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:1809.05641 · Physical Review A

Symmetric versus bosonic extension for bipartite states

Youning Li, Shilin Huang, Dong Ruan, Bei Zeng

Published Jan 18, 2019• 3 citations
arXivHTMLPDF

Abstract

A bipartite state ${\ensuremath{\rho}}^{AB}$ has a $k$-symmetric extension if there exists a ($k+1$)-partite state ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ with marginals ${\ensuremath{\rho}}^{A{B}_{i}}={\ensuremath{\rho}}^{AB},\ensuremath{\forall}i$. The $k$-symmetric extension is called bosonic if ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ is supported on the symmetric subspace of ${B}_{1}{B}_{2}...{B}_{k}$. Understanding the structure of symmetric and bosonic extension has various applications in the theory of quantum entanglement, quantum key distribution, and the quantum marginal problem. In particular, bosonic extension gives a tighter bound for the quantum marginal problem based on separability. In general, it is known that a ${\ensuremath{\rho}}^{AB}$ admitting symmetric extension may not have bosonic extension. In this work, we show that, when the dimension of the subsystem $B$ is 2 (i.e., a qubit), ${\ensuremath{\rho}}^{AB}$ admits a $k$-symmetric extension if and only if it has a $k$-bosonic extension. Our result has an immediate application to the quantum marginal problem and indicates a special structure for qubit systems based on group representation theory.

DOI
10.1103/physreva.99.012332

Related papers

Symmetric extension of two-qubit states

Jianxin Chen, Zhengfeng Ji, David W. Kribs +2 more

A bipartite state ${\ensuremath{\rho}}_{AB}$ is symmetric extendible if there exists a tripartite state ${\ensuremath{\rho}}_{AB{B}^{\ensuremath{'}}}$ whose $AB$ and $A{B}^{\ensuremath{'}}$ marginal states are both identical to ${\ensuremath{\rho}}_{AB}$. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of Einstein-Podolsky-Rosen pairs, one-way distillation of secure keys, quantum marginal problems, and antidegradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state ${\ensuremath{\rho}}_{AB}$; specifically, $\mathrm{tr}({\ensuremath{\rho}}_{B}^{2})\ensuremath{\ge}\mathrm{tr}({\ensuremath{\rho}}_{AB}^{2})\ensuremath{-}4\sqrt{det{\ensuremath{\rho}}_{AB}}$. As a special case we solve the bosonic three-representability problem for the two-body reduced density matrix.

arXiv:1310.3530•Physical Review A•Sep 17, 2014•41 citations

Compatible quantum correlations: Extension problems for Werner and isotropic states

Peter D. Johnson, Lorenza Viola

We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global $N$-partite quantum state. Intuitively, we say that the multipartite state ``joins'' the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible $N$-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in $d$ dimensions and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is $m$-$n$ sharability of a Werner state across $N$ subsystems, which may be seen as a variant of the $N$-representability problem to the case where subsystems are partitioned into two groupings of $m$ and $n$ parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-$n$ sharability for arbitrary $d$. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.

arXiv:1305.1342•Physical Review A•Sep 27, 2013•29 citations

Power of symmetric extensions for entanglement detection

Miguel Navascués, Masaki Owari, Martin Bodo Plenio

In this paper, we present progress on the study of the symmetric extension criterion for separability. First, we show that a perturbation of order $O(1/N)$ is sufficient and, in general, necessary to destroy the entanglement of any state admitting an $N$ Bose-symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from $N$ Bose-symmetric extensions with positive partial transpose (PPT) decreases at least as fast as $O(1/{N}^{2})$. From these results, we derive upper bounds on the time and space complexity of the weak membership problem of separability when attacked via algorithms that search for PPT-symmetric extensions. Finally, we show how to estimate the error we incur when we approximate the set of separable states by the set of (PPT) $N$-extendable quantum states in order to compute the maximum average fidelity in pure state estimation problems, the maximal output purity of quantum channels, and the geometric measure of entanglement.

