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Symmetric separability criterion for number conserving mixed states

Zhanyu Ma, Eran Sela

Published Oct 18, 2021• 2 citations
OpenAlex

Abstract

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, $\Delta S_m$. Whenever $\Delta S_m>0$, 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 $\Delta S_m$ in thermal 1D systems combining high temperature expansion and conformal field theory. We discuss routes towards experimental observation.

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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), $\mathrm{\ensuremath{\Delta}}{S}_{m}$, 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.

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Aspects of Entanglement Negativity in One Dimensional Critical Systems

Andrea Coser

In this thesis we study entanglement in one-dimensional critical quantum many-body systems and in particular we will focus on disconnected regions. Given a system in a pure state, to quantify the amount of entanglement between a multicomponent subsystem and the rest of the full system we can use as an entanglement measure the renowned entanglement entropy. However, if we are interested in the entanglement shared among the disconnected regions, the entanglement entropy fails to be a good quantifier. The reason is that the state of the subsystem is in general mixed once the rest is traced out, and the entanglement entropy is a good measure of entanglement only for pure states. A good measure of entanglement in mixed states is the logarithmic negativity, which is the quantitative version of Peres' criterion of separability. The main advantage of the negativity with respect to other entanglement measures is the simplicity of its definition in terms of the density matrix describing the quantum state. Since the definition does not require any variational calculus, it is much more easily computable than any other entanglement measure, and therefore we can obtain some results also in complicated settings such as many-body systems. We will be mostly interested in the configuration where the full system is divided into two parts, A and B. If we do not have access to the degrees of freedom in B, we can describe subsystem A through its reduced density matrix, where the degrees of freedom in B have been traced out. The residual subsystem in A will in general be left in a mixed state. Subsystem A is then divided again into two parts, A_1 and A_2, and we will be interested in the entanglement shared by these two components. Knowing the full density matrix, the logarithmic negativity can be easily computed from the eigenvalues of its partial transpose with respect to the degrees of freedom living in one of the two subsystems A_1 or A_2. However, the full density matrix of a many-body state is in general unaccessible, even numerically, since the size of the matrix grows exponentially with the size of the system. If we concentrate on critical systems whose low-energy physics can be described by a quantum field theory, we can resort to its powerful tools to compute the entanglement properties. In particular the main tool is the replica trick. The entanglement entropy is obtained from the moments of the reduced density matrix, while the negativity can be computed from the moments of the partial transpose. However, even the computation of the integer-order moments is not an easy task, and exact analytical results are known only for the simplest quantum field theories. Hence, throughout the thesis we will usually consider conformal field theories, which have an enhanced set of symmetries and therefore allow for some exact computations. In the Introduction, we will try to frame this work by briefly reviewing some of the main topics in the literature of quantum many-body systems and quantum field theories where entanglement plays a crucial role. We will also describe what are the main features that are requested to a `good' measure of entanglement, and we will review some of the main measures that have been considered so far in the context of quantum information. We will define the entanglement negativity and stress its main advantages, as well as its drawbacks. After reviewing some basic facts of conformal field theories, we will give some technical details on the computation of entanglement entropy and logarithmic negativity in general quantum field theories, which will be needed in the rest of the thesis. In Chap. 2 we will start the study of the entanglement of several disjoint disconnected regions by considering the entanglement that the union of these regions share with the rest of the system. We will compute the integer-order Rényi entropies of the subsystem from which the entanglement entropy can be obtained through the replica limit. Unfortunately, the exact analytic continuation to real order is still out of reach and therefore the entanglement entropy cannot be computed exactly. However numerical extrapolations allow to obtain some accurate estimates. In the remaining chapters we will focus on the entanglement shared between two non complementary disjoint regions, through the computations of the entanglement negativity in different settings. In Chap. 3, we will consider a global quantum quench starting from a conformal boundary state and evolving through a conformal evolution. A general formula for the mutual information and the logarithmic negativity for two adjacent and disjoint intervals is given in the spacetime limit. In Chap. 4 we focus on the XY chain and we recover a formula for the partial transpose as a sum of four fermionic auxiliary Gaussian density matrices, by generalizing some previous results for the reduced density matrix of spin systems and for the partial transpose of pure fermionc Gaussian states. Even if the computation of the negativity is still out of reach, we can obtain formulas for the integer-order moments of the partial transpose in terms of the correlation matrices relative to the two components and to the region in between them. In Chap. 5 we will study the entanglement negativity for a free Dirac fermion field. Starting from the lattice results, we will obtain a path integral representation of the partial transpose which can be easily generalized to the continuum. With this representation of the partial transpose, we can construct all its integer-order moments and compute them for the simple conformal field theory of the free fermion. Again, the computation of the negativity could not be accomplished due to technical difficulties in the analytic continuation to real order. The same computation has been extended also to the modular invariant Dirac fermion and the Ising model, and the obtained formulas coincide with the ones already present in the literature. This analysis draws an interesting connection between some terms appearing in the formulas for the moments of the partial transpose (as well as of the R\'enyi entropies) found for the lattice models and the ones found for the corresponding theories describing their scaling limit. Whenever possible, all the analytical results will be checked against numerical calculations performed on simple free chain models. In Chaps. 2 and 3 we will consider bosonic Gaussian states, specifically the harmonic chain in its ground state and out of equilibrium, while in Chaps. 4 and 5 we will consider fermionic systems, specifically the XX and Ising spin chains, and the tight-binding model. Finally, we will draw some conclusions and discuss some open problems.

