On the structure of the set of positive maps

TL;DR

This paper provides a comprehensive description of the set of positive maps between $C^*$-algebras and bounded operators, clarifying the origin of non-decomposable maps using tensor product theories.

Contribution

It offers a new framework based on tensor product theories to characterize positive maps and explains the emergence of non-decomposable maps.

Findings

Full description of positive maps set

Clarification of non-decomposable maps origin

New tensor-based characterization method

Abstract

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.

Equations

$$({\mathfrak{A}}_h,{\mathfrak{A}}^{+})$$

$$\pi (u)=\inf \{\sum _{i=1}^n\Vert x_i\Vert \Vert y_i\Vert :u=\sum _{i=1}^n{x}_i\otimes y_i\}.$$

$$(X{\otimes }_{\pi }Y{)}^{\ast }=\mathcal{L}(X,Y^{\ast }),$$

$$⟨\sum _{i=1}^n{x}_i\otimes y_i,S⟩=\sum _{i=1}^n⟨y_i,S{x}_i⟩.$$

$$\pi (x)=\inf \{\sum _{i=1}^n\Vert a_i\Vert \Vert b_i{\Vert }_1:x=\sum _{i=1}^n{a}_i\otimes b_i,a_i\in \mathfrak{A},b_i\in \mathfrak{T}\},$$

$$(\tilde{\varphi })(\sum _{i=1}^n{a}_i\otimes b_i)=\sum _{i=1}^n\mathrm{Tr}(\varphi (a_i)b_i^t),$$

$$C_p=conv(E_{+}\otimes F_{+})=\{\sum _{i=1}^n{x}_i\otimes y_i;x_i\in E_{+},y_i\in F_{+},n\in \mathrm{I}\mathrm{N}\}$$

$$C_i=\{t\in E\otimes F;⟨t,E_{+}^{\ast }\otimes F_{+}^{\ast }⟩\ge 0\}.$$

$$C_p\subset C_{\alpha }\subset C_i.$$

$$P_{\alpha }=\{T\in \mathcal{L}(\mathfrak{A},\mathcal{B}(\mathcal{H}));\tilde{T}(\sum _{i=1}^n{a}_i\otimes b_i)=\sum _{i=1}^n\mathrm{Tr}T(a_i)b_i^t\ge 0\mathrm{for}\mathrm{any}\sum _{i=1}^n{a}_i\otimes b_i\in C_{\alpha }\},$$

$$P_i\subset P_{\alpha }\subset P_p.$$

$$P_p=\{T\in \mathcal{L}(\mathfrak{A},\mathcal{B}(\mathcal{H}));\tilde{T}(\sum _{i=1}^n{a}_i\otimes b_i)=\mathrm{Tr}T(a_i)b_i^t\ge 0\mathrm{for}\mathrm{any}\sum _{i=1}^n{a}_i\otimes b_i\in C_p\}.$$

$$\mathrm{Tr}T(a)b\ge 0,$$

$$C_{cp}=(\mathfrak{A}{\otimes }_{\pi }\mathfrak{T}{)}_{+}.$$

$$<\phi \otimes \mathrm{Tr}a(\cdot ),\sum _{i,j}{x}_i^{\ast }{x}_j\otimes {\varrho }_i^{\ast }{\varrho }_j>=\sum _{i,j}\phi (x_i^{\ast }{x}_j)\mathrm{Tr}{\varrho }_i^{\ast }{\varrho }_{j}a\ge 0,$$

$$\tilde{T}(\sum _{i,j}^k(a_i\otimes b_i{)}^{\ast }(a_j\otimes b_j))=\sum _{i,j}^k\mathrm{Tr}T(a_i^{\ast }{a}_j)(b_i^{\ast }{b}_j{)}^t=\sum _{i,j}^k\sum _n((b_i{)}^t{e}_n,T(a_i^{\ast }{a}_j)(b_j{)}^t{e}_n)$$

$$(\text{ref}3.10)=\sum _{i,j}^k\sum _n\overline{(x,e_n)}(f_i,T(a_i^{\ast }{a}_j)f_j)(x,e_n)=\sum _{i,j}^k(f_i,T(a_i^{\ast }{a}_j)f_j).$$

$$(\text{ref}3.10)=\sum _{i,j}^k(c_{i}g,T(a_i^{\ast }{a}_j)c_{j}g)\ge 0$$

$$\sum _{i=1}^n\phi (a_i)\mathrm{Tr}{\varrho }_{i}a\ge 0,$$

$$P_i=\{T:\sum _i\mathrm{Tr}T(a_i){\varrho }_i\ge 0\mathrm{for}\sum _i{a}_i\otimes {\varrho }_i{\ge }^{st}0\}.$$

$$\sum _i\mathrm{Tr}{\varrho }_{\phi }{a}_i\mathrm{Tr}{\varrho }_{i}a\ge 0,$$

$$\sum _i(f,a_{i}f)(g,{\varrho }_{i}g)=\sum _i(f\otimes g,a_i\otimes {\varrho }_{i}f\otimes g)\ge 0,$$

