Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.

Dec 3, 20200 citations

A variational quantum algorithm for approximating convex roofs

George Androulakis, Ryan McGaha

Many entanglement measures are first defined for pure states of a bipartite Hilbert space, and then extended to mixed states via the convex roof extension. In this article we alter the convex roof extension of an entanglement measure, to produce a sequence of extensions that we call $f$ - $d$ extensions, for $d \in \mathbb{N}$ , where $f:[0,1]\to [0, \infty)$ is a fixed continuous function which vanishes only at zero. We prove that for any such function $f$ , and any continuous, faithful, non-negative function, (such as an entanglement measure), $\mu$ on the set of pure states of a finite dimensional bipartite Hilbert space, the collection of $f$ - $d$ extensions of $\mu$ detects entanglement, i.e. a mixed state $\rho$ on a finite dimensional bipartite Hilbert space is separable, if and only if there exists $d \in \mathbb{N}$ such that the $f$ - $d$ extension of $\mu$ applied to $\rho$ is equal to zero. We introduce a quantum variational algorithm which aims to approximate the $f$ - $d$ extensions of entanglement measures defined on pure states. However, the algorithm does have its drawbacks. We show that this algorithm exhibits barren plateaus when used to approximate the family of $f$ - $d$ extensions of the Tsallis entanglement entropy for a certain function $f$ and unitary ansatz $U(\theta)$ of sufficient depth. In practice, if additional information about the state is known, then one needs to avoid using the suggested ansatz for long depth of circuits.Many entanglement measures are first defined for pure states of a bipartite Hilbert space, and then extended to mixed states via the convex roof extension. In this article we alter the convex roof extension of an entanglement measure, to produce a sequence of extensions that we call $f$-$d$ extensions, for $d \in \mathbb{N}$, where $f:[0,1]\to [0, \infty)$ is a fixed continuous function which vanishes only at zero. We prove that for any such function $f$, and any continuous, faithful, non-negative function, (such as an entanglement measure), $\mu$ on the set of pure states of a finite dimensional bipartite Hilbert space, the collection of $f$-$d$ extensions of $\mu$ detects entanglement, i.e. a mixed state $\rho$ on a finite dimensional bipartite Hilbert space is separable, if and only if there exists $d \in \mathbb{N}$ such that the $f$-$d$ extension of $\mu$ applied to $\rho$ is equal to zero. We introduce a quantum variational algorithm which aims to approximate the $f$-$d$ extensions of entanglement measures defined on pure states. However, the algorithm does have its drawbacks. We show that this algorithm exhibits barren plateaus when used to approximate the family of $f$-$d$ extensions of the Tsallis entanglement entropy for a certain function $f$ and unitary ansatz $U(\theta)$ of sufficient depth. In practice, if additional information about the state is known, then one needs to avoid using the suggested ansatz for long depth of circuits.

arXiv:2203.02099Quantum Information and ComputationSep 1, 20222 citations

Insufficiency of Pure-State Ensembles in Characterizing Transformations of Entangled States under LOCC

C. L. Liu, Baoqing Sun, D. L. Zhou

The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions $E_f^{cr}(ρ) \geq E_f^{cr}(σ)$ for all convex roof entanglement measures $E_f^{cr}$ is insufficient to guarantee the existence of an LOCC transformation $Λ^L(\cdot)$ from $ρ$ to $σ$ ; and (ii) The inequalities $\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)$ for all entanglement monotones $E$ are not sufficient to ensure the existence of an LOCC transformation from $\{p_j, \ket{\varphi_j}\}$ to $\{q_l, \ket{ψ_l}\}$ .The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions $ E_f^{cr}(ρ) \geq E_f^{cr}(σ) $ for all convex roof entanglement measures $E_f^{cr}$ is insufficient to guarantee the existence of an LOCC transformation $Λ^L(\cdot)$ from $ρ$ to $σ$; and (ii) The inequalities $\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)$ for all entanglement monotones $E$ are not sufficient to ensure the existence of an LOCC transformation from $\{p_j, \ket{\varphi_j}\}$ to $\{q_l, \ket{ψ_l}\}$.

