PERRON-FROBENIUS THEORY FOR POSITIVE MAPS ON TRACE IDEALS. Dedicated to S. Doplicher and J.E. Roberts on the occasion of their 60th birthday

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1 PERRON-FROBENIUS THEORY FOR POSITIVE MAPS ON TRACE IDEALS R. SCHRADER ABSTRACT. This article provides sufficient conditions for positive maps on the Schatten classes Â Ô ½ Ô ½ of bounded operators on a separable Hilbert space such that a corresponding Perron-Frobenius theorem holds. With applications in quantum information theory in mind sufficient conditions are given for a trace preserving, positive map on  ½, the space of trace class operators, to have a unique, strictly positive density matrix which is left invariant under the map. Conversely to any given strictly positive density matrix there are trace preserving, positive maps for which the density matrix is the unique Perron-Frobenius vector. Dedicated to S. Doplicher and J.E. Roberts on the occasion of their 60th birthday 1. INTRODUCTION In the theory of quantum information the transmission through noisy channels plays an important role. Usually it is described by what physicists either call a quantum operation (see e.g. [25]) or a stochastic map ([1], see also [19]) or a super-operator (see e.g. [28]) and what mathematicians call a completely positive (trace preserving) map. In mathematics positive maps were first studied by Kadison in the context of -algebras [18] and completely positive maps by Stinespring [33]. In quantum physics it was first introduced by Haag and Kastler, who called it an operation [15]. They were then studied in more detail in [16]. [21, 22] contains an extensive discussion, how this concept naturally arises in quantum physics. Because they preserve positivity such maps naturally fit into a context where it makes sense to ask about a formulation of a corresponding Perron-Frobenius theory. As the name indicates, the earliest results relating positivity of a finite dimensional linear map to the non-degeneracy of an eigenvalue and the positivity of the resulting (Perron-Frobenius) eigenvector is due to Perron and Frobenius [27, 8, 9]. The first extension to the infinite dimensional was given by Jentzsch [17]. Since then there has been an extensive development (see e.g. [3, 23, 30]). The interest in quantum physics and quantum field theory originates from the observation, that very often the ground state of a quantum system with Hamilton operator À ¼ is nondegenerate and nowhere vanishing. Indeed, under suitable circumstances such a ground state can be viewed as the Perron-Frobenius vector for ÜÔ ØÀ Ø ¼, an observation first made by Glimm and Jaffe [12] (see also [29, 13] for an overview and with references to other articles). The Perron-Frobenius theory and related topics for positive linear maps on linear spaces of operators was first studied in [7, 36, 14, 2], where in the first article the space of operators was the -algebra of all linear operators on a finite dimensional Hilbert space. In the second and third articles the discussion was extended to the infinite dimensional case and in the fourth article the analysis was carried out in the context of a von Neumann algebra. The present article can be viewed as an extension of the discussion in [7] to the infinite dimensional case. Indeed, we will consider positive maps on the Schatten classes Â Ô ½ Ô ½ and provide a Perron-Frobenius theory on such spaces. As in the usual context the notions of ergodicity and positivity improving will play an essential role. Since density matrices are elements of  ½ our main focus will be on the case Ô ½. In particular we will show that any compact, ergodic, completely positive and trace preserving map in  ½ has a unique density matrix invariant under this map (i.e. the eigenvalue equals 1) and which therefore can be viewed as the Perron-Frobenius vector for this map Mathematics Subject Classification. Primary 47B10, 47B65; Secondary 81P15, 82B10. schrader@physik.fu-berlin.de, Supported in part by DFG SFB 288 Differentialgeometrie und Quantenphysik. 1

2 The article is organized as follows. In Section 2 we will briefly review the concepts needed for our discussion. In Section 3 we will prove a Perron-Frobenius theorem (non-degeneracy of a certain eigenvalue), provided such an eigenvalue exists, a typical condition needed in infinite dimensional contexts. Section 4 will provide sufficient conditions for such an eigenvalue to exist. In Section 5 we will give examples. Also we will prove a converse in the sense that to any density matrix ¼ there are completely positive maps for which this density matrix is the Perron-Frobenius eigenvector. 