$$T(a)=\sum _k{V}_k^{\ast }a{V}_k,$$

$$\sum _{i,k}\mathrm{Tr}{V}_k^{\ast }{a}_i{V}_k{\varrho }_i=\sum _{i,k,l}(e_l,V_k^{\ast }{a}_i{V}_k{\varrho }_i{e}_l)=\sum _{i,k,l}(|f_k><g_k|e_l,a_i|f_k><g_k|{\varrho }_i{e}_l)$$

$$=\sum _{i,k,l}(f_k,a_i{f}_k)({\varrho }_i^{\ast }{g}_k,e_l)(e_l,g_k)=\sum _{i,k}(f_k,a_i{f}_k)(g_k,{\varrho }_i{g}_k)$$

$$=\sum _k(f_k\otimes g_k,\left(\sum _i{a}_i\otimes {\varrho }_i\right)f_k\otimes g_k).$$

$$\sum _{k,l,i}(e_l,V_k^{\ast }{a}_i{V}_k{\varrho }_i{e}_l)=\sum _{i,k,l}(\sum _m|f_m^k><g_m^k|e_l,a_i\sum _n|f_n^k><g_n^k|{\varrho }_i{e}_l)$$

$$=\sum _{i,k,l,m,n}(e_l,g_m^k)(f_m^k,a_i{f}_n^k)(g_n^k,{\varrho }_i{e}_l)=\sum _{k,m,n}(f_m^k\otimes g_n^k,\left(\sum _i{a}_i\otimes {\varrho }_i\right)f_n^k\otimes g_m^k).$$

$$C_p\subset C_d\subset C_{cp}\mathrm{and}{\mathrm{P}}_{\mathrm{cp}}\subset {\mathrm{P}}_{\mathrm{d}}\subset {\mathrm{P}}_{\mathrm{p}},$$

$$P_d=\{T:\sum _{i,j=1}^n\mathrm{Tr}T(a_i^{\ast }{a}_j)(b_i^{\ast }{b}_j{)}^t\ge 0\mathrm{and}\sum _{i,j=1}^n\mathrm{Tr}T(a_i^{\ast }{a}_j)(b_i^{\ast }{b}_j)\ge 0,$$

Full text

1. Introduction

Despite the facts that positive maps are an essential ingredient in a description of quantum systems and that they play an important role in mathematics [1], [2], a characterization of the structure of the set of all positive maps has been a long standing challenge in mathematical physics. The key reason behind that is the complexity of this structure - the structure of positive maps is drastically nontrivial even for the finite dimensional case. To illustrate this point it is enough to note that even the convex structure of the positive maps, $\Phi:{\mathcal{B}}({\mathcal{H}})\to{\mathcal{B}}({\mathcal{H}})$, is highly complicated even in very low dimensions of the Hilbert space ${\mathcal{H}}$.

In the sixties it was shown [3] (see also Størmer’s book [2] and references given there) that every positive map for the 2D case, i.e. for $\dim{\mathcal{H}}=2$, is decomposable. The first example of non-decomposable map was given by Choi [4], see also [5] and [6], for 3D case, i.e. for $\dim{\mathcal{H}}=3$. Since then, other examples of non-decomposable maps were constructed. In particular, by results of Woronowicz [6] and Størmer [3], if $\dim{\mathcal{H}}_{1}\cdot\dim{\mathcal{H}}_{2}\leq 6$, all positive maps $T:{\mathcal{B}}({\mathcal{H}}_{1})\to{\mathcal{B}}({\mathcal{H}}_{2})$ are decomposable but this is not true in higher dimensions. On the other hand, the emergence of non-decomposable maps may be considered as a huge obstacle in getting a canonical form for a positive map.

The present work being a continuation of our previous papers [7], [8], [9], [10], [11], [12] and [13], provides an analysis as well as a description of the structure of the set of positive maps. These results stem from the observation that the linear tensor product structure is not compatible with either topology (there are many cross-norms) or an order (there are many tensor cones).

The deep Grothendieck’s result gives the relation between linear maps on Banach spaces ${\mathcal{L}}(X,Y)$ and linear continuous functionals on the projective tensor product $(X\otimes_{\pi}Y_{*}).$ That is a landmark, a crucial ingredient of our approach. An analysis of various orders in $(X\otimes_{\pi}Y_{*})$ yields essential information about the structure of positive maps. In particular, as a byproduct, we will get an explanation of the origin of non-decomposable maps.

This paper is organized as follows: first we give necessary preliminaries in Section 2. Next, in Section 3, a description of various types of positive maps will be given. As a result, the structure of the set of all positive maps will be characterized. Conclusions and final remarks are given in Section 4.

2. Definitions and notations

For any $C^{*}$-algebra $\mathfrak{A}$, we denote the set of all self-adjoint (positive) elements of $\mathfrak{A}$ by $\mathfrak{A}_{h}$ ($\mathfrak{A}^{+}$). If $\mathfrak{A}$ is a unital $C^{*}$-algebra then a state on $\mathfrak{A}$ is a linear functional \phi:\mathfrak{A}\to{\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}} such that $\phi(a)\geq 0$ for every $a\geq 0$ ($a\in\mathfrak{A}^{+}$) and $\phi({\mathbb{I}})=1$, where ${\mathbb{I}}$ is the unit of $\mathfrak{A}$. The set of all states on $\mathfrak{A}$ will be denoted by ${\mathcal{S}}_{\mathfrak{A}}$. In particular

[TABLE]

is an ordered Banach space.

