# Cloning by positive maps in von Neumann algebras

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## Abstract

We investigate cloning in the general operator algebra framework in arbitrary dimension assuming only positivity instead of strong positivity of the cloning operation, generalizing thus results obtained so far under that stronger assumption. The weaker positivity assumption turns out quite natural when considering cloning in the general \(C^*\)-algebra framework.

## Keywords

Cloning states Positive maps von Neumann algebras## Mathematics Subject Classification (2000)

Primary 81R15 Secondary 81P50 46L30## 1 Introduction

Cloning and broadcasting of quantum states has recently become an important topic in Quantum Information Theory. Since its first appearance in [5, 14] in the form of a no-cloning theorem it has been investigated in various settings. The most interesting ones are the Hilbert space setup considered in [3, 8], and the setup of generic probabilistic models considered in [1, 2]. A common feature of these approaches consists in restricting attention to the finite-dimensional models; moreover, in the Hilbert space setup the map defining cloning or broadcasting is assumed to be completely positive. In [6] cloning and broadcasting are investigated in the general operator algebra framework, i.e. instead of the full algebra of all linear operators on a finite-dimensional Hilbert space an arbitrary von Neumann algebra on a Hilbert space of arbitrary dimension is considered; moreover, the cloning (broadcasting) operation is assumed to be a Schwarz (called also *strongly positive*) map instead of completely positive. The present paper can be viewed as a supplement to [6]; we follow the same approach weakening further the assumption on positivity of the cloning operation, namely, we assume only that it is positive. This weaker assumption is all we can hope for while considering the general problem of cloning in \(C^*\)-algebras (cf. [7]), thus the main theorem of the present paper (Theorem 8) is of importance for cloning in \(C^*\)-algebras. However, in our approach interesting results are obtained only for cloning, the problem of broadcasting in such a setup is still an open question.

It is probably worth mentioning that although cloning and broadcasting have their origins in quantum information theory, they are nevertheless purely mathematical objects concerning states on some \(C^*\)- or \(W^*\)-algebras, and thus their investigation is of independent mathematical interest. This is the approach taken in the present paper—we do not refer to any physical notions, however it is still possible (and hoped for) that the results obtained can find some physical applications.

The main results of this work are as follows. It is shown that all states cloneable by an operation are extreme points of the set of all states broadcastable by this operation, and a description of some algebra associated with the cloneable states is given. Moreover, for an arbitrary subset \(\Gamma \) of the normal states of a von Neumann algebra it is proved that the states in \(\Gamma \) are cloneable if and only if they have mutually orthogonal supports—the result obtained in [6] under the assumption that the cloning operation is a Schwarz map. Finally, the problem of uniqueness of the cloning operation is considered.

## 2 Preliminaries and notation

Let \(\fancyscript{M}\) be an arbitrary von Neumann algebra with identity \({\small 1}\!\!1\) acting on a Hilbert space \(\mathcal {H}\). The predual \(\fancyscript{M}_*\) of \(\fancyscript{M}\) is a Banach space of all *normal*, i.e. continuous in the \(\sigma \)-weak topology linear functionals on \(\fancyscript{M}\).

*state*on \(\fancyscript{M}\) is a bounded positive linear functional \(\rho :\fancyscript{M}\rightarrow \mathbb {C}\) of norm one. For a normal state \(\rho \) its

*support*, denoted by \(\mathrm{s }(\rho )\), is defined as the smallest projection in \(\fancyscript{M}\) such that \(\rho (\mathrm{s }(\rho ))=\rho ({\small 1}\!\!1)\). In particular, we have

*faithful*if for each positive element \(x\in \fancyscript{M}\) from the equality \(\rho _{\theta }(x)=0\) for all \(\theta \in \Theta \) it follows that \(x=0\). It is seen that the faithfulness of this family is equivalent to the relation \(e={\small 1}\!\!1\); moreover, if \(\rho _{\theta }(exe)=0\) for all \(\theta \in \Theta \) and \(exe\geqslant 0\) then \(exe=0\).

By a \(W^*\)-algebra of operators acting on a Hilbert space we shall mean a \(C^*\)-subalgebra of \(\mathbb {B}(\mathcal {H})\) with identity, closed in the weak-operator topology. A typical example (and in fact the only one utilized in the paper) is the algebra \(p\fancyscript{M}p\), where \(p\) is a projection in \(\fancyscript{M}\). For arbitrary \(\mathcal {R}\subset \mathbb {B}(\mathcal {H})\) we denote by \(W^*(\mathcal {R})\) the smallest \(W^*\)-algebra of operators on \(\mathcal {H}\) containing \(\mathcal {R}\).

