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arXiv:0704.2973 · Physical Review A

Entanglement fidelity and measurement of entanglement preservation in quantum processes

Yang Xiang, Shi‐Jie Xiong

Published Jul 2, 2007• 11 citations
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Abstract

The entanglement fidelity provides a measure of how well the entanglement between two subsystems is preserved in a quantum process. By using a simple model, we show that in some cases this quantity in its original definition fails in the measurement of entanglement preservation. On the contrary, the modified entanglement fidelity, obtained by using a proper local unitary transformation on a subsystem, is shown to exhibit behavior similar to that of the concurrence in quantum evolution.

DOI
10.1103/physreva.76.014301

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The entanglement fidelity provides a measure of how well the entanglement between two subsystems is preserved in a quantum process. By using a simple model, we show that in some cases this quantity in its original definition fails in the measurement of entanglement preservation. On the contrary, the modified entanglement fidelity, obtained by using a proper local unitary transformation on a subsystem, is shown to exhibit behavior similar to that of the concurrence in quantum evolution.
We define the concurrence hierarchy as d − 1 independent invariants under local unitary transformations in d -level quantum system. The first one is the original concurrence defined by Wootters (1998 Phys. Rev. Lett. 80 2245) and Hill and Wootters (1997 Phys. Rev. Lett. 78 5022) in a two-level quantum system and generalized to the d -level pure quantum state case. We propose to use this concurrence hierarchy as a measurement of entanglement. This measurement does not increase under local quantum operations and classical communication.
Theory and experiment both demonstrate that an entangled quantum state of two subsystems is neither a superposition of states of its subsystems nor a superposition of composite states but rather a coherent superposition of nonlocal correlations between incoherently mixed local states of the two subsystems. Thus, even if one subsystem happens to be macroscopic as in the entangled "Schrodinger's cat" state resulting from an ideal measurement, this state is not the paradoxical macroscopic superposition it is generally presumed to be. It is, instead, a "macroscopic correlation," a coherent quantum correlation in which one of the two correlated sub-systems happens to be macroscopic. This clarifies the physical meaning of entanglement: When a superposed quantum system A is unitarily entangled with a second quantum system B, the coherence of the original superposition of different states of A is transferred to different correlations between states of A and B, so the entangled state becomes a superposition of correlations rather than a superposition of states. This transfer preserves unitary evolution while permitting B to be macroscopic without entailing a macroscopic superposition. This resolves the "problem of outcomes" but is not a complete resolution of the measurement problem because the entangled state is still reversible.
Preserving entanglement is a crucial dynamical process for entanglement-based quantum computation and quantum-information processes, such as one-way quantum computing and quantum key distribution. However, the problem of quantifying the ability of an experimental process to preserve two-qubit entanglement in experimentally feasible ways is not well understood. Accordingly, herein, we consider the use of two measures, namely composition and robustness, for quantitatively characterizing the ability of a process to preserve entanglement, referred to henceforth as entanglement preservability. A fidelity benchmark is additionally derived to identify the ability of a process to preserve entanglement. We show that the measures and introduced benchmark are experimentally feasible and require only local measurements on single qubits and preparations of separable states. Moreover, they are applicable to all physical processes that can be described using the general theory of quantum operations, e.g., qubit dynamics in photonic and superconducting systems. Our method extends the existing tools for analyzing channels, e.g., channel resource theory, to quantify entanglement preservability for non-trace-preserving quantum processes. The results are of significant interest for applications in quantum-information processing in which entanglement preservation is required.
Simple models for measuring the characteristics of a system of two atoms that do not interact with each other are used to show that the nonselectivity of measurement, related to the indistinguishability of the quantum subsystems with respect to some general parameter, leads to entanglement of the atomic states of the subsystem selected in such a way. This entanglement is in no way related to some collective process and is only due to the effective measurement in an entangled basis. An analogy between the preparation of entangled states of a system of subsystems that do not interact with each other by a quantum reduction (a measurement) and a quantum evolution is found.