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arXiv:quant-ph/0703017

A complete characterization of mixed state entanglement using probability density functions

Shanthanu Bhardwaj, Ravishankar, V.

Published Mar 2, 2007• 0 citations
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Abstract

We propose that the entanglement of mixed states is characterised properly in terms of a probability density function $\mathcal{P}(\mathcal{E})$. 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 $\mathcal{P}(\mathcal{E})$ and show that it is characterised by a set of parameters, of which concurrence is but one particular combination. $\mathcal{P}(\mathcal{E})$ is manifestly invariant under $SU(2) \times SU(2)$ 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.

DOI
10.48550/arxiv.quant-ph/0703017

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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 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.
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.
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.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix rho\\rhorho is entirely determined by the diagonal elements of rho\\rhorho in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for rho\\rhorho having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
There has recently been much discussion regarding entanglement transformations in terms of local filtering operations and whether the optimal entanglement for an arbitrary two-qubit state could be realized. We introduce an experimentally realizable scheme for manipulating the entanglement of an arbitrary state of two polarization-entangled qubits. This scheme is then used to provide some perspective to the mathematical concepts inherent in this field with respect to a laboratory environment. Specifically, we look at how to extract enhanced entanglement from systems with a fixed rank, and, in the case where the rank of the density operator for the state can be reduced, show how the state can be made arbitrarily close to a maximally entangled pure state. In this context we also discuss bounds on entanglement in mixed states.