2. POSITIVE AND COMPLETELY POSITIVE MAPS We start by recalling and establishing some definitions and facts related to trace ideals and to the concept of positive and completely positive maps ( for more details on trace ideals see e.g. [31, 32] and on positive and completely positive maps e.g. [33, 34, 5, 6, 35, 26]). Let À be any complex, separable Hilbert space with scalar product denoted by. By Àµ we denote the -algebra of all bounded operators on À equipped with the norm. By Á ¾ Àµ we denote the identity map on À. For any ¾ Àµ we set µ ½¾ ¾ Àµ where denotes the adjoint. Also we let ÌÖ be the trace operation on À. For ½ Ô ½ let Â Ô Â Ô Àµ be the Schatten class consisting of all elements ¾ Àµ such that Ô is trace class. Â Ô is equipped with the norm Ô ÌÖ Ô µµ ½Ô making each Â Ô a complete Banach space. One has ¾ Â Ô if ¾  Ô, i.e. the space Â Ô is self-adjoint in Àµ, such that Ô Ô. Also we set  ½ Àµ again with the same norm, i.e. ½. We will frequently make use of the trivial identity Ô Ô valid for all ¾ Â Ô and all ½ Ô ½. Via the trace Â Õ is the dual of Â Ô with ½Ô ½Õ ½ for all ½ Ô ½, i. e. each continuous linear functional on Â Ô is given by an ¾ Â Õ in the the form ÌÖ µ ¾  Ô. In particular the spaces Â Ô are reflexive for all ½ Ô ½. We prefer to use the adjoint in describing linear functionals since for Ô ¾, in which case  ¾ is the space of Hilbert-Schmidt operators on À, the scalar product making it a Hilbert space is just given by ¾ ÌÖ µ. Also one has the Hölder inequality ÌÖ µ Õ Ô. Let Àµ ½ be the closed positive cone of all elements ¼ in Àµ. As usual we write ¼ and ¼ if the relations ³ ³ ¼ and ³ ³ ¼ respectively hold for all ¼ ³ ¾ À. Correspondingly we write (or ) and (or ) if ¼ and ¼ respectively. Obviously ¼ or ¼ implies ¼. As is common, is said to be positive definite if ¼. Set Ô Ô Àµ  Ô, a closed set in  Ô. Also ¼ in Â Ô if and only if ÌÖ µ ¼ for all ¼ ¾  Õ. Correspondingly ¼ in Â Ô if and only if ÌÖ µ ¼ for all ¼ ¼ in Â Õ ½Ô ½Õ ½ ½ Ô ½µ. In this sense the cones Ô and Õ are dual to each other. The closed set ½½ ¾ ½ ÌÖ ½ ½ in  ½ is the set of all density matrices in À. By definition a positive map in Â Ô is a linear map from  ½ into itself, which leaves Ô invariant. is called -positive if the induced map Å Á in Â Ô Å À µ also leaves the corresponding cone Ô À Å À µ of non-negative elements invariant. Here À is any Hilbert space of dimension ½ ½. Obviously if is -positive, then it is also ¼ -positive for any ½ ¼ as is ¼. If and ¼ are both -positive on Â Ô then so is their composite Æ ¼ and their sum ¼. So the -positive maps in Â Ô form a cone in the linear space of all linear maps in  Ô. If is -positive for all, then is called completely positive. Thus the map «««for any «¾ È Àµ is completely positive on Â Ô for all ½ Ô ½ as is any finite linear combination ««with «¾ Àµ. If «has an inverse «½ in Àµ then «½ «½. Also «Æ «¼ ««¼ and «¾ «for ««¼ ¾ Àµ and ¾. As it turns out at least for Ô ½ the induced linear map ÅÁÀ ¼ from  Խ Å À ¼ µ then also leaves the corresponding cone invariant for any separable Hilbert space À ¼ (see e.g. [22]). Since density matrices are in  ½, the case Ô ½ is of most interest in quantum physics. Then there are good physical reasons to consider completely positive maps rather than only positive maps (see e.g, [22]). Although the following lemma is well known in similar contexts, we still will provide the short proof.

3 Lemma 2.1. The following relation holds if the map in Â Ô ½ Ô ½µ is positive. (1) µ µ ¾ Â Ô Proof. We first consider the case when is self-adjoint, i.e.. Write with ¾ Ô ¼ and. Here are the positive and negative parts of, obtainable from the spectral representation of or more explicitly as ½¾ µ. Obviously Ô Ô. Since µ ¼, µ µ µ is selfadjoint. For arbitrary ¾ Àµ write ¾ Àµ with ½¾ µ ½¾ µ, such that ¾ Àµ are self-adjoint with ¾ Â Ô whenever ¾  Ô. More precisely we have the a priori bound Ô Ô Ô Ô. This gives the decomposition and hence by the linearity of (2) µ µ µ µ µ µ µ µ µ µ µ µ µ µ with µ Ô µ µ Ô µ µ Ô µ µ Ô µ µ Ô. In particular (2) shows that indeed (1) holds. We also note the following. The relation for ¾ Â Ô implies µ µ µ, whenever the map in Â Ô is positive. This in turn gives (3) µ µ valid for any ¾ Â Ô and any positive map in  Ô. We do not know whether (or when) this relation continues to hold when the condition is dropped (see, however, (23) below for a weaker result, which will suffice for our purposes). Observe by comparison that for any matrix Ë with non-negative entries, the classical context for the Perron-Frobenius theorem, one has ËÞ ËÞ for Þ ¾. Here Þ is the vector whose components are the absolute values of the corresponding components of Þ. Also for real vectors Ü Ü and Ý Ý by definition Ü Ý if and only if Ü Ý holds for all. Furthermore one does not necessarily have µ µ for any ¾  Ô. However, since implies µ µ µ, the inequalities µ µ hold. Lemma 2.2. Any positive map in Â Ô ½ Ô ½µ is continuous, i.e. it satisfies Ô ½. (4) Here we denote by Ô the norm of any continuous linear map in  Ô, such that In particular (5) Ô ½ if is completely positive on  ½ (see e.g. [6, 22]). ÙÔ µ Ô Ô½ ÙÔ ÌÖ µµ ¾ ½½ Proof. We adapt a standard proof (see e.g. [26], p. 19) used in the context of positive maps on non-unital -algebras. We first claim that it suffices to prove boundedness on Ô. Indeed, this follows from the decomposition (2) and the related bounds. Assume that is not bounded on Ô. Then there are ¾ Ô with Ô ½ and µ Ô. Let È ½¾ ¾ Ô, such that ¼ ½ ¾ holds for all. We need the fact that (6) Ô ¼ Ô holds for any ¼ ¼ ¾ Ô. The case Ô ½ is trivial. When ½ Ô ½ take ³ to be a complete orthonormal basis in À diagonalizing. Then we have Ô Ô ³ ³ Ô ³ ¼Ô ³ ¼ Ô Ô ³ ¼ ³ Ô