A linear map $T:{\mathfrak{A}}_{1}\to{\mathfrak{A}}_{2}$ between $C^{*}$-algebras ${\mathfrak{A}}_{1}$ and ${\mathfrak{A}}_{2}$ is called positive if $T({\mathfrak{A}}_{1}^{+})\subseteq{\mathfrak{A}}_{2}^{+}$. For $k\in{\rm I\!N}$ we consider a map T_{k}:M_{k}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})\otimes{\mathfrak{A}}_{1}\to M_{k}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})\otimes{\mathfrak{A}}_{2} where M_{k}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}) denotes the algebra of $k\times k$ matrices with complex entries and $T_{k}=id_{M_{k}}\otimes T$. We say that $T$ is k-positive if the map $T_{k}$ is positive. The map $T$ is said to be completely positive (cp for short) if $T$ is k-positive for every $k\in{\rm I\!N}$. The set of all linear, bounded (unital) positive maps $T:\mathfrak{A}_{1}\to\mathfrak{A}_{2}$ will be denoted by ${\mathcal{L}}^{+}(\mathfrak{A}_{1},\mathfrak{A}_{2})$ (${\mathcal{L}}^{+}_{1}(\mathfrak{A}_{1},\mathfrak{A}_{2})$, respectively).

From now on we make the assumption that $\mathfrak{A}_{2}$ is equal to ${\mathcal{B}}({\mathcal{H}})$ for some Hilbert space ${\mathcal{H}}$. The important class of positive maps is formed by decomposable maps. They are defined as follows. Let $T\in{\mathcal{L}}^{+}(\mathfrak{A},{\mathcal{B}}({\mathcal{H}}))$. We say that $T$ is co-positive if $t\circ T$ is completely positive, where $t$ is the transpose map on ${\mathcal{B}}({\mathcal{H}})$. $T$ is decomposable if $T$ is the sum of a completely positive map and a co-positive map. Otherwise, $T$ is called an indecomposable map.

To speak about the structure of ${\mathcal{L}}^{+}({\mathfrak{A}},{\mathcal{B}}({\mathcal{H}}))$ we need some preliminaries. To this end, firstly, we note that in his pioneering work on Banach spaces, Grothendieck [14] observed the links between tensor products and mapping spaces. To describe this, we will select certain results from the theory of tensor products of Banach spaces. The point is that the synthesis of the linear structure of tensor products with a topology is not unique - namely, there are many “good” cross-norms (cf [15]) (the same can be said about the synthesis of the linear structure of tensor product and an order, see the next Section). However, among them, there is the projective norm which gives rise to the projective tensor product and this tensor product linearizes bounded bilinear mappings just as the algebraic tensor product linearizes bilinear mappings (see [16]).

Let $X$, $Y$ be Banach algebras. We denote by $X\odot Y$ the algebraic tensor product of $X$ and $Y$ (algebraic tensor product of two ∗-Banach algebras is defined as tensor product of two vector spaces with ∗-algebraic structure determined by the two factors; so the topological questions are not considered). We consider the following (projective) norm on $X\odot Y$

[TABLE]

We denote by $X\otimes_{\pi}Y$ the completion of $X\odot Y$ with respect to the projective norm $\pi$ and this Banach space will be referred to as the projective tensor product of the Banach spaces $X$ and $Y$. Denote by $\mathfrak{B}(X\times Y)$ the Banach space of bounded bilinear mappings $B$ from $X\times Y$ into the field of scalars with the norm given by $||B||=\sup\{|B(x,y)|;\|x\|\leq 1,\|y\|\leq 1\}$. Note (for all details see [16]), that with each bounded bilinear form $B\in\mathfrak{B}(X\times Y)$ there is an associated operator $L_{B}\in{\mathcal{L}}(X,Y^{*})$ defined by $\langle y,L_{B}(x)\rangle=B(x,y)$. The mapping $B\mapsto L_{B}$ is an isometric isomorphism between the spaces $\mathfrak{B}(X\times Y)$ and ${\mathcal{L}}(X,Y^{*})$. Hence, there is an identification

[TABLE]

such that the action of an operator $S:X\to Y^{*}$ as a linear functional on $X\otimes_{\pi}Y$ is given by

[TABLE]

We wish to complete the presented compilation by recalling the another of Størmer’s result (see [3]) which will be the crucial in our work. Moreover, it can serve as an illustration on the given material as well as to indicate that relation (2.2) is very relevant to an analysis of positive maps. To present the above result we need some preparations.