A projection \(p\) in a \(W^*\)-algebra \(\fancyscript{M}\) is said to be *minimal* if it majorizes no other nonzero projection in \(\fancyscript{M}\). A \(W^*\)-algebra \(\fancyscript{M}\) is said to be *atomic* if the supremum of all minimal projections in \(\fancyscript{M}\) equals the identity of \(\fancyscript{M}\).

*Jordan product*\(x\circ y\) as follows

*algebra*if it is weak-operator closed, contains an identity \(p\), i.e. \(pa=ap=a\) for each \(a\in \mathcal {A}\), and is closed with respect to the Jordan product, i.e. for any \(a,b\in \mathcal {A}\) we have \(a\circ b\in \mathcal {A}\).

Let \(\fancyscript{M}\) and \(\mathcal {N}\) be \(W^*\)-algebras. A linear map \(T:\fancyscript{M}\rightarrow \mathcal {N}\) is said to be *normal* if it is continuous in the \(\sigma \)-weak topologies on \(\fancyscript{M}\) and \(\mathcal {N}\), respectively. It is called *unital* if it maps the identity of \(\fancyscript{M}\) to the identity of \(\mathcal {N}\).

*multiplicative domain*of \(T\) as

*broadcasting*and

*cloning*of states.

*channel*. (This terminology is almost standard, because by a “channel” is usually meant the map \(K\) as above, however, with some additional assumption of complete- or at least two-positivity.) A state \(\rho \in \fancyscript{M}_*\) is

*broadcast*by channel \(K_*\) if \((\Pi _i K_*)(\rho )=\rho ,\,i=1,2\); in other words, \(\rho \) is broadcast by \(K_*\) if for each \(x\in \fancyscript{M}\)

*broadcastable*if there is a channel \(K_*\) that broadcasts each member of this family.

A state \(\rho \in \fancyscript{M}_*\) is *cloned* by channel \(K_*\) if \(K_*\rho =\rho \otimes \rho \). A family of states is said to be *cloneable* if there is a channel \(K_*\) that clones each member of this family.

## 3 Broadcasting

The discussion in this section has an auxiliary character and is in main part a repetition for *positive* maps of the reasoning from [6] performed there for *Schwarz* maps. Its main purpose is to analyze some properties of broadcasting channels employed in Sect. 4.

*all*normal states broadcastable by \(K_*\). We have \(\Gamma \subset \mathcal {B}(K_*)\), thus our main object of interest will be the set \(\mathcal {B}(K_*)\). In the rest of this section we assume that we are given a fixed channel \(K_*\). Define maps \(L,R:\fancyscript{M}\rightarrow \fancyscript{M}\) as

### **Lemma 1**

### *Proof*

### **Lemma 2**

- (i)
For each \(x\in \mathcal {F}(L^{(p)})\) we have \(x\otimes {\small 1}\!\!1\in \mathcal {A}\),

- (ii)
For each \(x\in \mathcal {F}(R^{(p)})\) we have \({\small 1}\!\!1\otimes x\in \mathcal {A}\).

### *Proof*

### **Lemma 3**

### **Proposition 4**

- (i)
\(\varphi \) belongs to \(\mathcal {B}(K_*)\),

- (ii)
\(\varphi \) is \(\mathfrak {S}\)-invariant,

- (iii)
\(\varphi =\varphi \circ \mathbb {E}\).

### *Proof*

(i)\(\Longrightarrow \)(ii). It follows from Lemma 1.

It turns out that the map \(K^{(p)}\) has a special form on the tensor product algebra \(\mathcal {F}(\mathfrak {S})\overline{\otimes }\mathcal {F}(\mathfrak {S})\) (this is the weak closure of the algebra of *operators* \(\big \{\sum _{i=1}^mx_i\otimes y_i:x_i,y_i\in \mathcal {F}(\mathfrak {S})\big \}\) acting on \(\mathcal {H}\otimes \mathcal {H}\)).

### **Proposition 5**

### *Proof*

The next result is well-known in the case \(p={\small 1}\!\!1\) (cf. [13, Lemma1]). Its proof for arbitrary \(p\) is similar, so we omit it.

### **Lemma 6**

For each \(\rho \in \mathcal {B}(K_*)\) we have \(s(\rho )\in \mathcal {F}(\mathfrak {S})\).

## 4 Cloning

### **Theorem 7**

- 1.
The states in \(\mathcal {C}(K_*)\) have mutually orthogonal supports, and are extreme points of \(\mathcal {B}(K_*)\).

- 2.
The algebra \(e\mathcal {F}(\mathfrak {S})e\) is an atomic abelian \(W^*\)-subalgebra of \(\mathcal {F}(\mathfrak {S})\), generated by \(\{\mathrm{s }(\rho ):\rho \in \mathcal {C}(K_*)\}\), and such that \(e\mathcal {F}(\mathfrak {S})e\subset \mathcal {F}(\mathfrak {S})'\) — the commutant of \(\mathcal {F}(\mathfrak {S})\).

