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arXiv:quant-ph/0107014

Can spectral and local information decide separability?

K. G. H. Vollbrecht, Michael M. Wolf

Published Jul 3, 2001• 0 citations
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Abstract

We discuss the discriminating power of separability criteria, which are based on the spectrum of a quantum state and its reductions. Common examples are entropic inequalities utilizing conditional Tsallis or Renyi entropies. We prove that these inequalities are implied by the reduction criterion for any positive value of the entropic parameters. We show however, that arbitrary sets of criteria based on spectral and local information can never be sufficient by establishing a separable, isospectral and locally undistinguishable counterpart for any Werner state in odd dimensions. For the case of two qubit systems we show that a simple controlled phase gate operation can produce an isospectral, entangled state out of a separable one, which has the same reductions.

DOI
10.48550/arxiv.quant-ph/0107014

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We discuss the discriminating power of separability criteria, which are based on the spectrum of a quantum state and its reductions. Common examples are entropic inequalities utilizing conditional Tsallis or Renyi entropies. We prove that these inequalities are implied by the reduction criterion for any positive value of the entropic parameters. We show however, that arbitrary sets of criteria based on spectral and local information can never be sufficient by establishing a separable, isospectral and locally undistinguishable counterpart for any Werner state in odd dimensions. For the case of two qubit systems we show that a simple controlled phase gate operation can produce an isospectral, entangled state out of a separable one, which has the same reductions.
We discuss conditional Rényi and Tsallis entropies for bipartite quantum systems of finite dimension. We investigate the relation between the positivity of conditional entropies and entanglement properties. It is in particular shown that any state having a negative conditional entropy with respect to any value of the entropic parameter is distillable since it violates the reduction criterion. Moreover, we show that the entanglement of Werner states in odd dimensions can neither be detected by entropic criteria nor by any other spectral criterion.
We discuss the entropic criterion for separability of compound quantum systems for general nonadditive entropic forms based on arbitrary concave functions f. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of f, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all q, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.
The quantum relative R\'enyi entropy of two noncommuting density matrices was recently defined from which its conditional entropy is deduced. This framework is here extended to the corresponding Tsallis relative entropy and to its conditional form. This expression of Tsallis conditional entropy is shown to witness entanglement beyond the method based on global and local spectra of composite quantum states. When the reduced density matrix happens to be a maximally mixed state, this conditional entropy coincides with the expression in terms of Tsallis entropies derived earlier by Abe and Rajagopal [Physica A 289, 157 (2001)]. The separability range in a noisy one-parameter family of WWW and Greenberger-Horne-Zeilinger states with three and four qubits is explored here and it is shown that the results inferred from negative Tsallis conditional entropy matches that obtained through Peres's partial transpose criterion for one-parameter family of WWW states, in one of its partitions. The criterion is shown to be nonspectral through its usefulness in identifying entanglement in isospectral density matrices.
Recently it was shown that if a given state fulfils the reduction criterion, it must also satisfy the known entropic inequalities. The natural question arises as to whether it is possible to derive some scalar inequalities stronger than the entropic ones, assuming that stronger criteria based on positive but not completely positive maps are satisfied. In the present paper we show that if certain conditions hold, the extended reduction criterion [H.-P. Breuer, Phys. Rev. Lett 97, 080501 (2006); W. Hall, J. Phys. A 40, 6183 (2007)] leads to some entropiclike inequalities, much stronger than their entropic counterparts. The comparison of the derived inequalities with other separability criteria shows that such an approach might lead to strong scalar criteria detecting both distillable and bound entanglement. In particular, in the case of SO(3)-invariant states it is shown that the present inequalities detect entanglement in regions, in which linear entanglement witnesses based on the extended reduction map fail. It should also be emphasized that in the case of $2\ensuremath{\bigotimes}d$ states the derived inequalities detect entanglement efficiently, while the extended reduction maps are useless, when acting on the qubit subsystem. Moreover, there is a natural way to construct a many-copy entanglement witnesses based on the derived inequalities so, in principle, there is a possibility of experimental realization. Some open problems and possibilities for further research are outlined.
The reduction criterion is a well-known necessary condition for separable states, and states violating this condition are entangled and also 1-distillable. In this paper we introduce a set of necessary conditions for separability of multipartite states, obtained from a set of positive but not completely positive maps. These conditions can be thought of as generalizations of the reduction criterion to multipartite systems. We use tripartite Werner states as an example to investigate the entanglement detecting powers of some of these conditions, and we also look at what these conditions mean in terms of distillation. Finally, we show that these maps can be used to give a partial solution to the subsystem problem, as described by Butterley et al.
Bipartite quantum states are classified into three categories: separable states, bound entangled states, and free entangled states. It is of great importance to characterize these families of states for the development of quantum information science. In this Letter, I show that the separable states and the bound entangled states have a common spectral property. More precisely, I prove that for undis-tillable-separable and bound entangled-states, the eigenvalue vector of the global system is majorized by that of the local system. This result constitutes a new sufficient condition for distillability of bipartite quantum states. This is achieved by proving that if a bipartite quantum state satisfies the reduction criterion for distillability, then it satisfies the majorization criterion for separability.