Let $\mathfrak{A}$ be a norm closed self-adjoint subspace of bounded operators on a Hilbert space $\mathcal{K}$ containing the identity operator on $\mathcal{K}$. $\mathfrak{T}$ will denote the set of trace class operators on a Hilbert space ${\mathcal{H}}$. Recall that ${\mathcal{B}}({\mathcal{H}})\ni x\mapsto x^{t}\in{\mathcal{B}}({\mathcal{H}})$ stands for the transpose map of ${\mathcal{B}}({\mathcal{H}})$ with respect to some orthonormal basis. The set of all linear bounded maps $\phi:\mathfrak{A}\to{\mathcal{B}}({\mathcal{H}})$ will be denoted by ${\mathcal{L}}(\mathfrak{A},{\mathcal{B}}({\mathcal{H}}))$. Finally, we denote by ${\mathfrak{A}}\odot{\mathfrak{T}}$ the algebraic tensor product of $\mathfrak{A}$ and $\mathfrak{T}$ where ${\mathfrak{A}}{\otimes}_{\pi}\mathfrak{T}$ means its Banach space closure under the projective norm defined by

[TABLE]

where $\|\cdot\|_{1}$ stands for the trace norm. Now, we can quote (see [17])

Lemma 2.1**.**

There is an isometric isomorphism $\phi\mapsto\tilde{\phi}$ between ${\mathcal{L}}({\mathfrak{A}},{\mathcal{B}}({\mathcal{H}}))$ and $({\mathfrak{A}}{\otimes}_{\pi}{\mathfrak{T}})^{*}$ given by

[TABLE]

where $\sum_{i=1}^{n}a_{i}\otimes b_{i}\in{\mathfrak{A}}\odot{\mathfrak{T}}$.

Furthermore, $\phi\in{\mathcal{L}}^{+}({\mathfrak{A}},{\mathcal{B}}({\mathcal{H}}))$ if and only if $\tilde{\phi}$ is positive on ${\mathfrak{A}}^{+}{\otimes}_{\pi}{\mathfrak{T}}^{+}$.

3. Structure of ${\mathcal{L}}^{+}({\mathfrak{A}},{\mathcal{B}}({\mathcal{H}}))$

We begin with some preliminaries on tensor products of ordered Banach spaces. We already note in the previous section that the synthesis of the linear structure of a tensor product with an order is not unique. Following Wittstock [18] we start with,

Definition 3.1**.**

([18]) Let $E$, $F$ be ordered linear spaces with proper cones $E_{+}$, $F_{+}$. We call a cone $C_{\alpha}\subset E\otimes F$ a tensor cone and write $E\otimes_{\alpha}F\equiv(E\otimes F,C_{\alpha})$ if the canonical bilinear mappings $\omega:E\times F\to E\otimes_{\alpha}F$ and $\omega^{*}:E^{*}\times F^{*}\to(E\otimes_{\alpha}F)^{*}$ are positive. Thus, $x\otimes y\in C_{\alpha}$ for all $x\in E_{+}$, $y\in F_{+}$; and $x^{*}\otimes y^{*}\in C_{\alpha}^{*}$ for all $x^{*}\in E_{+}^{*}$, $y^{*}\in F_{+}^{*}$, where $E_{+}^{*}$, $F_{+}^{*}$, and $C_{\alpha}^{*}$ denote the dual cones.

There are two distinguished cones:

Definition 3.2**.**

([18]) The projective cone $C_{p}$:

[TABLE]

and

Definition 3.3**.**

([18]) The injective cone $C_{i}$:

[TABLE]

One has, see Proposition 1.14 in [18]:

Proposition 3.4**.**

([18]) If $C_{\alpha}$ is a tensor cone, then

[TABLE]

Combining Grothendieck’s result, (2.3), (2.2) and Lemma 2.1 with the concept of tensor cones, we get the following selection recipe:

[TABLE]

where $C_{\alpha}$ is a tensor cone in ${\mathfrak{A}}\otimes_{\pi}\mathfrak{T}$.

Remark 3.5*.*

In other words, $P_{\alpha}$ is a result of selecting such linear mappings which have “nice” positive functionals, where positivity is defined by a selected tensor cone. It is worth pointing out that there are, in general, many tensor cones. Consequently, there are many classes of positive maps. Furthermore, Proposition 3.4 implies:

[TABLE]

To study the structure of ${\mathcal{L}}^{+}({\mathfrak{A}},{\mathcal{B}}({\mathcal{H}}))$ we proceed to a description of some selected cones $C_{\alpha}$ and positive maps in $P_{\alpha}$.

3.1. Positive maps determined by $C_{p}$

According to the recipe given by (3.4) we are interested in the following maps:

[TABLE]

It is a simple matter to observe that in (3.6) a linear bounded map $T$ should satisfy:

[TABLE]

for any $a\in{\mathfrak{A}}_{+}$ and $b\in\mathfrak{T}_{+}$. But, it is nothing else but the definition of a positive map. Consequently, the smallest tensor cone $C_{p}$ defines the largest class of positive maps - just the set of all positive maps. We note that this result can be inferred from Stormer’s paper, see Lemma 2.1 and/or [17].

