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:1903.05373 · Quantum

Separability for mixed states with operator Schmidt rank two

Gemma De las Cuevas, Tom Drescher, Tim Netzer

Published Dec 2, 2019• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2019-12-02-203

Related papers

Separable decompositions of 2xn states with operator Schmidt rank three

Perrine Vantalon, Nicolas Macris

We prove that every bipartite state $ρ$ of a qubit coupled to an $n$-level system whose operator Schmidt rank equals three can be written as a mixture of $\text{rank}(ρ)$ pure product states, and as a mixture of at most $n+1$ mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.

quant-pharXiv:2609.39669•Oct 1, 2026•0 citations

The distillability of entanglement of bipartite reduced density matrices of a tripartite state

Yize Sun, Lin Chen

Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.

Journal of Physics A Mathematical and Theoretical•Jun 1, 2020•4 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

Purifications of multipartite states: limitations and constructive methods

Gemma De las Cuevas, Norbert Schuch, David Pérez-Garcı́a +1 more

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.

arXiv:1308.1914•New Journal of Physics•Dec 11, 2013•89 citations

Separability for weakly irreducible matrices

Daniel Cariello

This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC $($PPT$)$ matrix can be decomposed into a sum of $s+1$ SPC $($PPT$)$ matrices of which the first $s$ are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC $($PPT$)$ matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC $($PPT$)$ matrices to the case of weakly irreducible SPC $($PPT$)$ matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).

Quantum Information and Computation•Nov 1, 2014•11 citations

A New Method of Classification of Pure Tripartite Quantum States

Jyoti Faujdar, Anoopa Joshi, Satyabrata Adhikari

The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states $σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}$, where $|χ_{1}\rangle=α|0\rangle+β|1\rangle$, $|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle$. We have shown that the state $σ_{AB}$ represent a classical state when $n$ is odd while it represent a non-classical state when $n$ is even. The purification of the state $σ_{AB}$ is studied and found that the purification is possible if the spectral decomposition of the density matrices $ρ_{1}$ and $ρ_{2}$ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity $\langleχ_{1}|χ_{2}\rangle$, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.

arXiv:1408.1539•Aug 7, 2014•0 citations
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.
We prove that every bipartite state ρρρ of a qubit coupled to an nnn-level system whose operator Schmidt rank equals three can be written as a mixture of rank(ρ)\text{rank}(ρ)rank(ρ) pure product states, and as a mixture of at most n+1n+1n+1 mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.
Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.
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.
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.
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC (((PPT))) matrix can be decomposed into a sum of s+1s+1s+1 SPC (((PPT))) matrices of which the first sss are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC (((PPT))) matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC (((PPT))) matrices to the case of weakly irreducible SPC (((PPT))) matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).
The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states σAB=p∣χ1⟩⟨χ1∣⊗ρ1+(1−p)∣χ2⟩⟨χ2∣⊗ρ2σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}σAB​=p∣χ1​⟩⟨χ1​∣⊗ρ1​+(1−p)∣χ2​⟩⟨χ2​∣⊗ρ2​, where ∣χ1⟩=α∣0⟩+β∣1⟩|χ_{1}\rangle=α|0\rangle+β|1\rangle∣χ1​⟩=α∣0⟩+β∣1⟩, ∣χ2⟩=β∣0⟩+(−1)nα∣1⟩|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle∣χ2​⟩=β∣0⟩+(−1)nα∣1⟩. We have shown that the state σABσ_{AB}σAB​ represent a classical state when nnn is odd while it represent a non-classical state when nnn is even. The purification of the state σABσ_{AB}σAB​ is studied and found that the purification is possible if the spectral decomposition of the density matrices ρ1ρ_{1}ρ1​ and ρ2ρ_{2}ρ2​ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity ⟨χ1∣χ2⟩\langleχ_{1}|χ_{2}\rangle⟨χ1​∣χ2​⟩, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.