4 proving (6). Here we have used the estimate (7) ³ ³ Ô ³ Ô ³ valid for any ¾ Àµ, any normalized ³ ¾ À and any ½ Ô (see e.g. [32], p.21). Since is positive (6) gives ½ ¾ µ Ô µ Ô, which is a contradiction. For any continuous linear map in Â Ô ½ Ô ½ let be the adjoint continuous linear map from Â Õ ½Õ ½Ô ½ into itself given by the relation ÌÖ µµ µ ÌÖ µµ for all ¾ Â Õ and ¾  Ô. In particular holds whenever ½ Ô ½. Also ««with ««½ «µ by the cyclicity of the trace. By definition a continuous linear map in  ½ is called trace preserving if ÌÖ µµ ÌÖ µ holds for all ¾  ½ and this in turn is equivalent to the relation Áµ Á. Obviously trace preserving maps leave the set ½½ of density matrices invariant. If in  ½ is completely positive and trace invariant, then by (5) ½ ½. All completely positive maps on  ½ have the Kraus representation [22], a consequence of a theorem of Stinespring [33] (see also [5] for a proof in the finite dimensional case): Given such a there is an at most denumerable set of elements ««¾Æ in Àµ satisfying (8) ¾Ã ««½ Á for any finite subset Ã Æ such that «, again with (9) «µ ¾Æ ««which now may be an infinite sum. This representation is not unique. Conversely each such «is completely positive. More precisely «is defined as follows. Let «Æ «½ «Æ µ. Then the «Æ form a Cauchy È sequence with respect to the norm ½ and «is defined as the limit (see e.g. [22]). If ««Á, such that ½ ½, then «is trace preserving in  ½. In the finite dimensional case Ñ À ½µ the representation may always be chosen such that the index runs through a finite set of the order at most Ñ Àµ ¾. An important and interesting feature of completely positive maps in the context of quantum physics is that they not necessarily map pure states, i.e. one-dimensional orthogonal projections, into pure states. Again we have the following representation for the adjoint of «(10) «µ ««¾Æ i.e. ««. So «is trace preserving if and only if «Áµ Áand then «½ ½. Observe that to any completely positive with ½ ½ or equivalently Áµ Á ( this condition is again natural in the context of quantum physics, see e.g. [22]) we may in a natural way associate a trace preserving, completely positive map given as µ µ ««where «Á Áµµ ½¾ ¼. An example to which we shall return below is when the «s are orthogonal projection operators which are pairwise orthogonal, i.e. satisfy (11) ««Æ «Æ «È Then «Áµ Á if and only if the «provide a decomposition of unity, i.e. if «¾Æ Á holds. As already È remarked «is then also trace preserving. More generally if the «are all selfadjoint with ¾Æ «¾ Á, then «is trace preserving. We remark that there are some situations (see e.g. [20] and which actually was the motivation for the present discussion) where one starts with completely positive maps of the following form. Let Å µ be a measure space, i.e. is countably additive. Suppose we are given a measurable map «µ from Å into Àµ such that (12) «µ ¾ µ ½

5 We then set (13) µ «µ«µ µ such that its adjoint takes the form (14) µ «µ «µ µ The condition corresponding to (8) is given as (15) Áµ «µ «µ µ Á for some ¼ ½. To cast in the form (9) such that (15) gives (8), let be any orthonormal basis in the separable Hilbert space Ä ¾ Å µ and set «µ«µ µ As an example one may choose Å to be a compact group, its Haar measure and «µ a unitary representation. Then the image of consists of all trace class operators which commute with all «µ. Also is trace preserving and an idempotent, i.e. ¾. Moreover is a conditional expectation when Ñ À ½, i.e. µµ µ µ µµ holds for each. 3. THE PERRON-FROBENIUS THEOREM Before we turn to a discussion of the Perron-Frobenius theorem we make some general remarks on Ô µ for the map in  Ô, in particular when is positive. Here Ô µ is the spectrum of, i.e. the set of all for which µ does not have a bounded inverse in  Ô. Let Ö Ô µ be the spectral radius of any bounded linear map in  Ô, i.e. (16) Ö Ô µ ÐÑ ½ ½ Ô Ô By a celebrated general result of Gelfand [11], one has ÙÔ Ô µ Ö Ô µ. If the map in  ½ is trace preserving and completely positive then Ö ½ µ ½ ½. Indeed, is then also completely positive and trace preserving such that by (5) ½ ½ holds for all. Lemma 2.1 allows us to draw the following conclusions on Ô µ for positive maps in  Ô. First we observe that in general Ô µ is not contained in the real axis. In fact, in the finite dimensional case Ô µ Ê if and only if and in the infinite dimensional case the same statement is valid when Ô ¾. Let Ô µ be the resolvent set of the map in  Ô, i.e. the complement in of Ô µ. If ¾ Ô µ, then to each ¾ Â Ô there is a unique ¼ ¾  Ô, such that µ ¼ µ. Taking the adjoint and using (1) gives µ ¼ µ. Since can be chosen arbitrary also can be chosen arbitrary giving a unique ¼, this shows that both sets Ô µ and Ô µ lie symmetric with respect to the real axis. Let ¾ µ be a real eigenvalue with eigenvector : µ. So by (1) is also an eigenvector with the same eigenvalue. With the decomposition ½ ¾ we see that both self-adjoint operators ½ and ¾ are eigenvectors, so any eigenspace for a real eigenvalue is spanned by self-adjoint elements. If in addition the map in  ½ is trace preserving and ½ an eigenvalue and a corresponding eigenvector, then from ÌÖ µ ÌÖ µµ ÌÖ µ we deduce ÌÖ µ ¼. In the finite dimensional case Ñ À ½ all spaces Â Ô are equal and of course Ô µ is independent of Ô. Since all norms Ô are equivalent, by its definition Ö Ô µ is also independent of Ô, as it should be. To formulate the Perron-Frobenius theorem in the present context, we make the following definition, which is the just an adaption of the usual definition. Since any positive map in Â Ô is bounded ÜÔ Ø is a well defined positive map in Â Ô for all Ø ¼. Also its inverse ÜÔ Ø is a well defined bounded map in  Ô, not necessarily positive. Definition 3.1. A positive map in Â Ô ½ Ô ½µ is positivity improving if µ ¼ for any ¼ ¼. is ergodic if for any ¼ ¼ there is Ø ¼ with ÜÔ Ø µ µ ¼.

