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arXiv:2110.09388 · Physical Review A

Symmetric inseparability and number entanglement in charge-conserving mixed states

Zhanyu Ma, Cheolhee Han, Yigal Meir, Eran Sela

Published Apr 11, 2022• 43 citations
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Abstract

We explore sufficient conditions for inseparability in mixed states with a globally conserved charge, such as a particle number. We argue that even separable states may contain entanglement in fixed charge sectors, as long as the state cannot be separated into charge-conserving components. As a witness of symmetric inseparability we study the number entanglement (NE), $\mathrm{\ensuremath{\Delta}}{S}_{m}$, defined as the entropy change due to a subsystem's charge measurement. Whenever $\mathrm{\ensuremath{\Delta}}{S}_{m}>0$, there exist inseparable charge sectors, having finite (logarithmic) negativity, even when the full state either is separable or has vanishing negativity. We demonstrate that the NE is not only a witness of symmetric inseparability, but also an entanglement monotone. Finally, we study the scaling of $\mathrm{\ensuremath{\Delta}}{S}_{m}$ in thermal one-dimensional systems combining high-temperature expansion and conformal field theory.

DOI
10.1103/physreva.105.042416

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We explore sufficient conditions for inseparability in mixed states with a globally conserved charge, such as a particle number. We argue that even separable states may contain entanglement in fixed charge sectors, as long as the state cannot be separated into charge-conserving components. As a witness of symmetric inseparability we study the number entanglement (NE), \ensuremathΔSm\mathrm{\ensuremath{\Delta}}{S}_{m}\ensuremathΔSm​, defined as the entropy change due to a subsystem's charge measurement. Whenever \ensuremathΔSm>0\mathrm{\ensuremath{\Delta}}{S}_{m}>0\ensuremathΔSm​>0, there exist inseparable charge sectors, having finite (logarithmic) negativity, even when the full state either is separable or has vanishing negativity. We demonstrate that the NE is not only a witness of symmetric inseparability, but also an entanglement monotone. Finally, we study the scaling of \ensuremathΔSm\mathrm{\ensuremath{\Delta}}{S}_{m}\ensuremathΔSm​ in thermal one-dimensional systems combining high-temperature expansion and conformal field theory.
In the presence of a globally conserved charge NNN, a natural question is whether a given separable state can be separated into charge-conserving components. We dub this problem the Symmetric Separability Problem (SSP). On random states, the SSP is answered negatively with probability one for almost all NNN. Using a witness to the failure of symmetric separability, namely the number entanglement (NE) introduced in arXiv:2110.09388, we show that most symmetric and separable states are actually far from being symmetrically separable, with the NE featuring Gaussian concentration around a strictly positive mean value. We discuss some consequences of our results for quantum tasks in the presence of a superselection rule or in the absence of a common reference frame. Progress is made on the question of the size of the separable space constrained by NNN. We also touch upon the question of the complexity of SSP, and multiparty entanglement.
Sufficient conditions for inseparability of mixed states include (logarithmic) negativity. Here we explore systems with a globally conserved charge, such as a particle number. We argue that separable states contain entanglement in fixed charge sectors as long as the state can not be separated into symmetric classical components. As a witness of symmetric inseparability we study the entropy change due to a subsystem's charge measurement, ΔSm\Delta S_mΔSm​. Whenever ΔSm>0\Delta S_m>0ΔSm​>0, there exist inseparable charge sectors, having finite negativity, even when the full state is either separable or has vanishing negativity. We study the scaling of ΔSm\Delta S_mΔSm​ in thermal 1D systems combining high temperature expansion and conformal field theory. We discuss routes towards experimental observation.
We study quantum information aspects of the fermionic entanglement negativity recently introduced [H. Shapourian et al., Phys. Rev. B 95, 165101 (2017)] based on the fermionic partial transpose. In particular, we show that it is an entanglement monotone under the action of local quantum operations and classical communications, which preserves the local fermion-number parity, and satisfies other common properties expected for an entanglement measure of mixed states. We present fermionic analogs of tripartite entangled states such as WWW and Greenberger-Horne-Zeilinger states and compare the results of bosonic and fermionic partial transpose in various fermionic states, where we explain why the bosonic partial transpose fails in distinguishing separable states of fermions. Finally, we explore a set of entanglement quantities which distinguish different classes of entangled states of a system with two and three fermionic modes. In doing so, we prove that vanishing entanglement negativity is a necessary and sufficient condition for separability of $N\ensuremath{\ge}2$ fermionic modes with respect to the bipartition into one mode and the rest. We further conjecture that the entanglement negativity of inseparable states which mix local fermion-number parity is always nonvanishing.
In the presence of symmetry, entanglement measures of quantum many-body states can be decomposed into contributions from distinct symmetry sectors. Here we investigate the decomposability of negativity, a measure of entanglement between two parts of a generally open system in a mixed state. While the entanglement entropy of a subsystem within a closed system can be resolved according to its total preserved charge, we find that negativity of two subsystems may be decomposed into contributions associated with their charge imbalance. We show that this charge-imbalance decomposition of the negativity may be measured by employing existing techniques based on creation and manipulation of many-body twin or triple states in cold atomic setups. Next, using a geometrical construction in terms of an Aharonov-Bohm-like flux inserted in a Riemann geometry, we compute this decomposed negativity in critical one-dimensional systems described by conformal field theory. We show that it shares the same distribution as the charge-imbalance between the two subsystems. We numerically confirm our field theory results via exact calculations for noninteracting particles based on a double-Gaussian representation of the partially transposed density matrix.
We study the charge resolved structure of the pairwise mode entanglement produced by the Schwinger effect. Using the U(1)U(1)U(1) electric charge of quantum electrodynamics, we decompose the entanglement of the reduced density matrices of momentum modes into individual charge sectors, applying the symmetry resolved entanglement entropy to the pure bipartition reductions and the charge imbalance resolved negativity to the mixed particle-antiparticle pair. For a constant electric field we obtain closed form results for every mode pairing, with all field dependence controlled by the single pair production probability ∣β∣2=e−πμ|β|^2=e^{-πμ}∣β∣2=e−πμ. For the particle, particle and antiparticle antiparticle pairs the entanglement entropy within each charge sector is field independent, so that the field acts only by redistributing statistical weight among the sectors; for the mixed particle antiparticle pair the populated sector additionally carries a field dependent entropy. The vacuum created pair is the richest case: its resolved negativity is distributed over three imbalance sectors and is transported from one sector to the next as the field increases, the sectors contributing equally only at the critical field. The entanglement across a fixed momentum bipartition, by contrast, is frozen at its initial value because pair creation conserves the sector charge. Symmetry resolution thus refines the total entanglement measures into a sector resolved picture, exposing where in charge space the entanglement of the Schwinger mechanism resides and how the electric field moves it, information beyond that contained in the total measures or in particle number distributions.
We study the application of our recent holographic entanglement negativity conjecture for mixed states of adjacent subsystems in conformal field theories with a conserved charge. In this context we obtain the holographic entanglement negativity for zero and finite temperature mixed state configurations in d-dimensional conformal field theories dual to bulk extremal and non extremal charged AdSd+1AdS_{d+1}AdSd+1​ black holes. Our results conform to quantum information theory expectations and constitute significant consistency checks for our conjecture.