arXiv:0906.2731•Physical Review A•Nov 6, 2009•80 citations

Universal Entanglers for Bosonic and Fermionic Systems

Joel Klassen, Jianxin Chen, Bei Zeng

A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles $1,2$ with a Hilbert space $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$, a UE exists when $\min{(d_1,d_2)}\geq 3$ and $(d_1,d_2)\neq (3,3)$. It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq 8$). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.

arXiv:1305.7489•Leibniz international proceedings in informatics•Jan 1, 2013•1 citation

On state versus channel quantum extension problems: exact results for $U\;\otimes \;U\;\otimes \;U$ symmetry

Peter D. Johnson, Lorenza Viola

We develop a framework which unifies seemingly different extension (or "joinability") problems for bipartite quantum states and channels. This includes well known extension problems such as optimal quantum cloning and quantum marginal problems as special instances. Central to our generalization is a variant of the Choi-Jamiolkowski isomorphism between bipartite states and dynamical maps which we term the "homocorrelation map": while the former emphasizes the preservation of the positivity constraint, the latter is designed to preserve statistical correlations, allowing direct contact with entanglement. In particular, we define and analyze state-joining, channel-joining, and local-positive joining problems in three-party settings exhibiting collective UxUxU symmetry, obtaining exact analytical characterizations in low dimension. Suggestively, we find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree, while quantum channels are limited in the degree to which their measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces an upper bound on the extent of positive correlation across a bipartite system at a given time, as well as on the extent of negative correlation between the state of a same system across two instants of time. We argue that these general statistical bounds inform the quantum joinability limitations, and show that they are in fact sufficient for the three-party UxUxU-invariant setting.

arXiv:1405.1062•Journal of Physics A Mathematical and Theoretical•Dec 22, 2014•5 citations

Spectrum conditions for symmetric extendible states

Geir Ove Myhr, Norbert Lütkenhaus

We analyze bipartite quantum states that admit a symmetric extension. Any such state can be decomposed into a convex combination of states that allow a pure symmetric extension. A necessary condition for a state to admit a pure symmetric extension is that the spectra of the local and global density matrices are equal. This condition is also sufficient for two qubits but not for any larger systems. Using this condition, we present a conjectured necessary and sufficient condition for a two-qubit state to admit symmetric extension, which we prove in some special cases. The results from symmetric extension carry over to degradable and antidegradable channels and we use this to prove that all degradable channels with qubit output have a qubit environment.

