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Physical Review A

Entanglement versus negative domains of Wigner functions

Joachim Dahl, Holger Mack, A. Wolf, Wolfgang P. Schleich

Published Oct 23, 2006• 58 citations
Publisher

Abstract

We show that $s$ waves, that is wave functions that only depend on a hyperradius, are entangled if and only if the corresponding Wigner functions exhibit negative domains. We illustrate this feature using a special class of $s$ waves which allows us to perform the calculations analytically. This class includes a Gaussian, a maximally entangled as well as a ``shell'' state.

DOI
10.1103/physreva.74.042323

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We show that sss waves, that is wave functions that only depend on a hyperradius, are entangled if and only if the corresponding Wigner functions exhibit negative domains. We illustrate this feature using a special class of sss waves which allows us to perform the calculations analytically. This class includes a Gaussian, a maximally entangled as well as a ``shell'' state.
We derive explicit expressions for the Wigner function of wave functions in DDD dimensions which depend on the hyperradius---that is, of sss waves. They are based either on the position or the momentum representation of the sss wave. The corresponding Wigner function depends on three variables: the absolute value of the DDD-dimensional position and momentum vectors and the angle between them. We illustrate these expressions by calculating and discussing the Wigner functions of an elementary sss wave and the energy eigenfunction of a free particle.
We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth by Van Herstraeten and Cerf [Phys. Rev. A 104, 042211 (2021)] and Hertz et al. [J. Phys. A: Math. Theor. 50, 385301 (2017)]. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to $1+ln\ensuremath{\pi}$. We prove this conjecture for the qubits formed by Fock states $|0\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}$ that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for the conjecture's validity. Lastly, we elaborate on the states which are in accordance with our condition.
Two measures of nonclassicality, the entanglement potential and the negativity of the Wigner distribution function defined by the volume of its negative domains, are compared based on an investigation of the nonclassicality for Fock states and Schrödinger cat states in a decoherence process. Both the entanglement potential and the total negative probability are reduced in the linear loss process and the partial negative distribution of the Wigner function is wiped out for large losses while the entanglement potential is always positive. We give a bound condition and find that, though not yet mathematically proven in general, the upper bound of 50% is the maximum allowed loss for the survival of the negative distribution of the Wigner function.
We investigate the multipartite entanglement in the phase space using the negativity of Wigner function (NWF) and in the Hilbert space using the negativity of partial transpose (NPT). We analyze comparatively these quantities and the different types of entanglements that are present in two major classes — GHZ and [Formula: see text] — made of coherent states. We show that the negativity of Wigner function can be used as a good measure of genuine entanglement in multipartite systems. However, the negativity of partial transpose is a good quantifier for only the bipartite entanglement in tripartite systems.