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arXiv:0801.1545 · Physical Review A

Characteristics and benchmarks of entanglement of mixed states: The two-qubit case

Shanthanu Bhardwaj, V. Ravishankar

Published Feb 25, 2008• 7 citations
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Abstract

We propose that the entanglement of mixed states is characterized properly in terms of a probability density function ${\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})$. There is a need for such a measure since the prevalent measures (such as concurrence and negativity) for two-qubit systems are rough benchmarks and not monotones of each other. Focusing on the two-qubit states, we provide an explicit construction of ${\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})$ and show that it is characterized by a set of parameters of which concurrence is but one particular combination. ${\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})$ is manifestly invariant under $\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)$ transformations. It can, in fact, reconstruct the state up to local operations---with the specification of at most four additional parameters. Finally, the measure resolves the controversy regarding the role of entanglement in quantum computation in NMR systems.

DOI
10.1103/physreva.77.022322

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We propose that the entanglement of mixed states is characterized properly in terms of a probability density function P\ensuremathρ(E){\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})P\ensuremathρ​(E). There is a need for such a measure since the prevalent measures (such as concurrence and negativity) for two-qubit systems are rough benchmarks and not monotones of each other. Focusing on the two-qubit states, we provide an explicit construction of P\ensuremathρ(E){\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})P\ensuremathρ​(E) and show that it is characterized by a set of parameters of which concurrence is but one particular combination. P\ensuremathρ(E){\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})P\ensuremathρ​(E) is manifestly invariant under SU(2)\ifmmode×\else\texttimes\fiSU(2)\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)SU(2)\ifmmode×\else\texttimes\fiSU(2) transformations. It can, in fact, reconstruct the state up to local operations---with the specification of at most four additional parameters. Finally, the measure resolves the controversy regarding the role of entanglement in quantum computation in NMR systems.
We propose that the entanglement of mixed states is characterised properly in terms of a probability density function P(E)\mathcal{P}(\mathcal{E})P(E). There is a need for such a measure since the prevalent measures (such as \textit{concurrence} and \textit{negativity}) are rough benchmarks, and not monotones of each other. Considering the specific case of two qubit mixed states, we provide an explicit construction of P(E)\mathcal{P}(\mathcal{E})P(E) and show that it is characterised by a set of parameters, of which concurrence is but one particular combination. P(E)\mathcal{P}(\mathcal{E})P(E) is manifestly invariant under SU(2)×SU(2)SU(2) \times SU(2)SU(2)×SU(2) transformations. It can, in fact, reconstruct the state up to local operations - with the specification of at most four additional parameters. Finally the new measure resolves the controversy regarding the role of entanglement in quantum computation in NMR systems.
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.
The paper is an elaborate comparative assessment of the most relevant entropy-based quantities to be applied in quantifying quantum entanglement in pure, mixed, bipartite, and multipartite quantum systems. The study fills the existing gap in the literature in which most of the previous literature examines individual measures, but a lot of knowledge is lacking on what these measures are regarding their strengths, weaknesses, and the operational usefulness against each other. The study can prove the behaviour of each measure under noise, decoherence, dimensional scaling, and variation of state-type by analysing von Neumann entropy, the Tsallis entropy, entropy of entanglement of formation, concurrence and logarithmic negativity. Numerical studies with pure two-qubit states, Werner states, GHZ states, W states and mixed random states demonstrate specific sensitivity properties and computational limitations which ensure that no single measure can be a universal quantifier. The paper goes on to say that entropy-only measures do not work in mixed regimes because of a dominance in classical correlations. The results have several important implications to quantum communication, quantum simulation, and quantum computing in the NISQ-era, where the choice of measurement has a direct impact on benchmarking, error-correction scheme, and protocol performance based on entanglement. The paper ends with a recommendation of contextual, purpose-based scheme of selecting entanglement measures and suggests a further study on scalable benchmarking, hybrid quantifiers, and operational performance-based entanglement measurements. These observations reinforce the theoretical premise of entanglement studies and enable the advancement of more effective quantum technologies in the field of practice.
Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n) × U(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n = 2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU(2) × SU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.