Non-Positive Partial Transpose Subspaces Can be as Large as Any Entangled Subspace

Abstract

It is known that, in an $(m\otimes n)$-dimensional quantum system, the maximum dimension of a subspace that contains only entangled states is (m-1)(n-1). We show that the exact same bound is tight if we require the stronger condition that every state with range in the subspace has non-positive partial transpose. As an immediate corollary of our result, we solve an open question that asks for the maximum number of negative eigenvalues of the partial transpose of a quantum state. In particular, we give an explicit method of construction of a bipartite state whose partial transpose has (m-1)(n-1) negative eigenvalues, which is necessarily maximal, despite recent numerical evidence that suggested such states may not exist for large m and n.

4 pages, v2 contains minor updates such as typo fixes and additional references

References in corpus (3)

• Negative eigenvalues of partial transposition of arbitrary bipartite states
• Qubit-qudit states with positive partial transpose
• Entanglement is not a lower bound for geometric discord

Cited by in corpus (29)

• On the partial transpose of fermionic Gaussian states
• Quantum Codes of Maximal Distance and Highly Entangled Subspaces
• Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition
• One-shot entanglement distillation beyond local operations and classical communication
• Indistinguishability of bipartite states by positive-partial-transpose operations in the many-copy scenario
• The Inverse Eigenvalue Problem for Entanglement Witnesses
• A unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder inequalities
• Entangled subspaces and generic local state discrimination with pre-shared entanglement
• Inertias of entanglement witnesses
• Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses
• Entanglement properties of positive operators with ranges in completely entangled subspaces
• Measurement-based local quantum filters and their ability to transform quantum entanglement
• Fully non-positive-partial-transpose genuinely entangled subspaces
• Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis
• Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties
• Maximum entanglement of mixed symmetric states under unitary transformations
• The Non-m-Positive Dimension of a Positive Linear Map
• A characterization of entangled two-qubit states via partial-transpose-moments
• Comparing Geometric Discord and Negativity for Bipartite States
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures
• Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities
• Unextendible and strongly uncompletable product bases
• $k$-Positive Maps: New Characterizations and a Generation Method
• Genuinely entangled subspaces beyond strongly nonlocal unextendible biseparable bases
• Entangled Subspaces through Algebraic Geometry
• Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits
• Spectral characterizations of entanglement witnesses
• Useful variants and perturbations of completely entangled subspaces and spans of unextendible product bases