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Physical Review A

Nonseparability tests by noncommutativity of excitations

Tamás Vértesi, R. Englman

Published Feb 16, 2007• 3 citations
Publisher

Abstract

Elementary quantum mechanical principles are applied to bipartite or multipartite systems to test the nonseparability of subsystems (or nonadditivity in the system Hamiltonian) through interchanging the ordering of subsystem (local) operators that cause transitions between energy eigenstates, whether applied to pure or mixed states. A tentative measure for quantifying nonseparability is proposed, based on complete sets of subsystem transition operators. This serves as an upper bound for experimental determinations of the intersubsystem coupling. Two- and three-qubit systems are numerically tested to establish criteria for the noninvariance of transition probabilities in them with respect to the sequence of excitations, and it is conjectured that sequence noninvariance can be used as a probe for subsystems' nonseparability.

DOI
10.1103/physreva.75.022315

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Elementary quantum mechanical principles are applied to bipartite or multipartite systems to test the nonseparability of subsystems (or nonadditivity in the system Hamiltonian) through interchanging the ordering of subsystem (local) operators that cause transitions between energy eigenstates, whether applied to pure or mixed states. A tentative measure for quantifying nonseparability is proposed, based on complete sets of subsystem transition operators. This serves as an upper bound for experimental determinations of the intersubsystem coupling. Two- and three-qubit systems are numerically tested to establish criteria for the noninvariance of transition probabilities in them with respect to the sequence of excitations, and it is conjectured that sequence noninvariance can be used as a probe for subsystems' nonseparability.
The description of the complex separability structure of quantum states in terms of partially ordered sets has been recently put forward. In this work, we address the question of how to efficiently determine these structures for unknown states. We propose an experimentally accessible and scalable iterative methodology that identifies, on solid statistical grounds, sufficient conditions for nonseparability with respect to certain partitions. In addition, we propose an algorithm to determine the minimal partitions (those that do not admit further splitting) consistent with the experimental observations. We test our methodology experimentally on a 20-qubit IBM quantum computer by inferring the structure of the 4-qubit Smolin and an 8-qubit W state. In the first case, our results reveal that, while the fidelity of the state is low, it nevertheless exhibits the partitioning structure expected from the theory. In the case of the W state, we obtain very disparate results in different runs on the device, which range from nonseparable states to very fragmented minimal partitions with little entanglement in the system. Furthermore, our work demonstrates the applicability of informationally complete positive operator-valued measurements for practical purposes on current noisy intermediate-scale quantum devices.
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We propose a measure to quantify correlations in a bipartite quantum system of two quibits by assessing the minimum difference between outcome states of a subsystem by performing a local measurement on the other subsystem. This maximum similarity measure is a monotone function of the concurrence for pure states of two qubits; for mixed states it accounts for entanglement, dissonance, and classical correlations. Besides, we found a closed formula for evaluating the similarity degree of an arbitrary mix state of two two-dimensional systems.
We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence.Our measure allows one to quantify coherence without having to perform tomography of the quantum state and can be directly calculated from measurement expectation values.This definition of coherence allows the decomposition into contributions corresponding to the nonclassical correlations between the subsystems and localized on each subsystem.The method can also be applied to cases where the full set of measurement operators is unavailable.An estimator using the truncated measurement operators can be used to obtain lower bound to the genuine value of coherence.We illustrate the method for several bipartite systems and show the singular behavior of the coherence measure in a spin-1 chain, characteristic of a quantum phase transition.
We consider the concept of "the permutationally invariant (PI) part of a density matrix," which has proven very useful for both efficient quantum state estimation and entanglement characterization of N-qubit systems. We show here that the concept is, in fact, basis dependent but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part ρ(PI) of a general (mixed) N-qubit state ρ, we obtain (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multipartite separability criteria (one of which entails a sufficient criterion for genuine N-partite entanglement) that can be experimentally determined by just 2N+1 measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of N qubits, and (iv) an explicit example that shows there are, for increasing N, genuinely N-partite entangled states lying closer and closer to the maximally mixed state. Moreover, we show that if the PI part of a state is k nonseparable, then so is the actual state. We further argue to add as requirement on any multipartite entanglement measure E that it satisfy E(ρ)≥E(ρ(PI)), even though the operation that maps ρ→ρ(PI) is not local.