### *Proof*

- 1.Since \(\mathcal {C}(K_*)\subset \mathcal {B}(K_*)\) we may use the analysis of the preceding section. In particular, we adopt the setup and notation introduced there. For each \(\rho \in \mathcal {C}(K_*)\) we have \(K_*\rho =\rho \otimes \rho \), so taking into account Proposition 5 we obtain the equalityfor all \(x,y\in \mathcal {F}(\mathfrak {S})\). The equality above yields that for each \(z\in \mathcal {F}(\mathfrak {S})\) and any \(\rho \in \mathcal {C}(K_*)\) we have$$\begin{aligned} \rho (x)\rho (y)&=\rho \otimes \rho (x\otimes y)=(K_*\rho )(x\otimes y)=\rho (K(x\otimes y))\nonumber \\&=\rho (pK(x\otimes y)p)=\rho (K^{(p)}(x\otimes y))=\rho (x\circ y) \end{aligned}$$(7)Let \(x\) be an arbitrary selfadjoint element of \(\mathcal {F}(\mathfrak {S})\). For each \(\rho \in \mathcal {C}(K_*)\) we have by (8)$$\begin{aligned} \rho (z^2)=\rho (z)^2. \end{aligned}$$(8)which yields the equality$$\begin{aligned} \rho \big ((x-\rho (x){\small 1}\!\!1)^2\big )=\rho \big (x^2-2\rho (x)x-\rho (x)^2{\small 1}\!\!1\big ) =\rho (x^2)-\rho (x)^2=0, \end{aligned}$$i.e.$$\begin{aligned} \mathrm{s }(\rho )\big (x-\rho (x){\small 1}\!\!1\big )^2\mathrm{s }(\rho )=0, \end{aligned}$$Since for an element \(x\) of a \(JW^*\)-algebra \(x+x^*\) and \(x-x^*\) are also elements of this algebra the equality above holds for all \(x\in \mathcal {F}(\mathfrak {S})\) as well. Let \(\rho \) and \(\varphi \) be two distinct states from \(\mathcal {C}(K_*)\). Then by (9)$$\begin{aligned} \mathrm{s }(\rho )x=\rho (x)\mathrm{s }(\rho ). \end{aligned}$$(9)which after taking adjoints yields the equality$$\begin{aligned} \mathrm{s }(\rho )\mathrm{s }(\varphi )=\rho (\mathrm{s }(\varphi ))\mathrm{s }(\rho )\qquad \text {and}\qquad \mathrm{s }(\varphi )\mathrm{s }(\rho )=\varphi (\mathrm{s }(\rho ))\mathrm{s }(\varphi ), \end{aligned}$$Consequently,$$\begin{aligned} \mathrm{s }(\rho )\mathrm{s }(\varphi )=\mathrm{s }(\varphi )\mathrm{s }(\rho ). \end{aligned}$$showing that either \(\mathrm{s }(\rho )=\mathrm{s }(\varphi )\) or \(\mathrm{s }(\rho )\) and \(\mathrm{s }(\varphi )\) are orthogonal. If \(\mathrm{s }(\rho )=\mathrm{s }(\varphi )\) then on account of (9) we would have$$\begin{aligned} \rho (\mathrm{s }(\varphi ))\mathrm{s }(\rho )=\varphi (\mathrm{s }(\rho ))\mathrm{s }(\varphi )=\mathrm{s }(\varphi )\mathrm{s }(\rho ), \end{aligned}$$for each \(x\in \mathcal {F}(\mathfrak {S})\), i.e.$$\begin{aligned} \rho (x)\mathrm{s }(\rho )=\mathrm{s }(\rho )x=\mathrm{s }(\varphi )x=\varphi (x)\mathrm{s }(\varphi )=\varphi (x)\mathrm{s }(\rho ) \end{aligned}$$Let \(\mathbb {E}\) be the projection onto \(\mathcal {F}(\mathfrak {S})\) defined by formula (5). We have \(\rho =\rho \circ \mathbb {E}\) and \(\varphi =\varphi \circ \mathbb {E}\), thus equality (10) yields$$\begin{aligned} \rho |\mathcal {F}(\mathfrak {S})=\varphi |\mathcal {F}(\mathfrak {S}). \end{aligned}$$(10)contrary to the assumption that \(\rho \) and \(\varphi \) are distinct. Consequently, \(\rho \) and \(\varphi \) have orthogonal supports. Now take an arbitrary \(\rho \in \mathcal {C}(K_*)\), and assume that$$\begin{aligned} \rho =\rho \circ \mathbb {E}=\varphi \circ \mathbb {E}=\varphi \end{aligned}$$for some \(0<\lambda <1\) and \(\varphi _1,\varphi _2\in \mathcal {B}(K_*)\). Then$$\begin{aligned} \rho =\lambda \varphi _1+(1-\lambda )\varphi _2, \end{aligned}$$showing that \(\varphi _1(\mathrm{s }(\rho ))=\varphi _2(\mathrm{s }(\rho ))=1\), which means that \(\mathrm{s }(\varphi _1),\mathrm{s }(\varphi _2)\leqslant \mathrm{s }(\rho )\). From equality (9) we obtain for \(x\in \mathcal {F}(\mathfrak {S})\)$$\begin{aligned} 1=\rho (\mathrm{s }(\rho ))=\lambda \varphi _1(\mathrm{s }(\rho ))+(1-\lambda )\varphi _2(\mathrm{s }(\rho )), \end{aligned}$$giving the relation$$\begin{aligned} \varphi _{1,2}(x)=\varphi _{1,2}(\mathrm{s }(\rho )x)=\rho (x)\varphi _{1,2}(\mathrm{s }(\rho ))=\rho (x), \end{aligned}$$We have \(\rho =\rho \circ \mathbb {E}\) and \(\varphi _{1,2}=\varphi _{1,2}\circ \mathbb {E}\), and thus$$\begin{aligned} \rho |\mathcal {F}(\mathfrak {S})=\varphi _{1,2}|\mathcal {F}(\mathfrak {S}). \end{aligned}$$showing that \(\rho \) is an extreme point of \(\mathcal {B}(K_*)\).$$\begin{aligned} \rho =\rho \circ \mathbb {E}=\varphi _{1,2}\circ \mathbb {E}=\varphi _{1,2}, \end{aligned}$$
- 2.From equality (9) we obtain that \(\mathrm{s }(\rho )\in \mathcal {F}(\mathfrak {S})'\) for all \(\rho \in \mathcal {C}(K_*)\), and thatfor all \(x\in \mathcal {F}(\mathfrak {S})\), which means that \(e\mathcal {F}(\mathfrak {S})=e\mathcal {F}(\mathfrak {S})e\) is a \(W^*\)-algebra generated by \(\{\mathrm{s }(\rho ):\rho \in \mathcal {C}(K_*)\}\). By virtue of Lemma 6 we have \(s(\rho )\in \mathcal {F}(\mathfrak {S})\) for each \(\rho \in \mathcal {C}(K_*)\) thus \(e\mathcal {F}(\mathfrak {S})e\) is a subalgebra of \(\mathcal {F}(\mathfrak {S})\), and \(e\mathcal {F}(\mathfrak {S})e\subset \mathcal {F}(\mathfrak {S})'\). Finally, \(\mathrm{s }(\rho )\) is a minimal projection in \(e\mathcal {F}(\mathfrak {S})e\). Indeed, for any projection \(f\in \mathcal {F}(\mathfrak {S})\) equality (8) yields$$\begin{aligned} ex=\sum _{\rho \in \mathcal {C}(K_*)}\rho (x)\mathrm{s }(\rho ), \end{aligned}$$Now if \(q\) is a projection in \(e\mathcal {F}(\mathfrak {S})e\) such that \(q\leqslant \mathrm{s }(\rho )\) and \(q\ne \mathrm{s }(\rho )\) we cannot have \(\rho (q)=1\), thus \(\rho (q)=0\), and the faithfulness of \(\rho \) on the algebra \(\mathrm{s }(\rho )\fancyscript{M}\mathrm{s }(\rho )\) yields \(q=0\). Consequently, algebra \(e\mathcal {F}(\mathfrak {S})e\) being generated by minimal projections is atomic. \(\square \)$$\begin{aligned} \rho (f)=0\text { or }1. \end{aligned}$$

### **Theorem 8**

- (i)
\(\Gamma \) is cloneable,

- (ii)
The states in \(\Gamma \) have mutually orthogonal supports.

### *Proof*

The implication (i)\(\Longrightarrow \)(ii) follows from Theorem 7.

The above theorem yields an interesting corollary. Namely, a usual assumption about a channel is its complete (or at least two-) positivity, the assumption which we have tried to avoid in this work. It turns out that a stronger form of positivity gives the same cloneable states.

### **Corollary**

*Let*\(\Gamma \)

*be an arbitrary subset of normal states on a von Neumann algebra*\(\fancyscript{M}\).

*The following conditions are equivalent*

- (i)
\(\Gamma \)

*is cloneable by a completely positive channel,* - (ii)
\(\Gamma \)

*is cloneable by a positive channel.*

### *Proof*

Finally, let us say a few words about the uniqueness of the cloning operation.

### **Proposition 9**

*(*it turns out that in our case \(\mathbb {E}\) is actually a conditional expectation

*)*.

### *Proof*

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