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Discrete Wigner distribution for two qubits: a characterization of entanglement properties

Riccardo Franco, Vittorio Penna

Published Jan 1, 2005• 13 citations
OpenAlex

Abstract

Abstract. We study the properties of the discrete Wigner distribution for two qubits introduced by Wotters [1]. In particular, we analyze the entanglement properties within the Wigner distribution picture. We formulate the separability criteria coming from the Partial Transposition (PT) criterion, the Local Uncertainty Relations (LUR), and the Generalized Uncertainty Principle (GUP) in terms of discrete Wigner functions. Based on PT criterion, we establish the relation between non-negativity of Wigner function and the separability of the corresponding state. A mathematical object of great importance is the covariance matrix relevant to the axis operators. We show that for a separable state the correlations encoded in this object have an upper limit. This allows us to derive a separability criterion –stronger than the one based on LUR– in terms of the covariance matrix. PACS numbers: 03.67.Mn, 03.65.Wj, 42.50.Dv § To whom correspondence should be addressed riccardo.franco@polito.itDiscrete Wigner distribution for two qubits: a characterization... 2

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Discrete Wigner distribution for two qubits: a characterization of entanglement properties

Riccardo Franco, Vittorio Penna

We study the properties of the discrete Wigner distribution for two qubits introduced by Wotters. In particular, we analyse the entanglement properties within the Wigner distribution picture by considering the negativity of the Wigner function (WF) and the correlations of the marginal distribution. We show that a state is entangled if at least one among the values assumed by the corresponding discrete WF is smaller than a certain critical (negative) value. Then, based on the partial transposition criterion, we establish a relation between the separability of a density matrix and the non-negativity of the WFs relevant both to such a density matrix and to the partially transposed thereof. Finally, we derive a simple inequality—involving the covariance-matrix elements of a given WF—which appears to provide a separability criterion stronger than the one based on the local uncertainty relations.

arXiv:quant-ph/0509050•Journal of Physics A Mathematical and General•May 3, 2006•16 citations

Bipartite Entanglement, Partial Transposition and the Uncertainty Principle for Finite-Dimensional Hilbert Spaces

Y. B. Band, Pier A. Mello

We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter $r$ of a Bell state $|Φ^{+} \rangle$ and the completely incoherent state, $\hatρ_r = r |Φ^{+} \rangle \langle Φ^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}$: we find that for $r_0 < r < 1$, where $r_0$ is obtained as a function of the dimensionality $N$, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.

arXiv:1601.04481•Jan 18, 2016•0 citations

Investigating quantum coherence from discrete Wigner function

Lin Yin, Ming‐Da Huang, Yu Ya-Fei +1 more

Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.

Acta Physica Sinica•Jan 1, 2017•2 citations

Bound Entangled Gaussian States

Reinhard F. Werner, Michael M. Wolf

We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. For the separability, and the positivity of the partial transpose, we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state. It is shown that, for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose implies separability. However, this implication fails with two oscillators on each side, as we show by constructing a five parameter family of bound entangled Gaussian states.

arXiv:quant-ph/0009118•Physical Review Letters•Apr 16, 2001•459 citations

Wigner Functions and Separability for Finite Systems

Pittenger, A O, Rubin, M H

A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.

•Jan 19, 2005•44 citations

Separability Criteria from Uncertainty Relations

Otfried Gühne, Maciej Lewenstein

Abstract. We explain several separability criteria which rely on uncertainty relations. For the derivation of these criteria uncertainty relations in terms of variances or entropies can be used. We investigate the strength of the separability conditions for the case of two qubits and show how they can improve entanglement witnesses.

•Jan 1, 2004•10 citations
Abstract. We study the properties of the discrete Wigner distribution for two qubits introduced by Wotters [1]. In particular, we analyze the entanglement properties within the Wigner distribution picture. We formulate the separability criteria coming from the Partial Transposition (PT) criterion, the Local Uncertainty Relations (LUR), and the Generalized Uncertainty Principle (GUP) in terms of discrete Wigner functions. Based on PT criterion, we establish the relation between non-negativity of Wigner function and the separability of the corresponding state. A mathematical object of great importance is the covariance matrix relevant to the axis operators. We show that for a separable state the correlations encoded in this object have an upper limit. This allows us to derive a separability criterion –stronger than the one based on LUR– in terms of the covariance matrix. PACS numbers: 03.67.Mn, 03.65.Wj, 42.50.Dv § To whom correspondence should be addressed riccardo.franco@polito.itDiscrete Wigner distribution for two qubits: a characterization... 2
We study the properties of the discrete Wigner distribution for two qubits introduced by Wotters. In particular, we analyse the entanglement properties within the Wigner distribution picture by considering the negativity of the Wigner function (WF) and the correlations of the marginal distribution. We show that a state is entangled if at least one among the values assumed by the corresponding discrete WF is smaller than a certain critical (negative) value. Then, based on the partial transposition criterion, we establish a relation between the separability of a density matrix and the non-negativity of the WFs relevant both to such a density matrix and to the partially transposed thereof. Finally, we derive a simple inequality—involving the covariance-matrix elements of a given WF—which appears to provide a separability criterion stronger than the one based on the local uncertainty relations.
We first show that partial transposition for pure and mixed two-particle states in a discrete NNN-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter rrr of a Bell state ∣Φ+⟩|Φ^{+} \rangle∣Φ+⟩ and the completely incoherent state, ρ^r=r∣Φ+⟩⟨Φ+∣+(1−r)I^N2\hatρ_r = r |Φ^{+} \rangle \langle Φ^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}ρ^​r​=r∣Φ+⟩⟨Φ+∣+(1−r)N2I^​: we find that for r0<r<1r_0 < r < 1r0​<r<1, where r0r_0r0​ is obtained as a function of the dimensionality NNN, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. For the separability, and the positivity of the partial transpose, we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state. It is shown that, for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose implies separability. However, this implication fails with two oscillators on each side, as we show by constructing a five parameter family of bound entangled Gaussian states.
A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.