arXiv:quant-ph/0601201

Separable balls around the maximally mixed state for a 3-qubit system

Roland Hildebrand

Published Jan 30, 2006• 2 citations

Abstract

We obtain a new lower bound on the radius of the largest ball of separable unnormalized states around the identity matrix for a 3-qubit system. This also enables us to improve the corresponding lower bounds for multi-qubit systems. These bounds are approximately 5% better than the previously known ones. As a by-product, we compute the radius of the largest ball that fits into the triple projective tensor product of the unit ball in R^3.

DOI: 10.48550/arxiv.quant-ph/0601201

Related papers

Entangled states close to the maximally mixed state

Roland Hildebrand

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

Separable balls around the maximally mixed multipartite quantum states

Leonid Gurvits, Howard Barnum

arXiv:quant-ph/0302102Physical Review AOct 14, 200398 citations

Better bound on the exponent of the radius of the multipartite separable ball

Leonid Gurvits, Howard Barnum

arXiv:quant-ph/0409095Physical Review ASep 16, 200544 citations

Largest separable balls around the maximally mixed bipartite quantum state

Leonid Gurvits, Howard Barnum

arXiv:quant-ph/0204159Physical Review ADec 12, 2002279 citations

Maximally entangled mixed symmetric states of two qubits

Eduardo Serrano-Ensástiga, John Martin

Jan 1, 20210 citations

Size of the separable neighborhood of the maximally mixed bipartite quantum state

Leonid Gurvits, Howard Barnum

Apr 26, 20024 citations

arXiv:quant-ph/0601201

Separable balls around the maximally mixed state for a 3-qubit system

Roland Hildebrand

Published Jan 30, 2006• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/0601201

Related papers

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

Separable balls around the maximally mixed multipartite quantum states

Leonid Gurvits, Howard Barnum

We show that for an m-partite quantum system, there is a ball of radius ${2}^{\ensuremath{-}(m/2\ensuremath{-}1)}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an \ensuremath{\epsilon} below which mixtures of \ensuremath{\epsilon} of any density matrix with $1\ensuremath{-}\ensuremath{\epsilon}$ of the maximally mixed state will be separable. The \ensuremath{\epsilon} thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which nuclear magnetic resonance with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 23 qubits (compared to 13 qubits via earlier results). A ball of radius 1 is obtained for multipartite states separable over the reals.We show that for an m-partite quantum system, there is a ball of radius ${2}^{\ensuremath{-}(m/2\ensuremath{-}1)}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an \ensuremath{\epsilon} below which mixtures of \ensuremath{\epsilon} of any density matrix with $1\ensuremath{-}\ensuremath{\epsilon}$ of the maximally mixed state will be separable. The \ensuremath{\epsilon} thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which nuclear magnetic resonance with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 23 qubits (compared to 13 qubits via earlier results). A ball of radius 1 is obtained for multipartite states separable over the reals.

arXiv:quant-ph/0302102Physical Review AOct 14, 200398 citations

Better bound on the exponent of the radius of the multipartite separable ball

Leonid Gurvits, Howard Barnum

We show that for an $m$ -qubit quantum system, there is a ball of radius asymptotically approaching $\ensuremath{\kappa}{2}^{\ensuremath{-}\ensuremath{\gamma}m}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent $\ensuremath{\gamma}=0.5(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}3∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2\ensuremath{-}1)\ensuremath{\approx}0.292\phantom{\rule{0.2em}{0ex}}481\phantom{\rule{0.2em}{0ex}}25$ much smaller in magnitude than the best previously known exponent, from our earlier work, of $1∕2$ . For normalized $m$ -qubit states, we get a separable ball of radius $\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}(1+\ensuremath{\gamma})m}\ensuremath{\equiv}\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{6}^{\ensuremath{-}m∕2}$ (note that $\ensuremath{\kappa}=\sqrt{3}$), compared to the previous $2\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}3m∕2}$. This implies that with parameters realistic for current experiments, nuclear magnetic resonance (NMR) with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for $m$ -partite systems of fixed local dimension ${d}_{0}$ , although approaching our earlier exponent as ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.We show that for an $m$-qubit quantum system, there is a ball of radius asymptotically approaching $\ensuremath{\kappa}{2}^{\ensuremath{-}\ensuremath{\gamma}m}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent $\ensuremath{\gamma}=0.5(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}3∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2\ensuremath{-}1)\ensuremath{\approx}0.292\phantom{\rule{0.2em}{0ex}}481\phantom{\rule{0.2em}{0ex}}25$ much smaller in magnitude than the best previously known exponent, from our earlier work, of $1∕2$. For normalized $m$-qubit states, we get a separable ball of radius $\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}(1+\ensuremath{\gamma})m}\ensuremath{\equiv}\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{6}^{\ensuremath{-}m∕2}$ (note that $\ensuremath{\kappa}=\sqrt{3}$), compared to the previous $2\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}3m∕2}$. This implies that with parameters realistic for current experiments, nuclear magnetic resonance (NMR) with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for $m$-partite systems of fixed local dimension ${d}_{0}$, although approaching our earlier exponent as ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.