6 If is ergodic then ÜÔ Øµ µ ÜÔ Ø µ µ ¼ for all Ø Ø. Obviously is ergodic if it is positivity improving. A simple necessary criterion for ergodicity is given by Lemma 3.1. If is ergodic then ÃÖ Ô ¼. Proof. Assume that there is ¼ ¼ with µ ¼, such that ÜÔ Øµ µ for all Ø. If is not positive definite, then we have a contradiction, so let ¼. We claim there is ¼ ¼ ¼ which is not positive definite with ¼. In fact, we may take ¼ to be a onedimensional orthogonal projection onto an eigenvector of times the corresponding eigenvalue, which we choose not to be zero. This gives ¼ ¼ µ µ ¼ and we are back to the first situation. Some remarks concerning the finite dimensional case are in order. First we will show that when Ñ À ½ the present definition of ergodicity of is equivalent to the condition ½ µ Ñ À ½ µ ¼ for any ¼ ¼, a criterion used in [7]. Obviously this last condition implies ergodicity since ÜÔ Ø ½ µ ¾ Ñ ½ ص ½ µ for all Ñ and all Ø ¼. As for the converse assume there is ¼ ¼ such that ½ µ À ½ µ is not positive definite. So there is an orthogonal projection È ¼ Á and ¼ such that È µ È (see [7]). This gives ÜÔ Øµ È µ ÜÔ ØµÈ for all Ø ¼, so is not ergodic. We have the following necessary and simple criterion for to be ergodic. Lemma 3.2. If Ñ À ½ then Áµ ¼ for ergodic. The converse is not true as may be seen by taking to be given by (9), where the «s are taken to be a nontrivial decomposition of unity. Then Áµ Á whereas «µ «for all, such that is not ergodic. Proof. Assume that Áµ ¼ holds such that there is ³ ¾ À with ³ Áµ³ ¼. Then ³ µ³ ¼ holds for all selfadjoint due to the bound Á Áand therefore ³ µ³ ¼ for all ½ and all. This in turn implies ³ ÜÔ Øµ µµ³ ¼ for all Ø ¼ and all. Since ¼ by definition, there is ¼ with ¼ µ ¼ (see the decomposition (2)), so is not ergodic. Returning to the general case it is clear that «given by (9) is not ergodic if the «s satisfy condition (11). Also in the context Ô ½ and with a continuous, one parameter family of unitary operators Í Øµ defining a quantum mechanics on À in terms of its infinitesimal generator À, then Í Øµ (which describes the time evolution in the Heisenberg picture) for fixed Ø is not ergodic. If the completely positive map in Â Ô is given by (13) in terms of a unitary representation of a compact group, which contains at least one irreducible representation with finite multiplicity, then by Schur s Lemma is easily seen to be ergodic ( and even positivity improving) if and only if the representation is irreducible. Then also À is of course finite dimensional and the map is a conditional expectation. The following two lemmas are almost obvious. Lemma 3.3. A completely positive map in  ½ represented as «is positivity improving if and only for any ¼ ³ ¾ À the closed linear hull of the set of vectors «³ is all of À. Proof. It suffices to consider È µ, where È is any one-dimensional orthogonal projection (compare the proof of Lemma 3.1). Let ¼ ³ ¼ ¾ È À be a unit vector, so we have to consider (17) and the claim follows. ³ È µ³ ¾Æ ³ «³ ¼ ¾ This proof shows that in the infinite dimensional case no «with only finitely many nonvanishing «s can be positivity improving. In fact, for given È choose ¼ ³ to be orthogonal to all «³ ¼. Then (17) vanishes and therefore È µ is not positive definite. Let «µ be the algebra (not -algebra) generated by the «s.

