arXiv subjects

Daniel Cariello

Publications and source records attributed to Daniel Cariello.

13 recordsLinked to original sources

arXiv2024

On the structure of Completely Reducible States

The complete reducibility property for bipartite states reduced the separability problem to a proper subset of positive under partial transpose states and was used to prove several theorems inside and outside entanglement theory. So far only three types of bipartite states were proved to possess this property. In this work, we provide some procedures to create states with this property, we call these states by the name of completely reducible states. The convex combination of such states is the first procedure, showing that the set of completely reducible states is a convex cone. We also provide a complete description of the extreme rays of this set. Then we show that powers, roots and partial traces of completely reducible states result in states of the same type. Finally, we consider a shuffle of states that preserves this property. This shuffle allows us to construct states with the complete reducibility property avoiding the only three conditions known to date that imply this property. We conclude this paper by showing a connection between our results and the distillability problem. All these results were possible due to a simplification of the description of this property also presented here for the first time.

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arXiv2024

A reduction of the separability problem to SPC states in the filter normal form

It was recently suggested that a solution to the separability problem for states that remain positive under partial transpose composed with realignment (the so-called symmetric with positive coefficients states or simply SPC states) could shed light on entanglement in general. Here we show that such a solution would solve the problem completely. Given a state in $ \mathcal{M}_k\otimes\mathcal{M}_m$, we build a SPC state in $ \mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$ with the same Schmidt number. It is known that this type of state can be put in the filter normal form retaining its type. A solution to the separability problem in $\mathcal{M}_k\otimes\mathcal{M}_m$ could be obtained by solving the same problem for SPC states in the filter normal form within $\mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$. This SPC state can be built arbitrarily close to the projection on the symmetric subspace of $ \mathbb{C}^{k+m}\otimes\mathbb{C}^{k+m}$. All the information required to understand entanglement in $ \mathcal{M}_s\otimes\mathcal{M}_t$ $(s+t\leq k+m)$ lies inside an arbitrarily small ball around that projection. We also show that the Schmidt number of any state $γ\in\mathcal{M}_n\otimes\mathcal{M}_n$ which commutes with the flip operator and lies inside a small ball around that projection cannot exceed $\lfloor\frac{n}{2}\rfloor$.

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arXiv2022

A triality pattern in entanglement theory

In this work, we present new connections between three types of quantum states: positive under partial transpose states, symmetric with positive coefficients states and invariant under realignment states. First, we obtain a common upper bound for their spectral radii and a result on their filter normal forms. Then we prove the existence of a lower bound for their ranks and the fact that whenever this bound is attained the states are separable. These connections add new evidence to the pattern that for every proven result for one of these types, there are counterparts for the other two, which is a potential source of information for entanglement theory.

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arXiv2021

Schmidt rank constraints in Quantum Information Theory

Can vectors with low Schmidt rank form mutually unbiased bases? Can vectors with high Schmidt rank form positive under partial transpose states? In this work, we address these questions by presenting several new results related to Schmidt rank constraints and their compatibility with other properties. We provide an upper bound on the number of mutually unbiased bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ $(m\leq n)$ formed by vectors with low Schmidt rank. In particular, the number of mutually unbiased product bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ cannot exceed $m+1$, which solves a conjecture proposed by McNulty et al. Then we show how to create a positive under partial transpose entangled state from any state supported on the antisymmetric space and how their Schmidt numbers are exactly related. Finally, we show that the Schmidt number of operator Schmidt rank 3 states of $\mathcal{M}_m\otimes \mathcal{M}_n\ (m\leq n)$ that are invariant under left partial transpose cannot exceed $m-2$.

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arXiv2019

Inequalities for the Schmidt Number of Bipartite States

In this short note we show two completely opposite methods of constructing entangled states. Given a bipartite state $γ\in M_k\otimes M_k$, define $γ_S=(Id+F)γ(Id+F)$, $γ_A=(Id-F)γ(Id-F)$, where $F\in M_k\otimes M_k$ is the flip operator. In the first method, entanglement is a consequence of the inequality $\text{rank}(γ_S)<\sqrt{\text{rank}(γ_A)}$. In the second method, there is no correlation between $γ_S$ and $γ_A$. These two methods show how diverse is quantum entanglement. We prove that any bipartite state $γ\in M_k\otimes M_k$ satisfies $\displaystyle SN(γ)\geq\max \left\{ \frac{\text{rank}(γ_L)}{\text{rank}(γ)}, \frac{\text{rank}(γ_R)}{\text{rank}(γ)}, \frac{SN(γ_S)}{2}, \frac{SN(γ_A)}{2} \right\},$ where $SN(γ)$ stands for the Schmidt number of $γ$ and $γ_L,γ_R$ are the marginal states of $γ$. We also present a family of PPT states in $M_k\otimes M_k$, whose members have Schmidt number equal to $n$, for any given $1\leq n\leq \left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.

