Conditional entropies and their relation to entanglement criteria

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

doi:10.1063/1.1498490

Cited by 58 publications

(93 citation statements)

References 25 publications

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“…To get the final expression for the negativity in Eq (13), we substitute the expression for the integrals from Eq (27). The expressions are not very illuminating, and for the lack of an asymptotic expression, we present the numerical values in Table (I), and plot them in Fig.…”

Section: Evaluating the Integralsmentioning

confidence: 99%

“…In fact, it can be shown that there exists no closed form solution for the sum in Eq. (27). The arguments leading to this 'tragic' conclusion are presented next.…”

Section: Appendix A: a Mathematical Digressionmentioning

confidence: 99%

“…This retards the evaluation of the quantities in Table (I) drastically, unless a closed form is found for quantity in Eq. (27). Consequently, it would not only be interesting, but indeed essential to have a closed form of the above expression.…”

Section: Appendix A: a Mathematical Digressionmentioning

confidence: 99%

“…It is the latter that happens in our case, thereby proving that the series in Eq. (27) is not summable in closed form.…”

Section: Appendix A: a Mathematical Digressionmentioning

confidence: 99%

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Negativity of random pure states

Datta

2010

Phys. Rev. A

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This paper deals with the entanglement, as quantified by the negativity, of pure quantum states chosen at random from the invariant Haar measure. We show that it is a constant (0.72037) multiple of the maximum possible entanglement. In line with the results based on the concentration of measure, we find evidence that the convergence to the final value is exponentially fast. We compare the analytically calculated mean and standard deviation with those calculated numerically for pure states generated via pseudorandom unitary matrices proposed by Emerson et. al. [Science, 302, 3098, (2003)]. Finally, we draw some novel conclusions about the geometry of quantum states based on our result.

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Section: Evaluating the Integralsmentioning

confidence: 99%

“…In fact, it can be shown that there exists no closed form solution for the sum in Eq. (27). The arguments leading to this 'tragic' conclusion are presented next.…”

Section: Appendix A: a Mathematical Digressionmentioning

confidence: 99%

Section: Appendix A: a Mathematical Digressionmentioning

confidence: 99%

“…It is the latter that happens in our case, thereby proving that the series in Eq. (27) is not summable in closed form.…”

Section: Appendix A: a Mathematical Digressionmentioning

confidence: 99%

See 2 more Smart Citations

Negativity of random pure states

Datta

2010

Phys. Rev. A

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“…A quite remarkable criterion of this type is the reduction criterion, which is equivalent to the PPT criterion for 2× 2 and 2 × 3 cases and weaker for higher dimensions [8]. There are also other criteria that turned out to be directly related to the PPT criterion: The majorization criterion [9] and also entropic criteria [10] have been shown to be weaker than the PPT criterion [11,12]. Moreover, one can extend the PPT condition to a test based on a complete hierarchy of symmetric extensions, where each step constitutes a semidefinite program [13] (another complete family of semidefinite tests has been described in [14]).…”

Section: Introductionmentioning

confidence: 99%

Unifying several separability conditions using the covariance matrix criterion

et al. 2008

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We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria which have been proposed to complement the positive partial transpose criterion -the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others -are in fact a corollary of the covariance matrix criterion.

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Direct Measurement of Nonlinear Properties of Bipartite Quantum States

Bovino

Castagnoli

Ekert

et al. 2006

Open Syst. Inf. Dyn.

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Non-linear properties of quantum states, such as entropy or entanglement, quantify important physical resources and are frequently used in quantum information science. They are usually calculated from a full description of a quantum state, even though they depend only on a small number parameters that specify the state. Here we extract a non-local and a non-linear quantity, namely the Renyi entropy, from local measurements on two pairs of polarization entangled photons. We also introduce a "phase marking" technique which allows to select uncorrupted outcomes even with non-deterministic sources of entangled photons. We use our experimental data to demonstrate the violation of entropic inequalities. They are examples of a non-linear entanglement witnesses and their power exceeds all linear tests for quantum entanglement based on all possible Bell-CHSH inequalities. PACS numbers: Many interesting properties of composite quantum systems , such as entanglement or entropy, are not measured directly but are inferred, usually from a full specification of a quantum state represented by a density operator. However, it is interesting to note that some of these properties can be measured in the same way we measure and estimate average values of observables. Here we illustrate this by extracting a non-local quantity, the Renyi entropy of the composite system, from local measurements on two pairs of polarization entangled photons. This quantity is a non-linear function of the density operator. We then use our experimental data to demonstrate the violation of entropic inequalities, which can be also interpreted as the experimental demonstration of a non-linear entangle-ment witness. Consider a source which generates pairs of photons. The photons in each pair fly apart from each other to two distant locations A and B. Let us assume that the polarization of each pair is described by some density operator ̺, which is unknown to us. Following Schrödinger's remarks on relations between the information content of the total system and its subsystems [1], it has been proven that for separable states global von Neumann en-tropy is always not less then local ones [2]. Subsequently a number of entropic inequalities have been derived, satisfied by all separable states [3, 4, 5, 6]. The simplest one is based on the Renyi entropy, or the purity measure , Tr (̺ 2) and can be rewritten as Tr (̺ 2 A) ≥ Tr (̺ 2), Tr (̺ 2 B) ≥ Tr (̺ 2), (1) where ̺ A and ̺ B are the reduced density operators pertaining to individual photons. The inequalities (1) involve non-linear functions of density operators and are known to be stronger than all Bell-CHSH inequalities [3, 7]. There are entangled states which are not S 2 S 1 A B 1 2 3 4 FIG. 1: An outline of our experimental setup. Sources S1 and S2 emit pairs of polarization-entangled photons. The entangled pairs are emitted into spatial modes 1 and 3, and 2 and 4. One photon from each pair is directed into location A and the other into location B. At the two locations photons impinge on beam-splitters and ar...

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Conditional entropies and their relation to entanglement criteria

Cited by 58 publications

References 25 publications

Negativity of random pure states

Negativity of random pure states

Unifying several separability conditions using the covariance matrix criterion

Direct Measurement of Nonlinear Properties of Bipartite Quantum States

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