Upper Bound on the region of Separable States near the Maximally Mixed State

A lower bound on the amount of noise that must be added to a GHZ-like entangled state to make it separable (also called the random robustness) is found using the transposition condition. The bound is applicable to arbitrary numbers of subsystems, and dimensions of Hilbert space, and is shown to be exact for qubits. The new bound is compared with previous such bounds on this quantity, and found to be stronger in all cases. It implies that increasing the number of subsystems, rather than increasing their Hilbert space dimension, is a more effective way of increasing entanglement. An explicit decomposition into an ensemble of separable states, when the state is not entangled, is given for the case of qubits.A lower bound on the amount of noise that must be added to a GHZ-like entangled state to make it separable (also called the random robustness) is found using the transposition condition. The bound is applicable to arbitrary numbers of subsystems, and dimensions of Hilbert space, and is shown to be exact for qubits. The new bound is compared with previous such bounds on this quantity, and found to be stronger in all cases. It implies that increasing the number of subsystems, rather than increasing their Hilbert space dimension, is a more effective way of increasing entanglement. An explicit decomposition into an ensemble of separable states, when the state is not entangled, is given for the case of qubits.

arXiv:quant-ph/0002002Journal of Optics B Quantum and Semiclassical OpticsMay 11, 20007 citations

Robustness of entanglement

Guifré Vidal, R. Tarrach

In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system (qubit denotes quantum bit). As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system (qubit denotes quantum bit). As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.

arXiv:quant-ph/9806094Physical Review AJan 1, 1999600 citations

Minimal-error quantum state discrimination versus robustness of entanglement: More indistinguishability with less entanglement

Debarupa Saha, Kornikar Sen, Chirag Srivastava +1 more

We relate the the distinguishability of quantum states with their robustness of the entanglement, where the robustness of any resource quantifies how tolerant it is to noise. In particular, we identify upper and lower bounds on the probability of discriminating the states, appearing in an arbitrary multiparty ensemble, in terms of their robustness of entanglement and the probability of discriminating states of the closest separable ensemble. These bounds hold true, irrespective of the dimension of the constituent systems the number of parties involved, the size of the ensemble, and whether the measurement strategies are local or global. Additional lower bounds on the same quantity is determined by considering two special cases of two-state multiparty ensembles, either having equal entanglement or at least one of them being separable. The case of equal entanglement reveals that it is always easier to discriminate the entangled states than the ones in the corresponding closest separable ensemble, a phenomenon which we refer as "More indistinguishability with less entanglement". Furthermore, we numerically explore how tight the bounds are by examining the global discrimination probability of states selected from Haar-uniformly generated ensembles of two two-qubit states. We find that for two-element ensembles of unequal entanglements, the minimum of the two entanglements must possess a threshold value for the ensemble to exhibit "More indistinguishability with less entanglement".We relate the the distinguishability of quantum states with their robustness of the entanglement, where the robustness of any resource quantifies how tolerant it is to noise. In particular, we identify upper and lower bounds on the probability of discriminating the states, appearing in an arbitrary multiparty ensemble, in terms of their robustness of entanglement and the probability of discriminating states of the closest separable ensemble. These bounds hold true, irrespective of the dimension of the constituent systems the number of parties involved, the size of the ensemble, and whether the measurement strategies are local or global. Additional lower bounds on the same quantity is determined by considering two special cases of two-state multiparty ensembles, either having equal entanglement or at least one of them being separable. The case of equal entanglement reveals that it is always easier to discriminate the entangled states than the ones in the corresponding closest separable ensemble, a phenomenon which we refer as "More indistinguishability with less entanglement". Furthermore, we numerically explore how tight the bounds are by examining the global discrimination probability of states selected from Haar-uniformly generated ensembles of two two-qubit states. We find that for two-element ensembles of unequal entanglements, the minimum of the two entanglements must possess a threshold value for the ensemble to exhibit "More indistinguishability with less entanglement".