7 Lemma 3.4. A completely positive map in  ½ represented as «is ergodic if and only if every non-zero vector ³ in À is cyclic for «µ, i.e. the strong closure of the linear space «µ³ is all of À. These two results apply to all Kraus representations «of. Proof. We use the same notation as in the proof of the previous lemma. ««½ «¾ «for ½ ¾ µ and set. Then and again the claim follows. ³ ÜÔ Øµ È µ³ ½ Ø ¼ ³ «³ ¼ ¾ In addition write The next theorem is part of the Perron-Frobenius theorem in the present context. It is part because it assumes that Ö Ô µ is an eigenvalue, a standard assumption one has to make in the infinite dimensional context if no other information is available. Theorem 3.1. Let the positive map in Â Ô ½ Ô ½) be ergodic and assume that Ö Ô µ Ô. If Ö Ô µ is an eigenvalue of, then this eigenvalue is simple and the eigenvector may be chosen to be positive definite. Remark 3.1. According to usual terminology such a vector is called a Perron- Frobenius vector. When Ñ À ½ then Ö Ô µ is independent of Ô and is always an eigenvalue. This case has been covered by Evans and Hoegh-Krohn [7] with an application to quantum Monte-Carlo processes and by Groh [14]. At present we do not know whether in the infinite dimensional case the condition Ö Ô µ Ô can be dropped. For the proof of the theorem we need the following simple Lemma 3.5. For any ¼ and ¼ ¼ ¼ in Â Ô ½ Ô ½ the following strict inequality (18) holds. ¼ Ô ¼ Ô The condition ¼ cannot be weakened to ¼ as the choice È ¼ and ¼ É ¼ for two finite dimensional orthogonal projections with È É ¼ shows. Proof. The estimate (18) for the case Ô ½ is trivial, so let Ô ½. Since ¼ ¾ Â Ô is self-adjoint, there is a complete set orthonormal eigenvectors ³ with real eigenvalues of ¼. With ³ ¼ µ³ we obtain (19) ¼ Ô Ô Ô ³ ³ ³ ¼ ³ Ô Observe that (20) ³ ³ ³ ¼ ³ Ô ÑÜ ³ ³ Ô ³ ¼ ³ Ô µ ³ ³ ³ ¼ ³ µ Ô ³ ¼ µ³ Ô ³ ¼ µ Ô ³ holds, since ¼ and ¼ ¼. In the last estimate in (20) we used the estimate (7). Furthermore ³ ³ ¼ holds for all. Also ³ ¼ ³ ¼ for at least one. Otherwise we would also have ³ ¼ ³ ¼ ¼ for all ¼ by Schwarz inequality and hence ¼ ¼, contradicting the assumption. Thus the first inequality in (20) is actually strict for at least one. Inserting (20) into (19) and using this last observation proves (18). È ÖÓÓ Ó Ø Ì ÓÖÑ: Let ¼ be any eigenvector for with eigenvalue Ö Ô µ. By an observation made above, we may assume that is self-adjoint. Observe the following estimate, valid for any Ø ¼. (21) ÜÔ ØÖ Ô µ Ô ÜÔ ØÖ Ô µ Ô ÜÔ Øµ µ Ô ÜÔ Øµ µ Ô ÜÔ Øµ µ Ô ÜÔ Ø Ô Ô ÜÔ Ø Ô Ô

8 Here we have used the estimate (6) in combination with (3) for the map ÜÔ Ø. Since by assumption Ö Ô µ Ô, all inequalities in (21) actually have to be equalities. In particular for all Ø ¼ (22) ÜÔ Øµ µ Ô ÜÔ Øµ µ ÜÔ Øµ µ Ô ÜÔ Øµ µ Ô ÜÔ Øµ µ ÜÔ Øµ µ Ô If ¼ or ¼ there is nothing to prove so assume that ¼. By the ergodicity of we have ÜÔ Øµ µ ¼ for all large Ø ¼. But then (22) contradicts (18). So we must have either ¼ or ¼ and by replacing by if necessary we may without restriction assume that. Again by ergodicity we have ÜÔ ØÖ Ô µ ÜÔ Øµ µ ¼ for all large Ø such that ¼. The proof of non-degeneracy is now easy. Assume there are two linearly independent eigenvectors ½ ¼ and ¾ ¼, which we may normalize to ½ Ô ¾ Ô ½. Then ¼ ½ ¾ is also an eigenvector and by our previous discussion we must have either ½ ¾ or ¾ ½. This would imply that ½ Ô ¾ Ô or ¾ Ô ½ Ô respectively, again an easy consequence of (6) and its proof. This is a contradiction, so we must have ½ ¾, thus concluding the proof of the theorem. 4. EXISTENCE OF EIGENVALUES In this section we want in particular to establish conditions for positive maps in  Ô, which are sufficient to show that Ö Ô µ is an eigenvalue. We start with a preparation. For any ¾ Àµ we write Í with unique Í for its polar decomposition. More precisely Í is isometric on Ê ÃÖµ, i.e. Í Í ³ ³ for ³ ¾ Ê ÃÖµ, and Í ³ ¼ for ³ ¾ Ê µ ÃÖ. The next two lemmas replace the Kadison-Schwarz inequality (see e.g. the second reference in [18] and [34, 4, 24] for other early references) used in the context of -algebras in [7, 14] and in the context of von Neumann algebras in [2]. Since the spaces Â Ô Ô ½ are not algebras these two lemmas will provide the appropriate substitute. Lemma 4.1. Let the map in Â Ô ½ Ô ½ be 2-positive. Then for all ³ ³ ¼ ¾ À and all ¾ Â Ô (23) ³ ¼ µ³ ¾ ³ µ³³ ¼ Æ Í µ³ ¼ Also if not all terms in (23) are vanishing, then there is equality if and only if both relations µ µ (24) µ Æ Í µ and (25) hold. µ µ µ Æ Í µ ³ ¼ Æ Í µ³ ¼ ³ ³ ¼ µ³ ³ ¼ ¼ ³ µ ³ ¼ ³ ³ µ³³ ¼ ¼ For estimate (23) follows from (3), since then Í Í Í Í Í Í Í. commutes with such that Proof. Here and in the proof of the next lemma we will mimic and extend Exercise 3.4 in [26], page 39 in the present context. Set ½ ½¾ ¾ ½¾ Í such that ¾ ½ ¾ ½ ¾ ¾ Í Í Í µ are all in  Ô. Consider ½ ¾ ½ ¾ (26) ¼ ¼ ¼ ¼ ¼ Í µ in À À À Å ¾, which defines an element in Ô À Àµ. By assumption µ µ (27) ¾ ¼ Í µ µ Æ Í µ

