The problem of classification of decomposable (in the sense of Stormer) positive maps between matrix algebras is presented. We propose the new notion of "finite" version of decomposability ($k$-decomposabilty). The characterisation of $k$-decomposability on the Hilbert space level is done. In the case of low dimensional algebras the notion of local decomposability and its applications for the description of decomposable maps are discussed.
Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any decomposable map, that is neither completely positive nor completely copositive, will lose decomposability eventually after taking enough tensor powers. Moreover, we establish explicit bounds to quantify when this happens. To prove these results we use a symmetrization technique from the theory of entanglement distillation, and analyze when certain symmetric …
We address two sets of long-standing open questions in probability theory, from a computational complexity perspective: divisibility of stochastic maps, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic maps is an NP-complete problem, and extend this result to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochastic maps. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is…
We study dynamical semigroups of positive, but not completely positive maps on finite-dimensional bipartite systems and analyze properties of their generators in relation to non-decomposability and bound-entanglement. An example of non-decomposable semigroup leading to a 4x4-dimensional bound-entangled density matrix is explicitly obtained.
Sparse positive maps on qutrits with exact nondecomposability thresholds and PPT-entanglement transitions
Università degli Studi di Pavia（帕维亚大学）；INFN Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；Università di Trieste（的里雅斯特大学）；INFN Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；MIT（麻省理工学院）；CNR-Istituto di Fotonica e Nanotecnologie（国家研究委员会光子学与纳米技术研究所）
Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $Φ_{α, β,n} : M_{n}(\mathbb{C}) …
We investigate the space of quantum operations, as well as the larger space of maps which are positive, but not completely positive. A constructive criterion for decomposability is presented. A certain class of unistochastic operations, determined by unitary matrices of extended dimensionality, is defined and analyzed. Using the concept of the dynamical matrix and the Jamiolkowski isomorphism we explore the relation between the set of quantum operations (dynamics) and the set of density matrices acting on an extended Hilbert space (kinematics). An analogous relation is established between the …
Generating Non-Decomposable Maps with Differentiable Semidefinite Programming
Università degli Studi di Pavia（帕维亚大学）；INFN Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；Università di Trieste（的里雅斯特大学）；MIT（麻省理工学院）；INFN Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；CNR-Istituto di Fotonica e Nanotecnologie（国家研究委员会光子学与纳米技术研究所）
We present a generalization of the family of linear positive maps in $M_3$ proposed thirty years ago by Cho et al. (Linear Algebra Appl. ${\bf 171}$, 213 (1992)) as a generalization of the seminal Choi non-decomposable map. The necessary and sufficient conditions for decomposability are provided.
The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$, $C_i$ be unital C*-algebras and let $α_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$. We obtain conditions under which any positive map $β$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$, such that $ α_1 \otimes α_2 \geq β$, factorizes as $β= γ\otimes α_2$ for some positive map…
A spin-$j$ state can be represented by a symmetric tensor of order $N=2j$ and dimension $4$. Here, $j$ can be a positive integer, which corresponds to a boson; $j$ can also be a positive half-integer, which corresponds to a fermion. In this paper, we introduce regularly decomposable tensors and show that a spin-$j$ state is classical if and only if its representing tensor is a regularly decomposable tensor. In the even-order case, a regularly decomposable tensor is a completely decomposable tensor but not vice versa; a completely decomposable tensors is a sum-of-squares (SOS) tensor but not vi…
For two positive maps $ϕ_i:B(\mathcal{K}_i)\to B(\mathcal{H}_i)$, $i=1,2$, we construct a new linear map $ϕ:B(\mathcal{H})\to B(\mathcal{K})$, where $\mathcal{K}=\mathcal{K}_1\oplus\mathcal{K}_2\oplus\mathbb{C}$, $\mathcal{H}=\mathcal{H}_1\oplus\mathcal{H}_2\oplus\mathbb{C}$, by means of some additional ingredients such as operators and functionals. We call it a merging of maps $ϕ_1$ and $ϕ_2$. We discuss properties of this construction. In particular, we provide conditions for positivity of $ϕ$, as well as for $2$-positivity, complete positivity and nondecomposability. In particular, we show …
We analyze positivity of a tensor product of two linear qubit maps, $Φ_1 \otimes Φ_2$. Positivity of maps $Φ_1$ and $Φ_2$ is a necessary but not a sufficient condition for positivity of $Φ_1 \otimes Φ_2$. We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both $Φ_1$ and $Φ_2$ are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for $n$-tensor-stable positive qubit maps, i.e. such qubit maps $Φ$ that $Φ^{\otimes n}$ is positive. Particular cases of 2- and 3-tensor-stable positive qubit …
We study positive maps of B(K) into B(H) for finite-dimensional Hilbert spaces K and H. Our main emphasis is on how Choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps. Special attention will be given to entanglement properties and k-positive maps, in particular tensor products of 2-positive maps. The latter problem is directly related to the question of n-copy distillability of quantum states, for which we obtain a partial result.
