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:1308.1914 · New Journal of Physics

Purifications of multipartite states: limitations and constructive methods

Gemma De las Cuevas, Norbert Schuch, David Pérez-Garcı́a, J. I. Cirac

Published Dec 11, 2013• 89 citations
arXivHTMLPDF

Abstract

We analyze the description of quantum many-body mixed states using matrix product states and operators. We consider two such descriptions: (i) as a matrix product density operator of bond dimension D ; and (ii) as a purification that is written as a matrix product state of bond dimension D '. We show that these descriptions are inequivalent in the sense that D ' cannot be upper bounded by D only. Then we provide two constructive methods to obtain (ii) out of (i). The sum of squares (sos) polynomial method scales exponentially in the number of different eigenvalues, and its approximate version is formulated as a semidefinite program, which gives efficient approximate purifications whose D ' only depends on D . The eigenbasis method scales quadratically in the number of eigenvalues, and its approximate version is very efficient for rapidly decaying distributions of eigenvalues. Our results imply that a description of mixed states which is both efficient and locally positive semidefinite does not exist, but that good approximations do.

DOI
10.1088/1367-2630/15/12/123021

Related papers

Separability for mixed states with operator Schmidt rank two

Gemma De las Cuevas, Tom Drescher, Tim Netzer

The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable, and can be written as a sum of at most four positive semidefinite matrices per site. This implies that these states can only contain classical correlations, and very few of them, as measured by the entanglement of purification. In contrast, MPDOs of bond dimension three can contain an unbounded amount of classical correlations.

arXiv:1903.05373•Quantum•Dec 2, 2019•9 citations

Efficient Description of Many-Body Systems with Matrix Product Density Operators

Jiří Guth Jarkovský, András Molnár, Norbert Schuch +1 more

Matrix product states form a powerful ansatz for the simulation of a wide range of one-dimensional quantum systems that are in a pure state.Their power stems from the fact that they faithfully approximate states with a low amount of entanglement, the "area law."However, in order to accurately capture the physics of realistic systems, one generally needs to apply a mixed-state description.In this work, we establish the mixed-state analog of this characterization.We show that one-dimensional mixed states with a low amount of entanglement, quantified by the entanglement of purification, can be efficiently approximated by matrix product density operators.

arXiv:2003.12418•PRX Quantum•Sep 3, 2020•37 citations

Mixed states in one spatial dimension: Decompositions and correspondence with nonnegative matrices

Gemma de las Cuevas, Tim Netzer

We study six natural decompositions of mixed states in one spatial dimension: the matrix product density operator form, the local purification form, the separable decomposition (for separable states), and their three translational invariant analogs. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorizations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorization, the nonnegative factorization, and the positive semidefinite factorization. We also show that a symmetric version of these decompositions corresponds to the symmetric factorization, the completely positive factorization, and the completely positive semidefinite transposed factorization. We leverage this correspondence to characterize the six decompositions of mixed states.

arXiv:1907.03664•Journal of Mathematical Physics•Apr 1, 2020•13 citations

A Sum-of-Squares Hierarchy with Quadratic Convergence for Quantum Channel Coding

Hoang Ta, Hoang Anh Tran

Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.

quant-pharXiv:2609.09629•Sep 10, 2026•0 citations

Fundamental limitations in the purifications of tensor networks

G. De las Cuevas, Toby S. Cubitt, J. I. Cirac +2 more

We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.

arXiv:1512.05709•Journal of Mathematical Physics•Jul 1, 2016•40 citations

Matrix product purifications for canonical ensembles and quantum number distributions

Thomas Barthel

Matrix product purifications (MPPs) are a very efficient tool for the simulation of strongly correlated quantum many-body systems at finite temperatures. When a system features symmetries, these can be used to reduce computation costs substantially. It is straightforward to compute an MPP of a grand-canonical ensemble, also when symmetries are exploited. This paper provides and demonstrates methods for the efficient computation of MPPs of canonical ensembles under utilization of symmetries. Furthermore, we present a scheme for the evaluation of global quantum number distributions using matrix product density operators (MPDOs). We provide exact matrix product representations for canonical infinite-temperature states, and discuss how they can be constructed alternatively by applying matrix product operators to vacuum-type states or by using entangler Hamiltonians. A demonstration of the techniques for Heisenberg spin-$1/2$ chains explains why the difference in the energy densities of canonical and grand-canonical ensembles decays as $1/L$.

arXiv:1607.01696•Physical review. B./Physical review. B•Sep 26, 2016•30 citations
We analyze the description of quantum many-body mixed states using matrix product states and operators. We consider two such descriptions: (i) as a matrix product density operator of bond dimension D ; and (ii) as a purification that is written as a matrix product state of bond dimension D '. We show that these descriptions are inequivalent in the sense that D ' cannot be upper bounded by D only. Then we provide two constructive methods to obtain (ii) out of (i). The sum of squares (sos) polynomial method scales exponentially in the number of different eigenvalues, and its approximate version is formulated as a semidefinite program, which gives efficient approximate purifications whose D ' only depends on D . The eigenbasis method scales quadratically in the number of eigenvalues, and its approximate version is very efficient for rapidly decaying distributions of eigenvalues. Our results imply that a description of mixed states which is both efficient and locally positive semidefinite does not exist, but that good approximations do.
The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable, and can be written as a sum of at most four positive semidefinite matrices per site. This implies that these states can only contain classical correlations, and very few of them, as measured by the entanglement of purification. In contrast, MPDOs of bond dimension three can contain an unbounded amount of classical correlations.
Matrix product states form a powerful ansatz for the simulation of a wide range of one-dimensional quantum systems that are in a pure state.Their power stems from the fact that they faithfully approximate states with a low amount of entanglement, the "area law."However, in order to accurately capture the physics of realistic systems, one generally needs to apply a mixed-state description.In this work, we establish the mixed-state analog of this characterization.We show that one-dimensional mixed states with a low amount of entanglement, quantified by the entanglement of purification, can be efficiently approximated by matrix product density operators.
We study six natural decompositions of mixed states in one spatial dimension: the matrix product density operator form, the local purification form, the separable decomposition (for separable states), and their three translational invariant analogs. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorizations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorization, the nonnegative factorization, and the positive semidefinite factorization. We also show that a symmetric version of these decompositions corresponds to the symmetric factorization, the completely positive factorization, and the completely positive semidefinite transposed factorization. We leverage this correspondence to characterize the six decompositions of mixed states.
Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.
We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.
Matrix product purifications (MPPs) are a very efficient tool for the simulation of strongly correlated quantum many-body systems at finite temperatures. When a system features symmetries, these can be used to reduce computation costs substantially. It is straightforward to compute an MPP of a grand-canonical ensemble, also when symmetries are exploited. This paper provides and demonstrates methods for the efficient computation of MPPs of canonical ensembles under utilization of symmetries. Furthermore, we present a scheme for the evaluation of global quantum number distributions using matrix product density operators (MPDOs). We provide exact matrix product representations for canonical infinite-temperature states, and discuss how they can be constructed alternatively by applying matrix product operators to vacuum-type states or by using entangler Hamiltonians. A demonstration of the techniques for Heisenberg spin-1/21/21/2 chains explains why the difference in the energy densities of canonical and grand-canonical ensembles decays as 1/L1/L1/L.