Qubit-qudit states with positive partial transpose

Abstract

We show that the length of a qubit-qutrit separable state is equal to the max(r,s), where r is the rank of the state and s is the rank of its partial transpose. We refer to the ordered pair (r,s) as the birank of this state. We also construct examples of qubit-qutrit separable states of any feasible birank (r,s). We determine the closure of the set of normalized two-qutrit entangled states of rank four having positive partial transpose (PPT). The boundary of this set consists of all separable states of length at most four. We prove that the length of any qubit-qudit separable state of birank (d+1,d+1) is d+1. We also show that all qubit-qudit PPT entangled states of birank (d+1,d+1) can be built in a simple way from edge states. If V is a subspace of dimension k<d in the tensor product of C^2 and C^d such that V contains no product vectors, we show that the set of all product vectors in the orthogonal complement of V is a vector bundle of rank d-k over the projective line. Finally, we explicitly construct examples of qubit-qudit PPT states (both separable and entangled) of any feasible birank.

13 pages, 2 tables

References in corpus (28)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Mixed State Entanglement and Quantum Error Correction
• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Three qubits can be entangled in two inequivalent ways
• Simple Proof of Security of the BB84 Quantum Key Distribution Protocol
• Separability criterion and inseparable mixed states with positive partial transposition
• Optimization of entanglement witnesses
• Secure key from bound entanglement
• Quantum inseparability as local pseudomixture
• Evidence for Bound Entangled States with Negative Partial Transpose
• Schmidt number witnesses and bound entanglement
• Separability in 2xN composite quantum systems
• Operational criterion and constructive checks for the separability of low rank density matrices
• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• Witnessing quantum discord in 2 x N systems
• Four-qubit entangled symmetric states with positive partial transpositions
• Three-by-three bound entanglement with general unextendible product bases
• Construction of entangled states with positive partial transposes based on indecomposable positive linear maps
• A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces
• Distillability and PPT entanglement of low-rank quantum states
• Closed formula for the relative entropy of entanglement in all dimensions
• Description of rank four PPT entangled states of two qutrits
• Searching for extremal PPT entangled states
• Optimal Decompositions of Barely Separable States
• Entanglement is not a lower bound for geometric discord
• Matrix Pencils and Entanglement Classification
• Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose

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• Dimensions, lengths and separability in finite-dimensional quantum systems
• Role of correlations in the two-body-marginal problem
• Geometry for separable states and construction of entangled states with positive partial transposes
• Multiqubit UPB: The method of formally orthogonal matrices
• Product states and Schmidt rank of mutually unbiased bases in dimension six
• Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$
• Maximal coin-position entanglement generation in a quantum walk for the third step and beyond regardless of the initial state
• Inertias of entanglement witnesses
• The role of phases in detecting three-qubit entanglement
• Boundary of the set of separable states
• Additivity of entanglement of formation via entanglement-breaking space
• Separability criterion for three-qubit states with a four dimensional norm
• Faces for two qubit separable states and the convex hulls of trigonometric moment curves
• A complete picture of the four-party linear inequalities in terms of the 0-entropy
• Trade-off relation among genuine three-qubit nonlocalities in four-qubit systems
• Length filtration of the separable states
• Separability for Weak Irreducible matrices