We provide a novel tool which may be used to construct new examples of positive maps in matrix algebras (or, equivalently, entanglement witnesses). It turns out that this can be used to prove positivity of several well known maps (such as reduction map, generalized reduction, Robertson map, and many others). Furthermore, we use it to construct a new family of linear maps and prove that they are positive, indecomposable and (nd)optimal.
The positive linear maps $\ad_s$ which send matrices $x$ to $s^*xs$ play important roles in quantum information theory as well as matrix theory. It was proved by Marciniak [Linear Multilinear Alg. 61 (2013), 970--975] that the map $\ad_s$ generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable and k-separable operators, due to the Jamiolkowski-Choi isomorphism. Generalizations to a situation where no such simple isomorphism…
In this paper, we discuss positive maps induced by (irreducibly) covariant linear operators for finite groups. The application of group theory methods allows deriving some new results of a different kind. In particular, a family of necessary conditions for positivity, for such objects is derived, stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map. In the low-dimensional cases, for the permutation group $S(3)$ and the quaternion group $Q$, the necessary and sufficient conditions are given, together with the discussion on their decompo…
Positive maps are useful for detecting entanglement in quantum information theory. Any entangled state can be detected by some positive map. In this paper, the relation between positive block matrices and completely positive trace-preserving maps is characterized. Consequently, a new method for constructing decomposable maps from positive block matrices is derived. In addition, a method for constructing positive but not completely positive maps from Hermitian block matrices is also obtained.
The paper is devoted to the problem of classification of extremal positive maps acting between $B(K)$ and $B(H)$ where $K$ and $H$ are Hilbert spaces. It is shown that every positive map with the property that $\rank ϕ(P)\leq 1$ for any one-dimensional projection $P$ is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2-positive turns out to automatically completely positive. Finally we get the same conclusion for such extremal positive maps that $\rank ϕ(P)\leq 1…
Valid transformations between quantum states are necessarily described by completely positive maps, instead of just positive maps. Positive but not completely positive maps such as the transposition map cannot be implemented due to the existence of entanglement in composite quantum systems, but there are classes of states for which the positivity is guaranteed, e.g., states not correlated to other systems. In this paper, we introduce the concept of N-copy extension of maps to quantitatively analyze the difference between positive maps and completely positive maps. We consider implementations o…
Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a posit…
A class of linear positive, trace preserving maps in $M_n$ is given in terms of affine maps in $\bBR^{n^2-1}$ which map the closed unit ball into itself.