3.2. Positive maps determined by $C_{cp}$

We define

[TABLE]

Firstly, we note that as ${\mathfrak{A}}$ and $\mathfrak{T}$ are ∗-Banach algebras, it is easy to check that then ${\mathfrak{A}}\otimes_{\pi}\mathfrak{T}$ is also ∗-Banach algebra under the natural involution $(x\otimes y)^{*}=x^{*}\otimes y^{*}$, cf [15], Section IV.4. Thus $C_{cp}$ is a cone. Clearly $x\otimes y\in({\mathfrak{A}}\otimes\mathfrak{T})_{+}$ if $x\in{\mathfrak{A}}_{+}$ and $y\in\mathfrak{T}_{+}$. Furthermore, for $\varphi\in{\mathfrak{A}}^{*}_{+}$ and $\mathrm{Tr}a(\cdot)\in\mathfrak{T}^{*}_{+}$, $a\in{\mathcal{B}}({\mathcal{H}})_{+}$, one has

[TABLE]

where $x_{i}\in{\mathfrak{A}}$, $\varrho_{i}\in\mathfrak{T}$, and where we have used the Schur product theorem for the Hadamard product. Thus, $C_{cp}$ is a tensor cone, and we can use the recipe (3.4). So, for $a_{i}\in{\mathfrak{A}}$, $b_{i}\in\mathfrak{T}$:

[TABLE]

Let $(b_{j})^{t}=|f_{j}><x|\ (\in\mathfrak{T})$, where $f_{j}\in{\mathcal{H}}$, $x\in{\mathcal{H}}$ such that $||x||=1$. Then,

[TABLE]

Now, put $f_{i}=c_{i}g$, where $c_{i}\in{\mathcal{B}}({\mathcal{H}})$, $g\in{\mathcal{H}}$. Then,

[TABLE]

It follows immediately from Corollary IV.3.4 in [15] that $T$ is a completely positive map. Consequently, the cone $C_{cp}$ is selecting cp maps $P_{cp}$. Again, we note that a similar characterization of cp maps could be inferred from Størmer’s paper [17].

3.3. Positive maps determined by $C_{i}$

We have seen that the tensor cone $C_{i}$ is the largest tensor cone. It was already noted by Stinespring [19] - see the nicely elaborated the Stinespring example in [18] - that in general, $C_{i}$ is not equal to $C_{cp}$. Now we wish to describe the corresponding maps which will be denoted by $P_{i}$. We note that the condition defining the cone $C_{i}$ can be written as

[TABLE]

for any $\varphi\in{\mathfrak{A}}^{*}_{+}$, $a\in{\mathcal{B}}({\mathcal{H}})_{+}$, where we took ${\mathfrak{A}}\otimes\mathfrak{T}\ni t=\sum_{i=1}^{n}a_{i}\otimes\varrho_{i}$.

We say that $t=\sum_{i=1}^{n}a_{i}\otimes\varrho_{i}$ satisfying condition (3.13) is st-positive (st stands for simple tensor) and denote $t\geq^{st}0$. We note that the property $t\geq^{st}0$ is, in general, weaker than standard positivity $t\geq 0$, see the example below. Taking into account that $P_{i}\subset P_{cp}$ , cf Remark 3.5 and the previous subsection, we infer that the recipe (3.4) leads to very regular completely positive maps which will be denoted by

[TABLE]

To see this regularity explicitly as well as the difference between $\geq 0$ and $\geq^{st}0$, we give an example which can be considered as a continuation of the above mentioned Stinespring’s example.

Example 3.6**.**

Let ${\mathfrak{A}}$ be equal to ${\mathcal{B}}({\mathcal{H}})$ with ${\mathcal{H}};\ \dim{\mathcal{H}}=n<\infty$. From the condition (3.13) we have

[TABLE]

for any $\varrho_{\varphi}\in{\mathcal{B}}({\mathcal{H}})_{+}$, $a\in{\mathcal{B}}({\mathcal{H}})_{+}$, where $t$ was taken to be of the form $t=\sum_{i}a_{i}\otimes\varrho_{i}$ with $a_{i},\varrho_{i}\in{\mathcal{B}}({\mathcal{H}})$. Put $\varrho_{\varphi}=|f><f|$ and $a=|g><g|$ for $f,g\in{\mathcal{H}}$. Then, (3.15) leads to

[TABLE]

for any $f,g\in{\mathcal{H}}$. Thus $\sum_{i}a_{i}\otimes\varrho_{i}\geq^{st}0$ implies the positivity of $\sum a_{i}\otimes\varrho_{i}$ only on simple tensors of ${\mathcal{H}}\otimes{\mathcal{H}}$. We emphasize that this condition is weaker than the standard positivity $\sum_{i}a_{i}\otimes\varrho_{i}\geq 0$.

To explain the regularity imposed by the cone $C_{i}$ we recall that $P_{i}\subset P_{cp}$, cf Remark 3.5. In particular, $T\in P_{i}$ can be written in the form (usually called the Kraus decomposition),

[TABLE]

where $V_{k}\in{\mathcal{B}}({\mathcal{H}})$, see Theorem 4.1.8 in [2]. As a first observation, we consider the case: $V_{k}=|f_{k}><g_{k}|$ with $f_{k},g_{k}\in{\mathcal{H}}$. Then,

[TABLE]

[TABLE]

[TABLE]

Consequently, $T(\cdot)=\sum_{i}V_{k}^{*}(\cdot)V_{k}$ with $V_{k}=|f_{k}><g_{k}|$ is a regular cp map which is in $P_{i}$. It is worth pointing out that one-rank operators $V_{k}$ ensures applicability of st positivity and that such maps $T(\cdot)=\sum_{i}V_{k}^{*}(\cdot)V_{k}$ are sometimes called super positive maps, cf Definition 5.1.2 in [2]. Now, we turn to the case $V_{k}=\sum_{m}^{M}|f^{k}_{m}><g^{k}_{m}|$ with $M\geq 2$.