arXiv:quant-ph/0409095Physical Review ASep 16, 200544 citations

Largest separable balls around the maximally mixed bipartite quantum state

Leonid Gurvits, Howard Barnum

For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral ${l}_{p}$ norms for $1<~p<~\ensuremath{\infty},$ of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\mathrm{tr}{\ensuremath{\rho}}^{2}$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by nuclear magnetic resonance quantum computation implementations or other experimental situations.For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral ${l}_{p}$ norms for $1<~p<~\ensuremath{\infty},$ of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\mathrm{tr}{\ensuremath{\rho}}^{2}$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by nuclear magnetic resonance quantum computation implementations or other experimental situations.

arXiv:quant-ph/0204159Physical Review ADec 12, 2002279 citations

Maximally entangled mixed symmetric states of two qubits

Eduardo Serrano-Ensástiga, John Martin

In this work, a variation of the problem originally solved by Verstraete, Audenaert, and De Moor [Phys. Rev. A 64, 012316 (2001)] on what is the maximum entanglement that can be created in a two-qubit system by a global unitary transformation is considered and solved when permutation invariance in the state is imposed. The additional constraint of permutation symmetry appears naturally in the context of bosonic systems or spin states. We also characterise symmetric two-qubit states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. This allows us to determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that includes the set of SAS states. For three-qubit systems, a necessary condition for absolute separability of symmetric states is given, which leads us to upper bounds on the ball radii similar to those studied for the two-qubit system.In this work, a variation of the problem originally solved by Verstraete, Audenaert, and De Moor [Phys. Rev. A 64, 012316 (2001)] on what is the maximum entanglement that can be created in a two-qubit system by a global unitary transformation is considered and solved when permutation invariance in the state is imposed. The additional constraint of permutation symmetry appears naturally in the context of bosonic systems or spin states. We also characterise symmetric two-qubit states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. This allows us to determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that includes the set of SAS states. For three-qubit systems, a necessary condition for absolute separability of symmetric states is given, which leads us to upper bounds on the ball radii similar to those studied for the two-qubit system.

Jan 1, 20210 citations

Size of the separable neighborhood of the maximally mixed bipartite quantum state

Leonid Gurvits, Howard Barnum

For finite-dimensional bipartite quantum systems, we find the size of the largest balls, in spectral $l_p$ norms for $1 \\le p \\le \\infty$ , of separable (unentangled) matrices around the identity matrix. We discuss corollaries and applications to density matrices and bulk quantum information processing such as high-temperature nuclear magnetic resonance.For finite-dimensional bipartite quantum systems, we find the size of the largest balls, in spectral $l_p$ norms for $1 \\le p \\le \\infty$, of separable (unentangled) matrices around the identity matrix. We discuss corollaries and applications to density matrices and bulk quantum information processing such as high-temperature nuclear magnetic resonance.

Apr 26, 20024 citations

\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.

We show that for an

m

-qubit quantum system, there is a ball of radius asymptotically approaching $\ensuremath{\kappa}{2}^{\ensuremath{-}\ensuremath{\gamma}m}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent $\ensuremath{\gamma}=0.5(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}3∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2\ensuremath{-}1)\ensuremath{\approx}0.292\phantom{\rule{0.2em}{0ex}}481\phantom{\rule{0.2em}{0ex}}25$ much smaller in magnitude than the best previously known exponent, from our earlier work, of

1∕2

. For normalized

m

-qubit states, we get a separable ball of radius $\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}(1+\ensuremath{\gamma})m}\ensuremath{\equiv}\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{6}^{\ensuremath{-}m∕2}$ (note that $\ensuremath{\kappa}=\sqrt{3}$), compared to the previous $2\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}3m∕2}$. This implies that with parameters realistic for current experiments, nuclear magnetic resonance (NMR) with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for

m

-partite systems of fixed local dimension

{d}_{0}

, although approaching our earlier exponent as ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.

For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral

{l}_{p}

norms for $1<~p<~\ensuremath{\infty},$ of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\mathrm{tr}{\ensuremath{\rho}}^{2}$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by nuclear magnetic resonance quantum computation implementations or other experimental situations.

For finite-dimensional bipartite quantum systems, we find the size of the largest balls, in spectral

l_p

norms for

1 \\le p \\le \\infty

, of separable (unentangled) matrices around the identity matrix. We discuss corollaries and applications to density matrices and bulk quantum information processing such as high-temperature nuclear magnetic resonance.