9 Consider the following linear transformation Ì on ¾ (equipped with the natural scalar product) given as the ¾ ¾ matrix ³ µ³ ³ µ ³ ¼ (28) Ì ³ ¼ µ³ ³ ¼ Æ Í µ³ ¼ such that for all ½ ¾ ¾ (29) ½ ¾ Ì ½ ¾ ½ ³ ¾ ³ ¼ µ µ µ Æ Í µ ½ ³ ¾ ³ ¼ ¼ with the obvious notation for the scalar product in À À. A linear transformation in ¾ is ¼ if and only if its trace and its determinant are both ¼. Hence the bound (23) follows. This discussion also easily gives the last claim in Lemma 4.1. Indeed, if Ì ¼, its determinant is equal to zero if and only if there is a non-zero eigenvector of Ì with eigenvalue zero and this is the case if and only if equality in (23) holds and then such an eigenvector of Ì is given either as ³ ¼ Æ Í µ³ ¼ ³ ¼ µ³ or as ³ µ ³ ¼ ³ µ³ More precisely, if all matrix elements of Ì are non-vanishing then these two eigenvectors have non-zero entries and are proportional. If Ì ¼ with Ø Ì ¼ and if one diagonal element is vanishing, such that also the off diagonal elements are vanishing, then exactly one of these vectors is the null vector. In view of (29) this concludes the proof of Lemma 4.1. Lemma 4.2. Let the map in Â Ô ½ Ô ½ be 2-positive. Then the bound (30) µ Ô µ ½¾ Ô Æ Í µ ½¾ Ô holds for all ¾  Ô. In particular ¾ ÃÖ if ¾ ÃÖ or ¾ ÃÖ Æ Í. Proof. Choosing ³ ¼ Í µ ³ in (23) we have the estimate (31) ³ µ³ ³ µ³ ½¾ Í µ ³ Æ Í µí µ ³ ½¾ for all ³ ¾ À. Thus (30) follows trivially from (31) in case Ô ½. So let ½ Ô ½. Take ³ to be an orthonormal basis in À diagonalizing µ with eigenvalues ¼. Then we obtain µ Ô Ô Ô ³ µ³ Ô (32) ³ µ³ Ô¾ Í µ ³ Æ Í µí µ ³ Ô¾ ³ µ Ô ³ ½¾ Í µ ³ Æ Í µµ Ô Í µ ³ µ Ô¾ Ô Æ Í µ Ô¾ Ô Here we have used (31) for the first inequality. The second inequality is a consequence of Schwarz inequality and (7). In what follows this will become important, since we will encounter the situation when all inequalities are actually equalities. Also we have used the fact that with ³ being an orthonormal basis also Í µ ³ is a possibly incomplete set of orthonormal vectors and which is responsible for the last inequality. ¾ is called a peripheral eigenvalue of the positive map in Â Ô if and hence also is an eigenvalue with Ö Ô µ. ½¾

10 Theorem 4.1. Let the map in Â Ô ½ Ô ½ be 2-positive with Ö Ô µ Ô. Assume that ¾ is a peripheral eigenvalue with eigenvector. Then Ö Ô µ is also an eigenvalue with eigenvector. In particular, if is 2-positive and compact with Ö Ô µ Ô, then Ö Ô µ is an eigenvalue. Here and in contrast to the previous discussions the case Ô ½ is not covered by the present discussion. Since  Խ is a -algebra, this case, however, is covered by the discussion in [36, 14]. Then and similarly in the context of von Neumann algebras one can actually prove more. In fact, if the spectrum is rescaled via Ö Ô µ, then the peripheral eigenvalues form a discrete sub-group of the unit circle group, see [2] and the first reference in [14]. Proof. We want to prove that ¼ is also an eigenvector with eigenvalue Ö Ô µ, i.e. µ Ö Ô µ. Since µ Ö Ô µ (such that also Í µ µí and hence Í µ Í by the uniqueness of Í ) it suffices to show that (33) µ³ ³ holds for all, where the ³ are as in the previous lemma. Also by the previous lemma we have (34) Ö Ô µ Ô Ô µ Ô µ Ô µ ½¾ Ô Æ Í µ ½¾ Ô Ô Ô Here we have used the fact that Í Ô ½. By the assumption Ö Ô µ Ô we must have equality. In particular this implies that (35) µ Ô µ ½¾ Ô Æ Í µ ½¾ Ô Inspection of the proof of Lemma 4.2 shows that we must have equality in (31) when ³ ³ for all. The relations (24) and (25) in Lemma 4.1 then imply that we must have (36) and (37) ³ ¼ Æ Í µ³ ¼ µ³ ³ ¼ µ³ µ ³ ¼ ³ µ³ Æ Í µ³ ¼ ³ µ ³ ¼ µ³ for all with ³ ¼ Í µ³. Assume to be such that ¼. Since ³ ¾ Ê µ we have Í µ ³¼ ³ and µ ³ ¼ µ³ ³ for all such. Similarly µ³ Í µ µ³ ³ ¼. Hence we may rewrite (36) and (37) as (38) and (39) ³ ¼ Æ Í µ³ ¼ µ³ ¾ ³ ³ µ³ Æ Í µ³ ¼ ¾ ³¼ whenever ¼. Furthermore