Physical Review A

Negativity and momentum entanglement in Bardeen-Cooper-Schrieffer states at finite temperature

Chun Kit Chung, C. K. Law

Published Sep 12, 2008• 7 citations

Abstract

We analyze the momentum entanglement of homogeneous Bardeen--Cooper--Schrieffer (BCS) states between spin-up and spin-down particles at finite temperature. This is achieved by studying the structure of the partial transposition of the two-particle density matrix (denoted by ${\ensuremath{\rho}}_{2}^{{T}_{A}}$) in momentum space, as well as the corresponding negativity. We show that ${\ensuremath{\rho}}_{2}^{{T}_{A}}$ consists of infinitely many decoupled $2\ifmmode\times\else\texttimes\fi{}2$ submatrices, and momentum entanglement is a necessary and sufficient condition for a nonzero order parameter $\ensuremath{\Delta}$. From the distribution of eigenvalues of ${\ensuremath{\rho}}_{2}^{{T}_{A}}$, we discover that momentum entanglement is most significantly contributed by pairs with momenta slightly above a surface related to the Fermi energy. In addition, we propose an entanglement witness operator for detecting momentum entanglement in BCS states.

DOI: 10.1103/physreva.78.034302

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Physical Review A

Negativity and momentum entanglement in Bardeen-Cooper-Schrieffer states at finite temperature

Chun Kit Chung, C. K. Law

Published Sep 12, 2008• 7 citations

Publisher

Abstract

{\ensuremath{\rho}}_{2}^{{T}_{A}}

) in momentum space, as well as the corresponding negativity. We show that

{\ensuremath{\rho}}_{2}^{{T}_{A}}

consists of infinitely many decoupled

2\ifmmode\times\else\texttimes\fi{}2

submatrices, and momentum entanglement is a necessary and sufficient condition for a nonzero order parameter

\ensuremath{\Delta}

. From the distribution of eigenvalues of

{\ensuremath{\rho}}_{2}^{{T}_{A}}

, we discover that momentum entanglement is most significantly contributed by pairs with momenta slightly above a surface related to the Fermi energy. In addition, we propose an entanglement witness operator for detecting momentum entanglement in BCS states.

DOI: 10.1103/physreva.78.034302

Related papers

A study of momentum entanglement and negativity in Bardeen-Cooper-Schrieffer states at finite temperature

Chun Kit Chung, Chi Kwong Law

Bulletin of the American Physical SocietyMar 16, 20090 citations

Entanglement in SU(2)-invariant quantum systems: The positive partial transpose criterion and others

John Schliemann

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 $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ 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 $\ensuremath{\rho}$ 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.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 $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ 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 $\ensuremath{\rho}$ 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.

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hep-thCross-listedarXiv:2512.10091Dec 12, 2025

{\ensuremath{\rho}}_{2}^{{T}_{A}}

) in momentum space, as well as the corresponding negativity. We show that

{\ensuremath{\rho}}_{2}^{{T}_{A}}

consists of infinitely many decoupled

2\ifmmode\times\else\texttimes\fi{}2

submatrices, and momentum entanglement is a necessary and sufficient condition for a nonzero order parameter

\ensuremath{\Delta}

. From the distribution of eigenvalues of

{\ensuremath{\rho}}_{2}^{{T}_{A}}

\\rho

is entirely determined by the diagonal elements of

\\rho

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

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.

(T=0)

superfluid properties including the order parameter, chemical potential, quasiparticle excitation spectrum, momentum distribution, atomic compressibility, ground-state energy, and low-energy collective excitations. We show that a quantum phase transition occurs for nonzero angular momentum pairing, unlike the

s

-wave case where the BCS to BEC evolution is just a crossover. Third, we present a Gaussian fluctuation theory near the critical temperature

(T={T}_{\mathrm{c}})

, and we analyze the number of bound, scattering, and unbound fermions as well as the chemical potential. Finally, we derive the time-dependent Ginzburg-Landau functional near

{T}_{\mathrm{c}}

, and compare the Ginzburg-Landau coherence length with the zero-temperature average Cooper pair size.

T_c

. At fixed

T

, entanglement appears only above a critical field

E_{\text{entang}}

. For fermions, we observe a qualitatively different pattern: the fermionic logarithmic negativity is non-vanishing at finite temperature, and is monotonically suppressed by thermal noise. As a function of the electric field, it is non-monotonic, featuring a temperature-independent optimal field strength

E_*

and decreasing on both sides of the maximum. We give quantitative estimates for analog experiments, where our entanglement criteria convert directly into concrete temperature and electric field constraints. These findings identify realistic regimes where the quantum character of Schwinger physics may be tested in the laboratory.