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arXiv:0811.2907 · Physical Review A

Quantitative complementarity between local and nonlocal character of quantum states in a three-qubit system

Xinhua Peng, Jingfu Zhang, Jiangfeng Du, Dieter Suter

Published May 8, 2008• 12 citations
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Abstract

Local or nonlocal character of quantum states can be quantified and is subject to various bounds that can be formulated as complementarity relations. Here, we investigate the local vs nonlocal character of pure three-qubit states by a four-way interferometer. The complete entanglement in the system can be measured as the entanglement of a specific qubit with the subsystem consisting of the other two qubits. The quantitative complementarity relations are verified experimentally in an NMR quantum information processor.

DOI
10.1103/physreva.77.052107

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Local or nonlocal character of quantum states can be quantified and is subject to various bounds that can be formulated as complementarity relations. Here, we investigate the local vs nonlocal character of pure three-qubit states by a four-way interferometer. The complete entanglement in the system can be measured as the entanglement of a specific qubit with the subsystem consisting of the other two qubits. The quantitative complementarity relations are verified experimentally in an NMR quantum information processor.
Complementarity was originally introduced as a qualitative concept for the discussion of properties of quantum mechanical objects that are classically incompatible. More recently, complementarity has become a quantitative relation between classically incompatible properties, such as the visibility of interference fringes and ``which-way'' information, but also between purely quantum mechanical properties, such as measures of entanglement. We discuss different complementarity relations for systems of two-, three-, or nnn qubits. Using nuclear magnetic resonance techniques, we have experimentally verified some of these complementarity relations in a two-qubit system.
Based on the quantitative complementarity relations, we analyze thoroughly the properties of multipartite quantum correlation and entanglement in four-qubit pure states. It is found that, unlike the three-qubit case, the single residual correlation and the genuine correlations of three and four qubits are unable to quantify entanglement appropriately. More interestingly, from our qualitative and numerical analysis, it is conjectured that the sum of all residual correlations is a good quantity for characterizing the multipartite entanglement in the system.
We derive two complementarity relations that constrain the individual and bipartite properties that may simultaneously exist in a multi-qubit system. The first expression, valid for an arbitrary pure state of n qubits, demonstrates that the degree to which single particle properties are possessed by an individual member of the system is limited by the bipartite entanglement that exists between that qubit and the remainder of the system. This result implies that the phenomenon of entanglement sharing is one specific consequence of complementarity. The second expression, which holds for an arbitrary state of two qubits, pure or mixed, quantifies a tradeoff between the amounts of entanglement, separable uncertainty, and single particle properties that are encoded in the quantum state. The separable uncertainty is a natural measure of our ignorance about the properties possessed by individual subsystems, and may be used to completely characterize the relationship between entanglement and mixedness in two-qubit systems. The two-qubit complementarity relation yields a useful geometric picture in which the root mean square values of local subsystem properties act like coordinates in the space of density matrices, and suggests possible insights into the problem of interpreting quantum mechanics.
We undertake experimental detection of the entanglement present in arbitrary three-qubit pure quantum states on an NMR quantum information processor. Measurements of only four observables suffice to experimentally differentiate between the six classes of states which are inequivalent under stochastic local operation and classical communication. The experimental realization is achieved by mapping the desired observables onto Pauli zzz operators of a single qubit, which is directly amenable to measurement. The detection scheme is applied to known entangled states as well as to states randomly generated using a generic scheme that can construct all possible three-qubit states. The results are substantiated via direct full quantum state tomography as well as via negativity calculations and the comparison suggests that the protocol is indeed successful in detecting tripartite entanglement without requiring any a priori information about the states.
We point out that for an arbitrary number of identical particles, each defined on a Hilbert space of arbitrary dimension, there exists a whole ladder of relations of complementarity between certain local and nonlocal measurements corresponding to every conceivable grouping of the particles, e.g., the more accurately we can know (by a measurement) some joint property of three qubits (projecting the state onto a tripartite-entangled state), the less accurate some other property, local to the three qubits, becomes. We investigate the relation between these complementarity relations and a similar relation based on interference visibilities. We also show that the complementarity relations are particularly tight for particles defined on prime dimensional Hilbert spaces.
We propose to quantify three-qubit entanglement using global negativity along with KKK-way negativities, where K=2K=2K=2 and 3. The principle underlying the definition of KKK-way negativity for pure and mixed states of NNN subsystems is a positive partial transpose sufficient condition. However, KKK-way partial transpose with respect to a subsystem is defined so as to shift the focus to KKK-way coherences instead of KKK subsystems of the composite system. A quantum state of a three-qubit system is characterized by the coherences measured by global, two-way, and three-way negativities. For a canonical state of three-qubit system, entanglement measures for genuine tripartite entanglement, W-like entanglement, and bipartite entanglement can be related to two-way and three-way negativities.