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arXiv:0809.4469 · Quantum Information and Computation

On global effects caused by locally noneffective unitary operations

Sevag Gharibian, Hermann Kampermann, D. Bruss

Published Nov 1, 2009• 15 citations
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Abstract

Given a bipartite quantum state $\rho$ with subsystems $A$ and $B$ of arbitrary dimensions, we study the entanglement detecting capabilities of locally noneffective, or cyclic, unitary operations [Fu, Europhys. Lett., vol. 75]. Local cyclic unitaries have the special property that they leave their target subsystem invariant. We investigate the distance between $\rho$ and the global state after local application of such unitaries as a possible indicator of entanglement. To this end, we derive and discuss closed formulae for the maximal such distance achievable for three cases of interest: (pseudo)pure quantum states, Werner states, and two-qubit states. What makes this criterion interesting, as we show here, is that it surprisingly displays behavior similar to recent anomalies observed for non-locality measures in higher dimensions, as well as demonstrates an equivalence to the CHSH inequality for certain classes of two-qubit states. Yet, despite these similarities, the criterion is not itself a non-locality measure. We also consider entanglement detection in bound entangled states.

DOI
10.26421/qic9.11-12-6

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Given a bipartite quantum state ρ\rhoρ with subsystems AAA and BBB of arbitrary dimensions, we study the entanglement detecting capabilities of locally noneffective, or cyclic, unitary operations [Fu, Europhys. Lett., vol. 75]. Local cyclic unitaries have the special property that they leave their target subsystem invariant. We investigate the distance between ρ\rhoρ and the global state after local application of such unitaries as a possible indicator of entanglement. To this end, we derive and discuss closed formulae for the maximal such distance achievable for three cases of interest: (pseudo)pure quantum states, Werner states, and two-qubit states. What makes this criterion interesting, as we show here, is that it surprisingly displays behavior similar to recent anomalies observed for non-locality measures in higher dimensions, as well as demonstrates an equivalence to the CHSH inequality for certain classes of two-qubit states. Yet, despite these similarities, the criterion is not itself a non-locality measure. We also consider entanglement detection in bound entangled states.
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The state of a bipartite system may be changed by a cyclic operation applied on one of its subsystems. The change is a nonlocal effect, and can be detected only by measuring the two parts jointly. By employing the Hilbert-Schmidt metric, we can quantify such nonlocal effects via measuring the distance between the initial and final state. We show that this nonlocal property can be manifested not only by entangled states but also by the disentangled states which are classically correlated . Furthermore, we study the effect for the system of two qubits in detail. It is interesting that the nonlocal effect of disentangled states is limited by 1/√2, while the entangled states can exceed this limit and reach 1 for maximally entangled states.
In quantum mechanics, general measurements often cause disturbance which may be exploited to quantify entanglement, nonlocality or quantumness. Imagine a bipartite state ρ ab shared by two parties a and b , and a von Neumann measurement performed locally on party a which does not disturb the local state ρ a ≔tr b ρ ab , but nevertheless may disturb the global state ρ ab . This disturbance is an indication of some kind of correlations or global effect in ρ ab which cannot be accounted for locally. We propose to use the maximum disturbance on ρ ab caused by locally non-disturbing measurements as a figure of merit quantifying the global effect (nonlocality), and investigate its fundamental properties. For general two-qubit states and some higher-dimensional symmetric states, we present analytic formulas for their global effects.
The necessary and sufficient conditions for the equivalence of arbitrary nnn-qubit pure quantum states under local unitary (LU) operations derived in [B. Kraus, Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two nnn-qubit states (with n>2n>2n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide; however, the states are neither LU equivalent nor can be mapped into each other by general local operations and classical communication.
The state of a bipartite system may be changed by a cyclic operation applied on one of its subsystem. The change is a nonlocal effect, and can be detected only by measuring the two parts jointly. By employing the Hilbert-Schmidt metric, we can quantify such nonlocal effects via measuring the distance between initial and final state. By making use of the new nonlocal effect we can measure the entanglement of bipartite system. For qubit pair, we show that this measurement is equivalent to degree of entanglement and consists with the Bell inequality.
The relation between Bell-CHSH violation and factorization of Hilbert space is considered here. That is, a state which is local in the sense of the Bell-CHSH inequality under a certain factorization of the underlying Hilbert space can be Bell-CHSH nonlocal under a different factorization. While this question has been addressed with respect to separability, the relation of the factorization with Bell-CHSH violation has remained hitherto unexplored. We find here that there is a set containing density matrices, which do not exhibit Bell-CHSH violation under any factorization of the Hilbert space brought about by global unitary operations. Using the Cartan decomposition of [Formula: see text], we characterize the set in terms of a necessary and sufficient criterion based on the spectrum of density matrices. Sufficient conditions are obtained to characterize such density matrices based on their bloch representations. For some classes of density matrices, necessary and sufficient conditions are derived in terms of bloch parameters. Furthermore, an estimation of the volume of such density matrices is achieved in terms of purity. The criterion is applied to some well-known class of states in two qubits. Since both local filtering and global unitary operations influence the Bell-CHSH violation of a state, a comparative study is made between the two operations. The inequivalence of the two operations (in terms of increasing Bell-CHSH violation) is exemplified through their action on some classes of states.