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arXiv2018

Sinkhorn-Knopp Theorem for PPT states

Given a PPT state $A=\sum_{i=1}^nA_i\otimes B_i \in M_k\otimes M_k$ and a vector $v\in\Im(A)\subset\mathbb{C}^k\otimes\mathbb{C}^k$ with tensor rank $k$, we provide an algorithm that checks whether the positive map $G_A:M_k\rightarrow M_k$, $G_A(X)=\sum_{i=1}^n tr(A_iX)B_i$, is equivalent to a doubly stochastic map. This procedure is based on the search for Perron eigenvectors of completely positive maps and unique solutions of, at most, $k$ unconstrained quadratic minimization problems. As a corollary, we can check whether this state can be put in the filter normal form. This normal form is an important tool for studying quantum entanglement. An extension of this procedure to PPT states in $M_k\otimes M_m$ is also presented.

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arXiv2018

Sinkhorn-Knopp theorem for rectangular positive maps

In this work, we adapt Sinkhorn-Knopp theorem for rectangular positive maps $(T:M_k\rightarrow M_m)$. We extend their concepts of support and total support to these maps. We show that a positive map $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\rightarrow M_m$ is equivalent to a positive map with total support. Moreover, if $k$ and $m$ are coprime then $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\rightarrow M_m$ has support. This result provides a necessary and sufficient condition for the filter normal form, which is commonly used in Quantum Information Theory in order to simplify the task of detecting entanglement. Let $A=\sum_{i=1}^nA_i\otimes B_i\in M_k\otimes M_m$ be a state and $G_A: M_k\rightarrow M_m$ be the positive map $G_A(X)=\sum_{i=1}^nB_itr(A_iX)$. We show that $A$ can be put in the filter normal form if and only if $G_A: M_k\rightarrow M_m$ is equivalent to a positive map with total support. We prove that any state $A\in M_k\otimes M_m\simeq M_{km}$ such that $\dim(\ker(A))<k-1$, if $k=m$, and $\dim(\ker(A))<\min\{k,m\}$, if $k\neq m$, can be put in the filter normal form. Recently, a connection between the capacity of a rectangular positive map $T:M_k\rightarrow M_m$ and the capacity of a certain square positive map $\widetilde{T}:M_{mk}\rightarrow M_{mk}$ was noticed. Here, we obtain a deeper connection between these maps. As a corollary of our main results, we prove that $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $\widetilde{T}:M_{mk}\rightarrow M_{mk}$ is equivalent to a doubly stochastic map.

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arXiv2016

A gap for PPT entanglement

Let $W$ be a finite dimensional vector space over a field with characteristic not equal to 2. Denote by $\text{Sym}(V)$ and $\text{Skew-Sym}(V)$ the subspaces of symmetric and skew-symmetric tensors of a subspace $V$ of $W\otimes W$, respectively. In this paper we show that if $V$ is generated by tensors with tensor rank 1, $V=\text{Sym}(V)\oplus\text{Skew-Sym}(V)$ and $W$ is the smallest vector space such that $V\subset W\otimes W$ then $\dim(\text{Sym}(V))\geq\max\{\frac{2\dim(\text{Skew-Sym}(V))}{\dim(W)}, \frac{\dim(W)}{2}\}$. This result has a straightforward application to the separability problem in Quantum Information Theory: If $ρ\in M_k\otimes M_k\simeq M_{k^2}$ is separable then $\text{rank}(Id+F)ρ(Id+F)\geq\text{max}\{ \frac{2}{r}\text{rank}(Id-F)ρ(Id-F), \frac{r}{2}\},$ where $F\in M_k\otimes M_k$ is the flip operator, $Id\in M_k\otimes M_k$ is the identity and $r$ is the marginal rank of $ρ+FρF$. We prove the sharpness of this inequality. Moreover, we show that if $ρ\in M_k\otimes M_k$ is positive under partial transposition (PPT) and $\text{rank }(Id+F)ρ(Id+F)=1$ then $ρ$ is separable. This result follows from Perron-Frobenius theory. We also present a large family of PPT matrices satisfying $\text{rank}(Id+F)ρ(Id+F)\geq r\geq \frac{2}{r-1} \text{rank}(Id-F)ρ(Id-F)$. There is a possibility that an entangled PPT matrix $ρ\in M_k\otimes M_k$ satisfying $1<\text{rank}(Id+F)ρ(Id+F)<\frac{2}{r} \text{rank}(Id-F)ρ(Id-F)$ exists. However, the family referenced above shows that finding one shall not be trivial.