From the physical point of view entanglement witnesses define a universal tool for analysis and classification of quantum entangled states. From the mathematical point of view they provide highly nontrivial generalization of positive operators and they find elegant correspondence with the theory of positive maps in matrix algebras. We concentrate on theoretical analysis of various important notions like (in)decomposability, atomicity, optimality, extremality and exposedness. Several methods of construction are provided as well. Our discussion is illustrated by many examples enabling the reader…
We outline a new approach to the characterization as well as to the classification of positive maps. This approach is based on the facial structures of the set of states and of the cone of positive maps. In particular, the equivalence between Schroedinger's and Heisenberg's pictures is reviewed in this more general setting. Furthermore, we discuss in detail the structure of positive maps for two and three dimensional systems. In particular, the explicit form of decomposition of a positive map and the uniqueness of this decomposition for extremal positive maps for 2 dimensional case are describ…
The concept of the {\em half density matrix} is proposed. It unifies the quantum states which are described by density matrices and physical processes which are described by completely positive maps. With the help of the half-density-matrix representation of Hermitian linear map, we show that every positive map which is not completely positive is a {\em difference} of two completely positive maps. A necessary and sufficient condition for a positive map which is not completely positive is also presented, which is illustrated by some examples.
The time evolution of an initially uncorrelated system is governed by a completely positive (CP) map. More generally, the system may contain initial (quantum) correlations with an environment, in which case the system evolves according to a not-completely positive (NCP) map. It is an interesting question what the relative measure is for these two types of maps within the set of positive maps. After indicating the scope of the full problem of computing the true volume for generic maps acting on a qubit, we study the case of Pauli channels in an abstract space whose elements represent an equival…
We provide a partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions. This construction generalizes celebrated Choi example of a map which is positive but not completely positive. It is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements. Such maps provide basic tools to study quantum entanglement in multipartite systems.
The full description of the set of positive maps $T: \qA \to \cB(\cH)$ ($\qA$ a $C^*$-algebra) is given. The approach is based on the simple prescription for selecting various types of positive maps. This prescription stems from the Grothendieck theory of projective tensor products complemented by the theory of tensor connes. In particular, the origin of non-decomposable maps is clarified.
We construct entangled states with positive partial transposes using indecomposable positive linear maps between matrix algebras. We also exhibit concrete examples of entangled states with positive partial transposes arising in this way, and show that they generate extreme rays in the cone of all positive semi-definite matrices with positive partial transposes. They also have Schmidt numbers two.
Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition. We thus generalise this bipartite concept to the multipartite setting by introducing non-positive maps that are positive on the subset of bi-separable states, but can map to a non-positive element if applied to a genuine multipartite entangled state. We explicitl…
We prove a reduction theorem for capacity of positive maps of finite dimensional C*-algebras, thus reducing the computation of capacity to the case when the image of a nonscalar projection is never a projection.
We analyze the structure of the subset of states generated by unital completely positive quantum maps, A witness that certifies that a state does not belong to the subset generated by a given map is constructed. We analyse the representations of positive maps and their relation to quantum Perron-Frobenius theory.
We give a criterion for a positive mapping on the space of operators on a Hilbert space to be indecomposable. We show that this criterion can be applied to two families of positive maps. These families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states.
A dynamical map is a map which takes one density operator to another. Such a map can be written in an operator-sum representation (OSR) using a spectral decomposition. The method of the construction applies to more general maps which need not be completely positive. The OSR not unique; there is a freedom to choose the set of operators in the OSR differently, yet still obtain the same map. Here we identify all maps which are equivalent to a given map. Whereas the freedom for completely positive maps is unitary, the freedom for maps which are not necessarily completely positive is pseudo-unitary…
Structural approximations to positive, but not completely positive maps are approximate physical realizations of these non-physical maps. They find applications in the design of direct entanglement detection methods. We show that many of these approximations, in the relevant case of optimal positive maps, define an entanglement breaking channel and, consequently, can be implemented via a measurement and state-preparation protocol. We also show how our findings can be useful for the design of better and simpler direct entanglement detection methods.
A necessary and sufficient condition for 1-distillability is formulated in terms of decomposable positive maps. As an application we provide insight into why all states violating the reduction criterion map are distillable and demonstrate how to construct such maps in a systematic way. We establish a connection between a number of existing results, which leads to an elementary proof for the characterisation of distillability in terms of 2-positive maps.