we recall that we made use of Schwarz inequality in the estimate (32). For this to be an equality we claim that we actually must have (40) ³ µ³ ³ ¼ Æ Í µ³ ¼ for all. Indeed, the Schwarz inequality in this context is an equality if and only if both sides of (40) are proportional with a common proportionality factor for all. By the equality (35) this proportionality factor has to be equal to 1, thus proving (40). If ¼ then ³ ¼ ¼ and hence by (40) µ³ ¼. Thus we have established (33) in the case when ¼. If ¼ then the left hand sides of (38) and (39) are both non-vanishing since then ³ ¼ ¼. Therefore ³ is an eigenvector of µ with eigenvalue ³ µ³ ¾ ³ ¼ Æ Í µ³ ¼ ¾ ³ µ³

11 This gives (33) in the case when ¼, thus concluding the proof of the theorem, since for compact operators the spectrum is pure point. The use of (7) in the second inequality in (32) does not give any additional mileage when Ô ½ since the ³ are eigenvectors of µ and the ³ ¼ eigenvectors of Æ Í and hence (7) gives an equality for the case at hand. The strategy of the proof was motivated by the following observation in the classical context. With the notation introduced after (3) assume that ËÞ Þ. This gives Þ Þ ËÞ ËÞ. Taking its usual Hilbert space norm in we obtain Þ ËÞ Ë Þ. So if Ë we must have ËÞ Ë Þ. The next result extends Theorem 3.1. Theorem 4.2. Let the map in Â Ô ½ Ô ½ be 2-positive and ergodic with Ö Ô µ Ô. Assume that ¾ is a peripheral eigenvalue. Then is a simple eigenvalue and the corresponding eigenvector is a normal operator. Proof. Assume there are two eigenvectors ½ and ¾ in Â Ô with eigenvalue, normalized to ½ Ô ½ Ô ¾ Ô ¾ Ô ½. By Theorem 3.1, its proof and by Theorem 4.1 we have that ¼ ½ ¾ is an eigenvector with eigenvalue Ö Ô µ. In particular both Í ½ and Í ¾ are unitaries. Since ½ ÜÔ ¾ is also an eigenvector for the same eigenvalue for all ¼ ¾ by the same argument we must have (41) The function ½ that (42) ½ ÜÔ ¾ ½ ÜÔ ¾ Ô ½ ÜÔ ¾ Ô is easily seen to be continuous. Hence there is ¼ such ½ ÜÔ ¼ ¾ Ô Take the square of (41) and use (42) to obtain Due to the relation Ê ½ À this gives Ñ ½ ÜÔ ¾ Ô ¼¾ ½ ÜÔ ¼ ¾ ¾ ½ ÜÔ ¾ ¾ Í ½ ÜÔ ¼ Í ¾ µ³ ¾ Í ½ ÜÔ Í ¾ µ³ ¾ which written out in turn gives (43) Í ½ ³ Í ¾ ³ ¼ Í ½ ³ Í ¾ ³ for all ¼ ¾ and all ³ ¾ À. We distinguish two possible cases. First if ½ ÜÔ ¼ ¾ ¼ and hence Í ½ ÜÔ ¼ Í ¾ ¼ there is nothing to prove. Otherwise Í Í ½ ½ ÜÔ ¼ Í ¾ is a unitary operator Áand hence by the spectral theorem for unitary operators there is ³ ¾ À such that ³ ͳµ ³ ͳ. Choose such that ÜÔ ¼ µ³ ͳ ³ ͳ. This contradicts (43), thus concluding the proof of the first part of the theorem. As for the last part Í is unitary, as already noted. Relation (39) shows that Í µ is also an eigenvector of with eigenvalue Ö Ô µ. Hence Í µ by uniqueness and normalization. So Í commutes with and therefore is normal with Í Í and, where we recall that is an eigenvector with eigenvalue. The next result is of relevance in quantum physics. We recall that for completely positive, trace preserving maps in  ½ one has Ö ½ µ ½ ½. So we immediately obtain the following existence result for Perron-Frobenius vectors. Corollary 4.1. Let the map in  ½ be completely positive, trace preserving, compact and ergodic. Then there is a unique density matrix ¼ which is left invariant under. A converse of this result is given by Theorem 5.1 below. The following result is of the type needed in the standard context of Monte Carlo simulations in lattice models of statistical mechanics. It states that, starting from any non-negative initial configuration, by iterations of an ergodic up-dating procedure (heat bath method) one eventually arrives at the desired equilibrium configuration which typically is given by a Gibbs distribution