quant-pharXiv:2512.21665Physical Review ADec 26, 20250 citations

An Extension of Entanglement Measures for Pure States

Xian Shi, Lin Chen

Abstract To quantify the entanglement is one of the most important topics in quantum entanglement theory. An entanglement measure is built from measures for pure states. Conditions when the entanglement measure is entanglement monotone and convex are presented, as well as the interpretation of smoothed one‐shot entanglement cost. Next, a difference between the measure under the local operation and classical communication and the separability‐preserving operations is presented. Then, the relation between the convex roof extended method and the way here for the entanglement measures built from the geometric entanglement measure for pure states, as well as the concurrence for pure states in two‐qubit systems are considered. It is also shown that the measure is monogamous for system.Abstract To quantify the entanglement is one of the most important topics in quantum entanglement theory. An entanglement measure is built from measures for pure states. Conditions when the entanglement measure is entanglement monotone and convex are presented, as well as the interpretation of smoothed one‐shot entanglement cost. Next, a difference between the measure under the local operation and classical communication and the separability‐preserving operations is presented. Then, the relation between the convex roof extended method and the way here for the entanglement measures built from the geometric entanglement measure for pure states, as well as the concurrence for pure states in two‐qubit systems are considered. It is also shown that the measure is monogamous for system.

arXiv:2007.13164Annalen der PhysikMar 17, 20217 citations

Strict entanglement monotonicity under local operations and classical communication

Yu Jing Guo

Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.

arXiv:1904.01183Physical Review AFeb 28, 201911 citations

Total correlations as multi-additive entanglement monotones

Gerardo A. Paz-Silva, John Henry Reina

We propose general conditions for a measure of total correlations as an entanglement monotone using its pure and mixed convex-roof extension. In doing so, we derive crucial theorems and propose a concrete candidate for a total correlation measure which is a multi-additive multipartite entanglement monotone. We perform numerical simulations that show, comparatively, the dependence of the proposed coherent total correlation measure on the size of the register for several different quantum states and for global and pairwise correlation measures. We also show how to simplify some of the conditions required for monotonicity, in particular that a function must be non-increasing under local operations and classical communications.We propose general conditions for a measure of total correlations as an entanglement monotone using its pure and mixed convex-roof extension. In doing so, we derive crucial theorems and propose a concrete candidate for a total correlation measure which is a multi-additive multipartite entanglement monotone. We perform numerical simulations that show, comparatively, the dependence of the proposed coherent total correlation measure on the size of the register for several different quantum states and for global and pairwise correlation measures. We also show how to simplify some of the conditions required for monotonicity, in particular that a function must be non-increasing under local operations and classical communications.

arXiv:quant-ph/0702177Journal of Physics A Mathematical and TheoreticalJan 6, 20095 citations

f

d

extensions, for

d \in \mathbb{N}

, where

f:[0,1]\to [0, \infty)

is a fixed continuous function which vanishes only at zero. We prove that for any such function

f

, and any continuous, faithful, non-negative function, (such as an entanglement measure),

\mu

on the set of pure states of a finite dimensional bipartite Hilbert space, the collection of

f

d

extensions of

\mu

detects entanglement, i.e. a mixed state

\rho

on a finite dimensional bipartite Hilbert space is separable, if and only if there exists

d \in \mathbb{N}

such that the

f

d

extension of

\mu

applied to

\rho

is equal to zero. We introduce a quantum variational algorithm which aims to approximate the

f

d

extensions of entanglement measures defined on pure states. However, the algorithm does have its drawbacks. We show that this algorithm exhibits barren plateaus when used to approximate the family of

f

d

extensions of the Tsallis entanglement entropy for a certain function

f

and unitary ansatz

U(\theta)

of sufficient depth. In practice, if additional information about the state is known, then one needs to avoid using the suggested ansatz for long depth of circuits.

E_f^{cr}(ρ) \geq E_f^{cr}(σ)

for all convex roof entanglement measures

E_f^{cr}

is insufficient to guarantee the existence of an LOCC transformation

Λ^L(\cdot)

from

ρ

σ

; and (ii) The inequalities

\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)

for all entanglement monotones

E

are not sufficient to ensure the existence of an LOCC transformation from

\{p_j, \ket{\varphi_j}\}

\{q_l, \ket{ψ_l}\}