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

Abstract

We study six natural decompositions of mixed states in one spatial dimension: the Matrix Product Density Operator (MPDO) form, the local purification form, the separable decomposition (for separable states), and their three translational invariant (t.i.) analogues. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorisations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorisation, the nonnegative factorisation, and the positive semidefinite factorisation. We also show that a symmetric version of these decompositions corresponds to the symmetric factorisation, the completely positive factorisation, and the completely positive semidefinite transposed factorisation, respectively. We leverage this correspondence to characterise the six decompositions of mixed states.

arXiv comment37 pages (single column). v2: some explanations improved (hopefully). Very close to published version

References in corpus (20)

• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm
• Tensor networks for complex quantum systems
• PEPS as ground states: degeneracy and topology
• The entanglement of purification
• A lower bound on the dimension of a quantum system given measured data
• Lifts of convex sets and cone factorizations
• A quantum version of Wielandt's inequality
• Purifications of multipartite states: limitations and constructive methods
• Matrix product operators and states: NP-hardness and undecidability
• Positive semidefinite rank
• Fundamental limitations in the purifications of tensor networks
• Irreducible forms of Matrix Product States: Theory and Applications
• Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning
• Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank
• Linear conic formulations for two-party correlations and values of nonlocal games
• Continuum limits of Matrix Product States
• Tensor decompositions on simplicial complexes with invariance
• Quantum version of Wielandt's Inequality revisited
• Separability for mixed states with operator Schmidt rank two

Cited by in corpus (8)

• Entanglement of formation of mixed many-body quantum states via Tree Tensor Operators
• Tensor decompositions on simplicial complexes with invariance
• Separability for mixed states with operator Schmidt rank two
• Generalised ansatz for continuous Matrix Product States
• Polynomial decompositions with invariance and positivity inspired by tensors
• Approximate tensor decompositions: disappearance of many separations
• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
• Locally Purified Maximally Mixed States At Scale: Entanglement Pruning and Symmetries