12 and which by construction of the up-dating is a Perron-Frobenius vector for the up-dating procedure. Corollary 4.2. Let the map in  ¾ be 2-positive, selfadjoint, compact and ergodic with Perron- Frobenius vector ¼ normalized to ¾ ½. Then for any ¾  ¾ (44) ÐÑ Ø½ ÜÔ Ø ¾ µµ µ ¾ If ¼ ¼ then the right hand side of (44) is non-vanishing. It would be interesting to find an analogous formulation when Ô ¾. Proof. We use familiar arguments. Let the form an orthonormal basis of eigenvectors for arranged such that ½ and ½. Here the s are the eigenvalues, i.e. µ. Since ½ ¾ is a simple eigenvalue, ½ ¾. By Plancherel s theorem we have È È with ¾ and ¾ ¾ ¾. Obviously holds such that (45) and the claim follows. ÜÔ Ø ¾ µµ µ Ø ½ µ ÜÔ Ø ¾ µµµ µ ¾ ¾ Ø ½ ¾ µ ¾ By a slight modification of the proof using the spectral representation of one may replace the compactness condition by the condition that ¾ is an isolated eigenvalue, i.e. ¾ is separated by a distance Ñ ¾ (the mass gap in physical language) from the remainder of the spectrum of, which otherwise may be arbitrary. Then the estimate (45) is still valid with ½ ¾ being replaced by Ñ ¾. 5. EXAMPLES In this section we will provide non-trivial examples when Ñ À ½. The issue is to show that all 4 conditions in Corollary 4.1 can be fulfilled simultaneously. In the finite dimensional case the compactness criterion is automatically satisfied and then it is easy to construct nontrivial examples. In general the existence of completely positive, trace preserving, compact maps in  ½ is clarified by È Lemma 5.1. Let ««¾Æ be such that ¾Æ ««Á and all «¾ Àµ are compact. Then the map «in  ½ is compact. Proof. We recall that «is the norm limit of «Æ as Æ ½. Since the space of compact operators is closed w.r.t. the norm topology it suffices to prove that all «Æ are compact. By assumption to each there is a sequence «of finite rank operators in Àµ such that ÐÑ ½ ««ÐÑ ½ ««¼ Since «is of finite rank «Æ with «Æ «½ «Æ µ is also of finite rank and in particular compact. Using the a priori estimate ½ ½ and a standard argument gives ÐÑ ½ «Æ «Æ ½ ¼ We are now prepared to provide non-trivial examples, which resulted from a discussion with D. Buchholz. Let ¾ È Æ be any orthonormal basis in À and ¼ any set of numbers which satisfy the condition ¾ ½ for all. In the Dirac notation we define the following family ««½½ of rank 1 operators «such that «

13 In other words «is the map ³ ³. We set By assumption we have «««µ ««¾ Á so «is completely positive, compact and trace preserving. We claim that it is also positivity improving by Lemma 3.3. Indeed, given ¼ ³ ¾ À, assume there is ¼ ³ ¼ ¾ À such that ¼ ³ ¼ «³ ³ ¼ ³ holds for all and. Since by assumption ¼ this contradicts the fact that the form an orthonormal basis. So this «satisfies all conditions of Corollary 4.1. The resulting Perron- Frobenius vector is given as the density matrix (46) ¼ ½ where the form È the components of the Perron-Frobenius vector with eigenvalue 1 for the matrix ¾, i.e. ¾. As a special case consider the choice ¼ for all with È ¾ ½. Then ¾. This discussion also gives the following converse to Corollary 4.1, which in its standard formulation forms one of the starting points for Monte Carlo simulations in lattice models of statistical mechanics and (euclidean) quantum field theory including lattice gauge theories. Theorem 5.1. Given an arbitrary density matrix ¼, there is a completely positive, trace preserving, compact and ergodic map in  ½ for which is the resulting Perron-Frobenius vector. In addition may be chosen to be idempotent (i.e. satisfies the characteristic equation Æ ½ µ ¼) and ½ µ ¼ ½. Proof. Take the s to form an orthonormal basis in which is diagonal, i.e. such that takes the form (46) and choose «with «as above with ½¾. Then we have µ ÌÖ for all ¾  ½, such that is idempotent. Also the kernel of consists of all elements which may be written in the form ÌÖ. Moreover let ¼ ½. Then for given ¼ the equation µ µ ¼ has a solution in of the form ½ ½ µ ÌÖ ¼ ½ ¼ As in the corresponding ( finite dimensional) standard context with prescribed Perron- Frobenius vector is of course not unique. We provide another example which in spirit is much closer to the usual local up-grading procedure in Monte Carlo simulations. È Set Ô ½ ½ µ for ¼ ½ and with the convention ¼ ¼. Then ¼ ½ Ô ½ for all and Ô ¼ for all and unless ½ and ½. Set and define Ì as Ì Ô ½¾ Ì µ Ì Ì An easy calculation shows that Ì is completely positive, trace preserving, compact and ergodic. Acknowledgments The author has profited from discussions with D. Buchholz, B. Kümmerer, M.B. Ruskai and E. Størmer.

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15 d) Cones of positive maps, Contemp. Math. 62 (1987), [35] M. Takesaki, Theory of Operator Algebras I, Springer, New York, [36] Y. Watatani and M. Enomoto, A Perron-Frobenius type theorem for positive linear maps on -algebras, Math. Japan. 24 (1979), INSTITUT FÜR THEORETISCHE PHYSIK, FREIE UNIVERSITÄT BERLIN, ARNIMALLEE 14, D BERLIN, GERMANY address: schrader@physik.fu-berlin.de