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arXiv:1906.04517 · Quantum

The Non-m-Positive Dimension of a Positive Linear Map

Nathaniel Johnston, Benjamin Lovitz, Daniel Puzzuoli

Published Aug 12, 2019• 5 citations
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Abstract

We introduce a property of a matrix-valued linear map Φ that we call its ``non-m-positive dimension'' (or ``non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of Im⊗Φ . Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map Im⊗Φ∗ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

DOI
10.22331/q-2019-08-12-172

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We introduce a property of a matrix-valued linear map Φ that we call its non-m-positive dimension'' (or non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of Im⊗Φ . Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map Im⊗Φ∗ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.
It is known that, in an $(m\ensuremath{\bigotimes}n)$-dimensional quantum system, the maximum dimension of a subspace that contains only entangled states is $(m\ensuremath{-}1)(n\ensuremath{-}1)$. 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 $(m\ensuremath{-}1)(n\ensuremath{-}1)$ negative eigenvalues, which is necessarily maximal, despite recent numerical evidence that suggested such states may not exist for large mmm and nnn.
This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the positivity of these forms and the structure of positive maps. We specifically explore the construction of indecomposable positive maps in matrix algebras, and their application as entanglement witnesses. Our analysis extends to the detection of Positive Partial Transpose (PPT) entangled states and the classification of edge PPT states in Mm(C)⊗Mn(C)M_m(\mathbb{C}) \otimes M_n(\mathbb{C})Mm​(C)⊗Mn​(C). Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.
A bipartite subspace SSS is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer kkk, there is no PPT state supporting on the orthogonal complement of S⊗kS^{\otimes k}S⊗k. We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Applying this result, we demonstrate a simple criterion for verifying whether a set of bipartite quantum states is asymptotically indistinguishable by operations completely preserving positivity of partial transpose. This criterion is based on the observation that if the support of some bipartite state is strongly PPT-unextendible, then any set of bipartite quantum states which contains this state cannot be asymptotically distinguished by PPT operations. Utilizing this criterion, we further show that any entangled pure state and its orthogonal complement cannot be distinguished asymptotically by PPT operations. Meanwhile, we also investigate that the minimum dimension of strongly PPT-unextendible subspaces in a m⊗nm\otimes nm⊗n system is m+n−1m+n-1m+n−1, which involves an extension of the result that non-positive-partial-transpose (NPT) subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87: 064302 (2013)].
We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of non-positive partial transpose bound entanglement. As an application, we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task.
The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on Cd⊗Cd\mathbb{C}^d \otimes \mathbb{C}^dCd⊗Cd alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for (d−1)(d-1)(d−1)-block positive operators, we prove that LSp⊆ASNd−1\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}LSp​⊆ASNd−1​ for p≤⌊d2/2⌋p\leq \lfloor d^2/2\rfloorp≤⌊d2/2⌋, where LSp\mathrm{LS}_pLSp​ denotes the set of states whose largest eigenvalue does not exceed the sum of their ppp smallest eigenvalues, and ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ denotes the set of states whose Schmidt number remains at most d−1d-1d−1 under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​. The global unitary requirement is nevertheless essential: for every d≥3d\geq3d≥3, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.
A bipartite subspace SSS is called strongly positive-partial-transpose (PPT) unextendible if for every positive integer kkk, there is no PPT operator supporting on the orthogonal complement of ${S}^{\ensuremath{\bigotimes}k}$. We show that a subspace is strongly PPT unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Based on these, we are able to propose a simple criterion for verifying whether a set of bipartite orthogonal quantum states is indistinguishable by PPT operations in the many-copy scenario. Utilizing this criterion, we further point out that any entangled pure state and its orthogonal complement cannot be distinguished by PPT operations in the many-copy scenario. On the other hand, we investigate that the minimum dimension of strongly PPT-unextendible subspaces in an $m\ensuremath{\bigotimes}n$ system is $m+n\ensuremath{-}1$, which involves a generalization of the result that non-positive-partial-transpose subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87, 064302 (2013)].