arXiv:0812.3667•Physical Review A•Jun 9, 2009•24 citations
A bipartite state \ensuremathρAB{\ensuremath{\rho}}^{AB}\ensuremathρAB has a kkk-symmetric extension if there exists a (k+1k+1k+1)-partite state \ensuremathρAB1B2...Bk{\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}\ensuremathρAB1​B2​...Bk​ with marginals \ensuremathρABi=\ensuremathρAB,\ensuremath∀i{\ensuremath{\rho}}^{A{B}_{i}}={\ensuremath{\rho}}^{AB},\ensuremath{\forall}i\ensuremathρABi​=\ensuremathρAB,\ensuremath∀i. The kkk-symmetric extension is called bosonic if \ensuremathρAB1B2...Bk{\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}\ensuremathρAB1​B2​...Bk​ is supported on the symmetric subspace of B1B2...Bk{B}_{1}{B}_{2}...{B}_{k}B1​B2​...Bk​. Understanding the structure of symmetric and bosonic extension has various applications in the theory of quantum entanglement, quantum key distribution, and the quantum marginal problem. In particular, bosonic extension gives a tighter bound for the quantum marginal problem based on separability. In general, it is known that a \ensuremathρAB{\ensuremath{\rho}}^{AB}\ensuremathρAB admitting symmetric extension may not have bosonic extension. In this work, we show that, when the dimension of the subsystem BBB is 2 (i.e., a qubit), \ensuremathρAB{\ensuremath{\rho}}^{AB}\ensuremathρAB admits a kkk-symmetric extension if and only if it has a kkk-bosonic extension. Our result has an immediate application to the quantum marginal problem and indicates a special structure for qubit systems based on group representation theory.
A bipartite state ${\ensuremath{\rho}}_{AB}$ is symmetric extendible if there exists a tripartite state ${\ensuremath{\rho}}_{AB{B}^{\ensuremath{'}}}$ whose ABABAB and $A{B}^{\ensuremath{'}}$ marginal states are both identical to ${\ensuremath{\rho}}_{AB}$. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of Einstein-Podolsky-Rosen pairs, one-way distillation of secure keys, quantum marginal problems, and antidegradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state ${\ensuremath{\rho}}_{AB}$; specifically, $\mathrm{tr}({\ensuremath{\rho}}_{B}^{2})\ensuremath{\ge}\mathrm{tr}({\ensuremath{\rho}}_{AB}^{2})\ensuremath{-}4\sqrt{det{\ensuremath{\rho}}_{AB}}$. As a special case we solve the bosonic three-representability problem for the two-body reduced density matrix.
We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global NNN-partite quantum state. Intuitively, we say that the multipartite state ``joins'' the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible NNN-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in ddd dimensions and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is mmm-nnn sharability of a Werner state across NNN subsystems, which may be seen as a variant of the NNN-representability problem to the case where subsystems are partitioned into two groupings of mmm and nnn parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-nnn sharability for arbitrary ddd. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.
In this paper, we present progress on the study of the symmetric extension criterion for separability. First, we show that a perturbation of order O(1/N)O(1/N)O(1/N) is sufficient and, in general, necessary to destroy the entanglement of any state admitting an NNN Bose-symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from NNN Bose-symmetric extensions with positive partial transpose (PPT) decreases at least as fast as O(1/N2)O(1/{N}^{2})O(1/N2). From these results, we derive upper bounds on the time and space complexity of the weak membership problem of separability when attacked via algorithms that search for PPT-symmetric extensions. Finally, we show how to estimate the error we incur when we approximate the set of separable states by the set of (PPT) NNN-extendable quantum states in order to compute the maximum average fidelity in pure state estimation problems, the maximal output purity of quantum channels, and the geometric measure of entanglement.
A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles 1,21,21,2 with a Hilbert space Cd1⊗Cd2\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}Cd1​⊗Cd2​, a UE exists when min⁡(d1,d2)≥3\min{(d_1,d_2)}\geq 3min(d1​,d2​)≥3 and (d1,d2)≠(3,3)(d_1,d_2)\neq (3,3)(d1​,d2​)=(3,3). It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d}Cd⊗Cd. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d}Cd⊗Cd if and only if d≥3d\geq 3d≥3 (resp. d≥8d\geq 8d≥8). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.
On state versus channel quantum extension problems: exact results for U  ⊗  U  ⊗  UU\;\otimes \;U\;\otimes \;UU⊗U⊗U symmetry
We develop a framework which unifies seemingly different extension (or "joinability") problems for bipartite quantum states and channels. This includes well known extension problems such as optimal quantum cloning and quantum marginal problems as special instances. Central to our generalization is a variant of the Choi-Jamiolkowski isomorphism between bipartite states and dynamical maps which we term the "homocorrelation map": while the former emphasizes the preservation of the positivity constraint, the latter is designed to preserve statistical correlations, allowing direct contact with entanglement. In particular, we define and analyze state-joining, channel-joining, and local-positive joining problems in three-party settings exhibiting collective UxUxU symmetry, obtaining exact analytical characterizations in low dimension. Suggestively, we find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree, while quantum channels are limited in the degree to which their measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces an upper bound on the extent of positive correlation across a bipartite system at a given time, as well as on the extent of negative correlation between the state of a same system across two instants of time. We argue that these general statistical bounds inform the quantum joinability limitations, and show that they are in fact sufficient for the three-party UxUxU-invariant setting.
We analyze bipartite quantum states that admit a symmetric extension. Any such state can be decomposed into a convex combination of states that allow a pure symmetric extension. A necessary condition for a state to admit a pure symmetric extension is that the spectra of the local and global density matrices are equal. This condition is also sufficient for two qubits but not for any larger systems. Using this condition, we present a conjectured necessary and sufficient condition for a two-qubit state to admit symmetric extension, which we prove in some special cases. The results from symmetric extension carry over to degradable and antidegradable channels and we use this to prove that all degradable channels with qubit output have a qubit environment.