arXiv:2402.05074Physical Review ANov 13, 20251 citation

Entangled states close to the maximally mixed state

Roland Hildebrand

This paper deals with the radius of the largest ball of separable mixed states around the maximally mixed state for multiqubit systems. This radius determines how close entangled states can be to the maximally mixed state. In Aubrun and Szarek (e-print arXiv:quant-ph/0503221) an upper bound on the radius was given, while in Gurvits and Barnum (e-print arXiv:quant-ph/0409095) a lower bound was provided. In this paper we improve both the upper and the lower bound, bringing the ratio of these bounds down to a constant $c=\sqrt{34∕27}\ensuremath{\approx}1.122$, as opposed to a term of order $\sqrt{m\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}m}$ for the best bounds known previously, where $m$ is the number of qubits in the system. We construct concrete examples of separable states on the boundary to entanglement which realize the upper bounds. As a by-product, we compute the radii of the largest balls that fit into the projective tensor product of three and four unit balls in ${\mathbb{R}}^{3}$ and in the projective tensor product of an arbitrary number of unit balls in ${\mathbb{R}}^{n}$ for $n=2$ , 4, and 8.This paper deals with the radius of the largest ball of separable mixed states around the maximally mixed state for multiqubit systems. This radius determines how close entangled states can be to the maximally mixed state. In Aubrun and Szarek (e-print arXiv:quant-ph/0503221) an upper bound on the radius was given, while in Gurvits and Barnum (e-print arXiv:quant-ph/0409095) a lower bound was provided. In this paper we improve both the upper and the lower bound, bringing the ratio of these bounds down to a constant $c=\sqrt{34∕27}\ensuremath{\approx}1.122$, as opposed to a term of order $\sqrt{m\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}m}$ for the best bounds known previously, where $m$ is the number of qubits in the system. We construct concrete examples of separable states on the boundary to entanglement which realize the upper bounds. As a by-product, we compute the radii of the largest balls that fit into the projective tensor product of three and four unit balls in ${\mathbb{R}}^{3}$ and in the projective tensor product of an arbitrary number of unit balls in ${\mathbb{R}}^{n}$ for $n=2$, 4, and 8.

arXiv:quant-ph/0503194Physical Review AJun 26, 200722 citations

Optimal Ensemble Length of Mixed Separable States

Robert B. Lockhart

The optimal (pure state) ensemble length of a separable state, A, is the minimum number of (pure) product states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest, the lower bounds are sharp. A consequence of our results is that for all two particle systems, except possibly those with a qubit and those with a nine dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with each particle having the same Hilbert space, with probablity 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system with p qubits with probability 1 it is greater than 2 2p /(1 + 2p), which is almost the maximal rank squared. 1The optimal (pure state) ensemble length of a separable state, A, is the minimum number of (pure) product states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest, the lower bounds are sharp. A consequence of our results is that for all two particle systems, except possibly those with a qubit and those with a nine dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with each particle having the same Hilbert space, with probablity 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system with p qubits with probability 1 it is greater than 2 2p /(1 + 2p), which is almost the maximal rank squared. 1

Aug 17, 199914 citations

Robustness of entanglement for W and Greenberger–Horne–Zeilinger mixed states

Zhu Zhen, Guo-Lin Lv, Maoke Miao +1 more

Abstract We study the robustness of entanglement for W and GHZ mixed states of three and four qubits. By using the definition of the robustness of entanglement and constructing multi-partite fully separable states, we obtain the analytical upper bound of the robustness of entanglement. By analyzing the entanglement witness , we obtain the lower bound of the robustness of entanglement. Then we find that the upper bound and the lower bound are fully coincident, which shows that our results are accurate and valuable.Abstract We study the robustness of entanglement for W and GHZ mixed states of three and four qubits. By using the definition of the robustness of entanglement and constructing multi-partite fully separable states, we obtain the analytical upper bound of the robustness of entanglement. By analyzing the entanglement witness , we obtain the lower bound of the robustness of entanglement. Then we find that the upper bound and the lower bound are fully coincident, which shows that our results are accurate and valuable.

Quantum Information ProcessingDec 26, 20241 citation

\sqrt{m\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}m}

for the best bounds known previously, where

m

is the number of qubits in the system. We construct concrete examples of separable states on the boundary to entanglement which realize the upper bounds. As a by-product, we compute the radii of the largest balls that fit into the projective tensor product of three and four unit balls in

{\mathbb{R}}^{3}

and in the projective tensor product of an arbitrary number of unit balls in

{\mathbb{R}}^{n}

for

n=2

, 4, and 8.