[TABLE]

[TABLE]

We see at once that st. positivity of $\sum_{i}a_{i}\otimes\varrho_{i}$ does not ensure the positivity of (3.19). Consequently, such cp maps are not, in general, in $P_{i}$.

3.4. Positive maps determined by $C_{cp}\cap id\otimes t(C_{cp})$

We begin by an examination of the set $C_{d}\equiv C_{cp}\cap id\otimes t(C_{cp})$, where $t$ as before, stands for the transposition. We first note that $C_{d}$ is a cone. To deduce that $C_{d}$ is a tensor cone , we note that $x\otimes y\in C_{d}$ for all $x\in{\mathfrak{A}}_{+}$ and $y\in\mathfrak{T}_{+}$. Subsequently, we observe that one has $x^{*}\otimes y^{*}\in C^{*}_{d}$ for all $x^{*}\in{\mathfrak{A}}^{*}_{+}$ and $y^{*}\in\mathfrak{T}^{*}_{+}$. Thus, $C_{d}$ is a tensor cone. Clearly,

[TABLE]

where $P_{d}$ stands for those positive maps which are determined by the cone $C_{d}$, i.e.

[TABLE]

where $a_{i}\in{\mathfrak{A}}$, and $b_{i}\in\mathfrak{T}$.

Further, let us consider the set $C_{ccp}\equiv id\otimes t(C_{cp})$. It follows by similar arguments as those employed in Subsection 3.2 that $C_{ccp}$ is a tensor come. Hence, it is easy to check that

[TABLE]

where $P_{ccp}$ stands for the set of all co-positive maps. Consequently, (3.22) and (3.20) lead to

[TABLE]

In other words, the cone $C_{d}$ determines decomposable maps $P_{d}$.

To finish this subsection we have to examine the question whether $C_{d}$ is always non-trivial, i.e. whether the inclusion $C_{p}\subset C_{d}$ is the proper one. To answer this question we give:

Example 3.7**.**

We assume that ${\mathfrak{A}}={\mathcal{B}}({\mathcal{H}})$ and that $\dim{\mathcal{H}}\leq 3$. To study the non-triviality of $({\mathcal{B}}({\mathcal{H}})\otimes{\mathcal{B}}(H))_{+}\cap id\otimes t(({\mathcal{B}}({\mathcal{H}})\otimes{\mathcal{B}}(H))_{+})$ it is enough to examine one dimensional (orthogonal) projectors of the form $|f><f|$ with $f\in{\mathcal{H}}\otimes{\mathcal{H}}$. To see this we begin with two observations:

Observation 3.8**.**

For ${\mathcal{H}}\otimes{\mathcal{H}}\ni h=\sum_{i}f_{i}\otimes g_{i}$ one has

[TABLE]

Observation 3.9**.**

Any $h\in{\mathcal{H}}\otimes{\mathcal{H}}$ can be written in the form $h=\sum_{i}v_{i}\otimes e_{i}\ (=\sum e_{i}\otimes w_{i})$ where $v_{i},w_{i}\in{\mathcal{H}}$ and $\{e_{i}\}$ is a basis in ${\mathcal{H}}$.

Proofs of the above observations are easy and they are left to the reader.

Let $|f><f|$ be given. Then, the above Observations and (3.4) imply

[TABLE]

where $f=e_{k}\otimes f_{k}$, $\{e_{i}\}$ a basis in ${\mathcal{H}}$. To see that that the Woronowicz scheme for description of positive maps of low dimensional matrix algebras is reproduced, [6], we note, in the “matrix terms”, that

[TABLE]

and

[TABLE]

where, using Woronowicz’s notation, $Q$ stands for the operator $\{|f_{k}><f_{l}|\}_{kl}$. Consequently, to infer the triviality of the cone $C_{d}$ for dimension $2$ from the Woronowicz result, Theorem 1.1 and Section 2 in [6], it is enough to reproduce, in the considered context, Woronowicz’s argument leading to “simple vectors”. To this end, we proceed to show that, for ${\mathcal{H}};\dim{\mathcal{H}}=2$, $id\otimes t(h><h|)$ is positive only if $h=f\otimes g$, $f,g\in{\mathcal{H}}$, so when $|h><h|\in{\mathcal{B}}({\mathcal{H}})_{+}\otimes{\mathcal{B}}({\mathcal{H}})_{+}$. Let $x=\sum_{m}w_{m}\otimes e_{m}\in{\mathcal{H}}\otimes{\mathcal{H}}$, and $h=\sum_{i}v_{i}\otimes e_{i}\in{\mathcal{H}}\otimes{\mathcal{H}}$. Then,

[TABLE]

[TABLE]

where both observations were used.

For, ${\mathcal{H}};\dim{\mathcal{H}}=2$ one gets

[TABLE]

while for ${\mathcal{H}};\dim{\mathcal{H}}=3$

[TABLE]

[TABLE]

We are looking for $\{v_{i}\}$ such that (3.28) ( respectively (3.29) ) are satisfied for arbitrary $\{w_{i}\}$.