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arXiv2015

Completely Reducible maps in Quantum Information Theory

In order to compute the Schmidt decomposition of $A\in M_k\otimes M_m$, we must consider an associated self-adjoint map. Here, we show that if $A$ is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC) or invariant under realignment then its associated self-adjoint map is completely reducible. We give applications of this fact in Quantum Information Theory. We recover some theorems recently proved for PPT and SPC matrices and we prove these theorems for matrices invariant under realignment using theorems of Perron-Frobenius theory. We also provide a new proof of the fact that if $\mathbb{C}^{k}$ contains $k$ mutually unbiased bases then $\mathbb{C}^{k}$ contains $k+1$. We search for other types of matrices that could have the same property. We consider a group of linear transformations acting on $M_k\otimes M_k$, which contains the partial transpositions and the realignment map. For each element of this group, we consider the set of matrices in $M_k\otimes M_k\simeq M_{k^2}$ that are positive and remain positive, or invariant, under the action of this element. Within this family of sets, we have the set of PPT matrices, the set of SPC matrices and the set of matrices invariant under realignment. We show that these three sets are the only sets of this family such that the associated self-adjoint map of each matrix is completely reducible. We also show that every matrix invariant under realignment is PPT in $M_2\otimes M_2$ and we present a counterexample in $M_k\otimes M_k$, $k\geq 3$.

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arXiv2014

Does Symmetry Imply PPT Property?

Recently, in [1], the author proved that many results that are true for PPT matrices also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called SPC in [1] (definition 1.1). Before that, in [9], Tóth and Gühne proved that if a state is symmetric then it is PPT if and only if it is SPC. A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in $M_2\otimes M_2$ (theorem 4.3). This theorem is a consequence of the fact that every density matrix in $M_2\otimes M_m$, with tensor rank smaller or equal to 3, is separable (theorem 3.2). This theorem is a generalization of the same result found in [1] for tensor rank 2 matrices in $M_k\otimes M_m$. Although, in $M_3\otimes M_3$, there exists a SPC matrix with tensor rank 3 that is not PPT (proposition 5.2). We shall also provide a non trivial example of a family of matrices in $M_k\otimes M_k$, in which both, the SPC and PPT properties, are equivalent (proposition 6.2). Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability (proposition 6.4).

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arXiv2013

Separability for Weak Irreducible matrices

This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We proved that many results holds for both types. If these matrices have specific Hermitian Schmidt decompositions then the matrices are separable in a very strong sense. We proved that both types have what we call split decompositions. We defined the notion of weak irreducible matrix, based on the concept of irreducible state defined recently. These split decomposition theorems together with the notion of weak irreducible matrix, imply that these matrices are weak irreducible or a sum of weak irreducible matrices of the same type. The separability problem for these types of matrices can be reduced to the set of weak irreducible matrices of the same type. We also provided a complete description of weak irreducible matrices of both types. Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable, we found sharp inequalites providing separability for both types.

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arXiv2013

Basic sequences and spaceability in $\ell_p$ spaces

Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p \in [1,\infty]$) $Z(\ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(\ell_\infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X \setminus Z(X)$ and its algebraic genericity.

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arXiv2011

Maximal spaceability in topological vector spaces

In this paper we introduce a new technique to prove the existence of closed subspaces of maximal dimension inside sets of topological vector sequence spaces. The results we prove cover some sequence spaces not studied before in the context of spaceability and settle some questions on classical sequence spaces that remained open.

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