Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1309.2006 · Physical Review A

Separability from spectrum for qubit-qudit states

Nathaniel Johnston

Published Dec 23, 2013• 65 citations
arXivHTMLPDF

Abstract

The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states $\ensuremath{\rho}$ with the property that ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ is separable for all unitary matrices $U$. This problem has been solved when the local dimensions $m$ and $n$ satisfy $m=2$ and $n\ensuremath{\le}3$. We solve all remaining qubit-qudit cases (i.e., when $m=2$ and $n\ensuremath{\ge}4$ is arbitrary). In all of these cases we show that a state is separable from spectrum if and only if ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ has positive partial transpose for all unitary matrices $U$. This equivalence is in stark contrast with the usual separability problem, where a state having positive partial transpose is a strictly weaker property than it being separable.

DOI
10.1103/physreva.88.062330

Related papers

Is Separability from Spectrum Determined by the Partial Transpose?

Srinivasan Arunachalam, Nathaniel Johnston, Vincent Russo

The separability from spectrum problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU † is separable for all unitary matrices U. We provide evidence that ρ is separable from spectrum if and only if UρU † has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest. 1

•Jan 1, 2014•1 citation

Spectral criteria for absolute positive partial transpose (PPT) in qudit-qudit state systems

Liang Xiong, Nung-Sing Sze

Separability from the spectrum is an important and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudit-qudit states in the bipartite state space Hmn=Hm⊗Hn. First, we propose the necessary and sufficient conditions for absolute positive partial transpose states in the Hilbert space H4n. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that these sufficient conditions can be extended to the general Hmn case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in Hmn are not absolutely separable. These conditions only depend on 2m − 1 eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement, but also effective in practice. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.

Journal of Mathematical Physics•Jan 1, 2026•0 citations

Is absolute separability determined by the partial transpose?

Srinivasan Arunachalam, Nathaniel Johnston, Vincent Russo

The absolute separability problem asks for a characterization of the quantum states $\rho \in M_m\otimes M_n$ with the property that $U\rho U^\dagger$ is separable for all unitary matrices $U$. We investigate whether or not it is the case that $\rho$ is absolutely separable if and only if $U\rho U^\dagger$ has positive partial transpose for all unitary matrices $U$. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer--Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.

Quantum Information and Computation•May 1, 2015•21 citations

Is absolute separability determined by the partial transpose?

Srinivasan Arunachalam, Nathaniel Johnston, Russo, Vincent

The absolute separability problem asks for a characterization of the quantum states $ρ\in M_m\otimes M_n$ with the property that $UρU^\dagger$ is separable for all unitary matrices $U$. We investigate whether or not it is the case that $ρ$ is absolutely separable if and only if $UρU^\dagger$ has positive partial transpose for all unitary matrices $U$. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer-Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.

arXiv:1405.5853•May 22, 2014•3 citations

The geometry of absolute separability and other convex matrix properties from spectrum

Jennifer Ahiable, Naga Bhavya Teja Kothakonda, Andreas Winter

We investigate the geometric structure of the set of spectra of bipartite absolute separable states ($\mathrm{ASEP}_{m,n}$) and absolute positive partial transpose states ($\mathrm{APPT}_{m,n}$), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that $\mathrm{APPT}_{m,n}$ is a spectrahedron for all $m\leq n$: in particular, all its faces are exposed. While $\mathrm{ASEP}_{2,n}$ is a spectrahedron, too, for $\mathrm{ASEP}_{m,n}$ we can in general only show that it is a semialgebraic set for all $m,n\geq 3$. Furthermore, we provide a complete characterization of the faces and extreme points of $\mathrm{APPT}_{m,n}$ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension $mn-m-1$. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of $\mathrm{APPT}_{m,n}$ via an inscribed polytope $\mathcal{P}_{m,n}$ and conjecture that the maximal purity of $\mathrm{APPT}_{m,n}$ (along with its spectra) coincides with the polytope for arbitrary dimensions except when $m=n=2$. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of $\mathrm{APPT}_{m,n}$ and demonstrate numerically that the minimum entropy eventually coincides with the polytope $\mathcal{P}_{m,n}$ as the local system dimension $n$ increases. Finally, we show that the relative spectral volume of $\mathrm{APPT}_{m,n}$ decays exponentially in $n$ by a constant multiplicative factor of the relative volume of the inscribed polytope $\mathcal{P}_{m,n}$.

quant-pharXiv:2608.03390•Aug 5, 2026•0 citations

Criteria of absolutely separability from spectrum for qudit-qudits states

Liang Lin Xiong, Nung-Sing Sze

Separability from the spectrum is a significant and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudits-qudits states in the bipartite states space $\mathcal{H}_{mn}=\mathcal{H}_m \otimes \mathcal{H}_n$. Firstly, we propose the necessary and sufficient conditions for absolute separable states in the Hilbert space $\mathcal{H}_{4n}$. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that this sufficient condition can be extended to the general $\mathcal{H}_{mn}$ case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in $\mathcal{H}_{mn}$ are not absolutely separable. These conditions only depend on $2m-1$ eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.