We present a general scheme that allows for construction of scalar separability criteria from positive but not completely positive maps. The concept is based on a decomposition of every positive map $Λ$ into a difference of two completely positive maps $Λ_1$, $Λ_2$, i.e., $Λ=Λ_1-Λ_2$. The scheme may be also treated as a generalization of the known entropic inequalities, which are obtained from the reduction map. An analysis performed on few classes of states shows that the new scalar criteria are stronger than the entropic inequalities and furthermore, when derived from indecomposable maps all…
We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map, which is nontrivial because NCP maps may lack a full positivity domain. The definition is motivated by specific examples. We prove that for valid NCP maps, the eigenvalue spectrum of the corresponding dynamical matrix is not bounded. Based on this, we argue that in general the volume measure of qubit maps, including NCP maps, is not well defined.
We investigate completely positive maps for an open system interacting with its environment. The families of the initial states for which the reduced dynamics can be described by a completely positive map are identified within the framework of direct-sum decomposition of state space. They includes not only separable states with vanishing or nonvanishing quantum discord but also entangled states. A general expression of the families as well as the Kraus operators for the completely positive maps are explicitly given. It significantly extends the previous results.
We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entanglement can be detected by one of these witnesses, and this witness detects a unique set of entanglement among those. Therefore, they provide a minimal set of witnesses to detect all entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entanglement with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one parameter family of indecomposabl…
We construct a new class of positive indecomposable maps in the algebra of `d x d' complex matrices. These maps are characterized by the `weakest' positivity property and for this reason they are called atomic. This class provides a new reach family of atomic entanglement witnesses which define important tool for investigating quantum entanglement. It turns out that they are able to detect states with the `weakest' quantum entanglement.
We define extension maps as maps that extend a system (through adding ancillary systems) without changing the state in the original system. We show, using extension maps, why a completely positive operation on an initially entangled system results in a non positive mapping of a subsystem. We also show that any trace preserving map, either positive or negative, can be decomposed in terms of an extension map and a completely positive map.
A theorem of Blackwell about comparison between information structures in classical statistics is given an analogue in the quantum probabilistic setup. The theorem provides an operational interpretation for trace-preserving completely positive maps, which are the natural quantum analogue of classical stochastic maps. The proof of the theorem relies on the separation theorem for convex sets and on quantum teleportation.
We provide a new class of positive maps in matrix algebras. The construction is based on the family of balls living in the space of density matrices of n-level quantum system. This class generalizes the celebrated Choi map and provide a wide family of entanglement witnesses which define a basic tool for analyzing quantum entanglement.
By introducing an operator sum representation for arbitrary linear maps, we develop a generalized theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This theory of "linear quantum error correction" is applicable in cases where the standard and restrictive assumption of a factorized initial system-bath state does not apply.
We present an "uncertainty principle" for quantum channels, showing a relationship between the dimensions of the range of a channel and the range of its complement. We examine some interesting specific cases, and discuss consequences for privacy of quantum channels.
We study the concurrence of arbitrary dimensional bipartite quantum systems. By using a positive but not completely positive map, we present an analytical lower bound of concurrence. Detailed examples are used to show that our bound can detect entanglement better and can improve the well known existing lower bounds.
The phenomenological dissipation of the Bloch equations is reexamined in the context of completely positive maps. Such maps occur if the dissipation arises from a reduction of a unitary evolution of a system coupled to a reservoir. In such a case the reduced dynamics for the system alone will always yield completely positive maps of the density operator. We show that, for Markovian Bloch maps, the requirement of complete positivity imposes some Bloch inequalities on the phenomenological damping constants. For non-Markovian Bloch maps some kind of Bloch inequalities involving eigenvalues of the…
A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool are real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps which are not completely positive is given.
Let $D$ be a space of $2\times n$ matrices. Then the face of the cone of all completely positive maps from $M_2$ into $M_n$ given by $D$ is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in $D$ forms a subspace of $D$ together with zero and $D^\perp$ is spanned by rank one matrices.
We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New classes of positive maps are constructed. Some open questions are answered.