•Oct 12, 2015•0 citations
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.
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.
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.
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 analyze the entanglement properties for the not-completely-permutation-symmetric states of quantum systems composed of two subsystems with an equal but arbitrary finite local Hilbert space dimension. We investigate both pure and mixed states with such a symmetry obtained by relaxing the symmetry requirement of the axisymmetric states. For such states we discuss the entanglement classification with respect to stochastic local operations and classical communication and establish the entanglement quantitatively by means of concurrence and negativity. In particular, we determine the separability criterion in the frame of various methods, including the kkk-positive map witness, optimal Schmidt-number witnesses, and entanglement measures.
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 this thesis we study entanglement in one-dimensional critical quantum many-body systems and in particular we will focus on disconnected regions. Given a system in a pure state, to quantify the amount of entanglement between a multicomponent subsystem and the rest of the full system we can use as an entanglement measure the renowned entanglement entropy. However, if we are interested in the entanglement shared among the disconnected regions, the entanglement entropy fails to be a good quantifier. The reason is that the state of the subsystem is in general mixed once the rest is traced out, and the entanglement entropy is a good measure of entanglement only for pure states. A good measure of entanglement in mixed states is the logarithmic negativity, which is the quantitative version of Peres' criterion of separability. The main advantage of the negativity with respect to other entanglement measures is the simplicity of its definition in terms of the density matrix describing the quantum state. Since the definition does not require any variational calculus, it is much more easily computable than any other entanglement measure, and therefore we can obtain some results also in complicated settings such as many-body systems. We will be mostly interested in the configuration where the full system is divided into two parts, A and B. If we do not have access to the degrees of freedom in B, we can describe subsystem A through its reduced density matrix, where the degrees of freedom in B have been traced out. The residual subsystem in A will in general be left in a mixed state. Subsystem A is then divided again into two parts, A_1 and A_2, and we will be interested in the entanglement shared by these two components. Knowing the full density matrix, the logarithmic negativity can be easily computed from the eigenvalues of its partial transpose with respect to the degrees of freedom living in one of the two subsystems A_1 or A_2. However, the full density matrix of a many-body state is in general unaccessible, even numerically, since the size of the matrix grows exponentially with the size of the system. If we concentrate on critical systems whose low-energy physics can be described by a quantum field theory, we can resort to its powerful tools to compute the entanglement properties. In particular the main tool is the replica trick. The entanglement entropy is obtained from the moments of the reduced density matrix, while the negativity can be computed from the moments of the partial transpose. However, even the computation of the integer-order moments is not an easy task, and exact analytical results are known only for the simplest quantum field theories. Hence, throughout the thesis we will usually consider conformal field theories, which have an enhanced set of symmetries and therefore allow for some exact computations. In the Introduction, we will try to frame this work by briefly reviewing some of the main topics in the literature of quantum many-body systems and quantum field theories where entanglement plays a crucial role. We will also describe what are the main features that are requested to a `good' measure of entanglement, and we will review some of the main measures that have been considered so far in the context of quantum information. We will define the entanglement negativity and stress its main advantages, as well as its drawbacks. After reviewing some basic facts of conformal field theories, we will give some technical details on the computation of entanglement entropy and logarithmic negativity in general quantum field theories, which will be needed in the rest of the thesis. In Chap. 2 we will start the study of the entanglement of several disjoint disconnected regions by considering the entanglement that the union of these regions share with the rest of the system. We will compute the integer-order Rényi entropies of the subsystem from which the entanglement entropy can be obtained through the replica limit. Unfortunately, the exact analytic continuation to real order is still out of reach and therefore the entanglement entropy cannot be computed exactly. However numerical extrapolations allow to obtain some accurate estimates. In the remaining chapters we will focus on the entanglement shared between two non complementary disjoint regions, through the computations of the entanglement negativity in different settings. In Chap. 3, we will consider a global quantum quench starting from a conformal boundary state and evolving through a conformal evolution. A general formula for the mutual information and the logarithmic negativity for two adjacent and disjoint intervals is given in the spacetime limit. In Chap. 4 we focus on the XY chain and we recover a formula for the partial transpose as a sum of four fermionic auxiliary Gaussian density matrices, by generalizing some previous results for the reduced density matrix of spin systems and for the partial transpose of pure fermionc Gaussian states. Even if the computation of the negativity is still out of reach, we can obtain formulas for the integer-order moments of the partial transpose in terms of the correlation matrices relative to the two components and to the region in between them. In Chap. 5 we will study the entanglement negativity for a free Dirac fermion field. Starting from the lattice results, we will obtain a path integral representation of the partial transpose which can be easily generalized to the continuum. With this representation of the partial transpose, we can construct all its integer-order moments and compute them for the simple conformal field theory of the free fermion. Again, the computation of the negativity could not be accomplished due to technical difficulties in the analytic continuation to real order. The same computation has been extended also to the modular invariant Dirac fermion and the Ising model, and the obtained formulas coincide with the ones already present in the literature. This analysis draws an interesting connection between some terms appearing in the formulas for the moments of the partial transpose (as well as of the R\'enyi entropies) found for the lattice models and the ones found for the corresponding theories describing their scaling limit. Whenever possible, all the analytical results will be checked against numerical calculations performed on simple free chain models. In Chaps. 2 and 3 we will consider bosonic Gaussian states, specifically the harmonic chain in its ground state and out of equilibrium, while in Chaps. 4 and 5 we will consider fermionic systems, specifically the XX and Ising spin chains, and the tight-binding model. Finally, we will draw some conclusions and discuss some open problems.