arXiv:2408.11684•Aug 21, 2024•0 citations
The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states \ensuremathρ\ensuremath{\rho}\ensuremathρ with the property that U\ifmmode†\else\textdagger\fi\ensuremathρU{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}UU\ifmmode†\else\textdagger\fi\ensuremathρU is separable for all unitary matrices UUU. This problem has been solved when the local dimensions mmm and nnn satisfy m=2m=2m=2 and n\ensuremath≤3n\ensuremath{\le}3n\ensuremath≤3. We solve all remaining qubit-qudit cases (i.e., when m=2m=2m=2 and n\ensuremath≥4n\ensuremath{\ge}4n\ensuremath≥4 is arbitrary). In all of these cases we show that a state is separable from spectrum if and only if U\ifmmode†\else\textdagger\fi\ensuremathρU{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}UU\ifmmode†\else\textdagger\fi\ensuremathρU has positive partial transpose for all unitary matrices UUU. This equivalence is in stark contrast with the usual separability problem, where a state having positive partial transpose is a strictly weaker property than it being separable.
The separability from spectrum problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU † is separable for all unitary matrices U. We provide evidence that ρ is separable from spectrum if and only if UρU † has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in any such state, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the families of isotropic and Werner states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest. 1
Separability from the spectrum is an important and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudit-qudit states in the bipartite state space Hmn=Hm⊗Hn. First, we propose the necessary and sufficient conditions for absolute positive partial transpose states in the Hilbert space H4n. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that these sufficient conditions can be extended to the general Hmn case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in Hmn are not absolutely separable. These conditions only depend on 2m − 1 eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement, but also effective in practice. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.
The absolute separability problem asks for a characterization of the quantum states ρ∈Mm⊗Mn\rho \in M_m\otimes M_nρ∈Mm​⊗Mn​ with the property that UρU†U\rho U^\daggerUρU† is separable for all unitary matrices UUU. We investigate whether or not it is the case that ρ\rhoρ is absolutely separable if and only if UρU†U\rho U^\daggerUρU† has positive partial transpose for all unitary matrices UUU. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer--Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.
The absolute separability problem asks for a characterization of the quantum states ρ∈Mm⊗Mnρ\in M_m\otimes M_nρ∈Mm​⊗Mn​ with the property that UρU†UρU^\daggerUρU† is separable for all unitary matrices UUU. We investigate whether or not it is the case that ρρρ is absolutely separable if and only if UρU†UρU^\daggerUρU† has positive partial transpose for all unitary matrices UUU. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer-Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.
We investigate the geometric structure of the set of spectra of bipartite absolute separable states (ASEPm,n\mathrm{ASEP}_{m,n}ASEPm,n​) and absolute positive partial transpose states (APPTm,n\mathrm{APPT}_{m,n}APPTm,n​), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ is a spectrahedron for all m≤nm\leq nm≤n: in particular, all its faces are exposed. While ASEP2,n\mathrm{ASEP}_{2,n}ASEP2,n​ is a spectrahedron, too, for ASEPm,n\mathrm{ASEP}_{m,n}ASEPm,n​ we can in general only show that it is a semialgebraic set for all m,n≥3m,n\geq 3m,n≥3. Furthermore, we provide a complete characterization of the faces and extreme points of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension mn−m−1mn-m-1mn−m−1. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ via an inscribed polytope Pm,n\mathcal{P}_{m,n}Pm,n​ and conjecture that the maximal purity of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ (along with its spectra) coincides with the polytope for arbitrary dimensions except when m=n=2m=n=2m=n=2. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ and demonstrate numerically that the minimum entropy eventually coincides with the polytope Pm,n\mathcal{P}_{m,n}Pm,n​ as the local system dimension nnn increases. Finally, we show that the relative spectral volume of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ decays exponentially in nnn by a constant multiplicative factor of the relative volume of the inscribed polytope Pm,n\mathcal{P}_{m,n}Pm,n​.
Separability from the spectrum is a significant and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudits-qudits states in the bipartite states space Hmn=Hm⊗Hn\mathcal{H}_{mn}=\mathcal{H}_m \otimes \mathcal{H}_nHmn​=Hm​⊗Hn​. Firstly, we propose the necessary and sufficient conditions for absolute separable states in the Hilbert space H4n\mathcal{H}_{4n}H4n​. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that this sufficient condition can be extended to the general Hmn\mathcal{H}_{mn}Hmn​ case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in Hmn\mathcal{H}_{mn}Hmn​ are not absolutely separable. These conditions only depend on 2m−12m-12m−1 eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.