Let’s consider (3.28) in detail. As $\{w_{i}\}$ was arbitrary we may change $w_{1}$ for $\lambda w_{1}$, $\lambda\in{\rm I\!R}$ to arrive at

[TABLE]

where $a=(w_{1},v_{1})(v_{1},w_{1})$, $b=(w_{2},v_{1})(v_{2},w_{1})+\overline{(w_{2},v_{1})(v_{2},w_{1})}$, $c=(w_{2},v_{2})(v_{2},w_{2})$. (3.30) holds if $c\leq\frac{b^{2}}{4a}$. As $\{w_{i}\}$ are arbitrary, we may take $w_{2}\bot v_{1}$. Then, $b=0$. So $c=0$. As $w_{1}$ is arbitrary, then $v_{1}=\lambda v_{2}$, where \lambda\in{\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}. However, this implies that $h$ is a simple tensor.

We now turn to the case $\dim{\mathcal{H}}=3$. It is easily seen that the above arguments fail when we have the 3 dimensional case. Fortunately, Choi gave an explicit construction of a matrix U\in(M_{3}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})\otimes M_{3}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}))^{+} such that U\neq M_{3}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})^{+}\otimes M_{3}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})^{+}, see [20]; another example was given in [21]. So, it is enough to see that the cone $C_{d}$ is not trivial for ${\mathcal{H}};\dim{\mathcal{H}}=3$.

3.5. k-positive maps

To complete this section we wish to examine $k$-positive maps. Our first observation is that we can apply the strategy described in subsection $3.1$ with the following modifications:

[TABLE]

and

[TABLE]

Consequently, we will examine

[TABLE]

To proceed with the analysis of $k$-positive maps we remind the reader that any $A\in M_{n}({\mathfrak{A}})$ has the unique representation:

[TABLE]

where $a_{ij}\in{\mathfrak{A}}$, and $\{\epsilon_{ij}\}_{i,j=1}^{n}$ is a matrix unit, ie $(\epsilon_{ij})^{*}=\epsilon_{ji}$, $\epsilon_{ij}\epsilon_{kl}=\delta_{jk}\epsilon_{il}$, $\sum_{i=1}^{n}\epsilon_{ii}=1$ cf [22], Appendix $T$. The above representation of $A\in M_{n}({\mathfrak{A}})$ combined with Kaplan’s arguments, see Proposition 1.1 in [23], plus obvious modifications, give an order isomorphism $\mathfrak{I}:M_{n}(B({\mathcal{H}})_{*})\to M_{n}(B({\mathcal{H}}))_{*}$ given by

[TABLE]

where $[\varrho_{ij}]\in M_{n}(B({\mathcal{H}})_{*})$, $[a_{ij}]\in M_{n}(B({\mathcal{H}}))$. Then, using the recipe analogous to that given by 3.4 one has the following prescription for selection of $k$-positive maps:

[TABLE]

To verify the above procedure, $(*)$, it is enough to note that [\varrho_{ij}]\in M_{k}(B({\mathcal{H}})_{*})^{+}=B({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}^{n}\otimes{\mathcal{H}})_{*}^{+} iff $\varrho_{ij}=\sum_{l=1}^{k}\sigma_{li}^{*}\sigma_{lj}$, where the unique representation (3.34) together with properties of trace class and Hilbert-Schmidt operators were used. Hence

[TABLE]

So $[T(a^{p}_{ij})]_{ij=1}^{k}\geq 0$ and $T$ is $k$-positive.

Finally we note that dropping the restriction for the upper bound of summation in (3.36) one gets

[TABLE]

where $l$ is an arbitrary natural number. It is easily seen that this is exactly the condition imposed by the cone $C_{cp}$. In other words, this observation sheds some new light on the origin of the definition of CP-maps.

4. Conclusions and remarks

The principal significance of the Example 3.7 is that it clarifies the appearance of non-decomposable maps, i.e. if $C_{d}\neq C_{p}$ then there is a room for non-decomposable maps. We have seen that both cones $C_{d}$ and $C_{p}$ are equal to each other for ${\mathcal{H}};\dim{\mathcal{H}}=2$.

The next important point to note here is that when studying the structure of the set of positive maps we restrict ourselves to a few special cones. In general, there could be other tensor cones. Thus, in principle, there could be other classes of interesting positive maps.

In [13] the structure of the set of positive maps was examined using the concept of elementary maps. Although this concept was to some extent vague, it was indicated that general structural properties should play a significant role. Here, we have seen that the synthesis of Grothendieck’s idea with the order on tensor product is playing a crucial role.

It is worth pointing out that the presented scheme is offering another approach to the concept of elementary maps. To see this, from now on we make the assumption: ${\mathfrak{A}}={\mathcal{B}}({\mathcal{H}})$ with ${\mathcal{H}};\dim{\mathcal{H}}=n<\infty$, cf [13]. We emphasize that this case is essential for Quantum Information Theory.

