hayashi, and M. Horodecki, Phys. Rev. Lett. 100, 210504 (2008)] makes it possible to unify various and generally inequivalent approaches adopted in order to derive information-disturbance tradeoffs in quantum theory. We focus in particular on those tradeoffs, constituting the vast majority of the literature on the subject, where disturbance is defined either in terms of average output fidelity or of entanglement fidelity.
We address the class of positive operator-valued measures (POVMs) for qubit systems that are obtained by coupling the signal qubit with a probe qubit and then performing a projective measurement on the sole probe system. These POVMs, which represent the simplest class of qubit POVMs, depends on 3 + 3 + 2 = 8 free parameters describing the initial preparation of the probe qubit, the Cartan representative of the unitary coupling, and the projective measurement at the output, respectively. We analyze in some detail the properties of the POVM matrix elements, and investigate their values for given ranges of the free parameters. We also analyze in detail the tradeoff between information and disturbance for different ranges of the free parameters, showing, among other things, that (i) typical values of the tradeoff are close to optimality and (ii) even using a maximally mixed probe one may achieve optimal tradeoff.
Entanglement concentration can be achieved in a deterministic fashion [Morikoshi, Phys. Rev. Lett. 84, 3189 (2000); Phys. Rev. A 64, 022316 (2001)]. We present a protocol for deterministic extraction of an Einstein-Podolsky-Rosen pair from two partially entangled pairs via local operations and classical communication. This protocol consists of a generalized measurement described by a positive operator-valued measure (POVM), one-way classical communication, and a corresponding unitary operation or a choice between the two pairs. Based on the theory of majorization, we explicitly construct the required POVM, which can be realized by performing a unitary operation in the extended space and a conventional Von Neuman orthogonal measurement. By decomposing the evolution process from the initial state to the final state, we construct the quantum circuits for realizing the unitary operation with quantum Toffoli gates, and thus provide a physical means to realize the deterministic entanglement concentration. Our method for constructing quantum circuits differs from the usual method based on decomposition of unitary matrices, and is convenient for a large class of quantum processes involving generalized measurements.