We first note that in the recipe (3.4) one can restrict oneself to extremal functionals. As an extreme point in a convex set can be treated as an elementary constituent, we are getting another concept of an elementary map. Define

[TABLE]

for any $\sum_{i=1}^{n}a_{i}\otimes b_{i}\in C_{\alpha}$, where $\sum_{i=1}^{n}a_{i}\otimes b_{i}\in{\mathcal{B}}({\mathcal{H}})\odot{\mathcal{B}}({\mathcal{H}})$, $C_{\alpha}$ is a tensor cone in ${\mathcal{B}}({\mathcal{H}})\otimes_{\pi}{\mathcal{B}}({\mathcal{H}})$, and $\tilde{T}$ is assumed to be an extremal functional. Consequently, elementary maps can be written as:

[TABLE]

for any $\sum_{i=1}^{n}a_{i}\otimes b_{i}\in C_{\alpha}$, where $\sum_{i=1}^{n}a_{i}\otimes b_{i}\in{\mathcal{B}}({\mathcal{H}})\odot{\mathcal{B}}({\mathcal{H}})$, $C_{\alpha}$ is a tensor cone in ${\mathcal{B}}({\mathcal{H}})\otimes_{\pi}{\mathcal{B}}({\mathcal{H}})$ and $h\in{\mathcal{H}}\otimes{\mathcal{H}}$.

The interest of this remark is that it provides a recipe for constructing “elementary” maps with specified positivity with respect to the selected cone $C_{\alpha}$. Furthermore, in an application of the above scheme to $2$-positive maps one recoves the concept of atomic maps.

The next important point to note here is that the relation between a cp-map $T$ and the asociated $C_{cp}$-positive functional $\tilde{T}$ can be considered as the starting point for a generalization of Krauss (Stinespring) decomposition, for the corresponding order structures see Section 1.4 in [24].

To see this we recall that \left(M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})\otimes_{\pi}M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})_{*}),C_{cp}\right) is an involutive Banach algebra and the cone $C_{cp}$ can be used to define the concept of states, cf Chapter I in [15]. Therefore, there is a posibility to speak about GNS construction. Then employing the decomposition theory, as it was given in Chapter 4, in [25], see Section 4.1.1, one can decompose the $C_{cp}$-positive functional $\tilde{T}$, so also the corresponding map $T$. In other words, we are getting a generalized decomposition of a cp-map.

Our next remark is clarifying a general form of a positive map (for finite dimensional case). We note that for any self-adjoint (hermitan) functional $\tilde{T}$ on a $C^{*}$-algebra ${\mathfrak{A}}$ there is a unique pair $\tilde{T}_{+}$ and $\tilde{T}_{-}$ of positive functionals such that $\tilde{T}=\tilde{T}_{+}-\tilde{T}_{-}$ , cf Theorem 3.2.5 in [26]. We can use this for unique decomposition of a self-adjoint functional on M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})\otimes M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}). Subsequently, one can transfer this decomposition to M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})\otimes_{\pi}M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}). Thus, we arrive at: a $C_{p}$-positive functional $\tilde{T}$ has unique decomposition $\tilde{T}=\tilde{T}_{+}-\tilde{T}_{-}$ , where $\tilde{T}_{+}$, $\tilde{T}_{-}$ are $C_{cp}$-positive functionals.

Consequently we got, using our scheme, the following decomposition of a positive map $T$; $T=T_{+}-T_{-}$, where $T_{+}$, $T_{-}$ are cp-maps. To elaborate and clarify the decomposition formula of $T$ we need some preliminaries, cf. [27].

Let $a_{1},...,a_{k}$ ,and $c_{1},...,c_{i}$ be in ${\mathcal{B}}({\mathcal{H}},{\mathcal{K}})$ (${\mathcal{H}},{\mathcal{K}}$ stand for Hilbert spaces). If for each $x\in{\mathcal{H}}$, there exists an $l\times k$ complex matrix $\{(\alpha_{i,j}(x))\}$ such that

[TABLE]

$(c_{1},...,c_{l})$ is said to be a locally linear combination of $(a_{1},...,a_{k})$. If coefficients $\{\alpha_{i,j}(x)\}$ can be taken in such way that the norm $\|\alpha_{i,j}(x)\|\leq 1$, for every $x$, then $(c_{1},...,c_{l})$ is said to be a contractive linear combination of $(a_{1},...,a_{k})$. Employing Hou’s result (see Corollary 2.6 in [27]) we arrive at

Proposition 4.1**.**

A positive map T:M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}})\otimes M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}) can be uniquely written in the form $T=T_{+}-T_{-}$ where $T_{+}$, $T_{-}$ are cp-maps. In particular $T_{+}(a)=\sum_{r}c_{r}ac_{r}^{*}$ where c_{r}\in M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}). Furthermore, there exists \{d_{p}\}\subset M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}) such that $(d_{1},...,d_{l})$ is a contractive locally linear combination of $(c_{1},...,c_{k})$ and

[TABLE]

for all a\in M_{n}({\hbox{\hbox to0.0pt{\kern 6.33339pt\vrule height=6.14993pt\hss}\hbox{ \displaystyle\rm C}}}).

Finally we note that in [11] the identification of the injective cone $C_{i}$ with the cone $C_{cp}$ was incorrect.

5. Acknowledgments

The author would like to express his thanks to Louis E. Labuschagne and Marcin Marciniak for several helpful comments. Furthermore, he wishes to thank the University of Gdansk where a part of the research was done.