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Paschke Dilations Paschke Dilations Abraham Westerbaan Bas Westerbaan Radboud Universiteit Nijmegen Radboud Universiteit Nijmegen [email protected] [email protected] In 1973 Paschke defined a factorization for completely positive maps between C∗-algebras. In this paper we show that for normal maps between von Neumann algebras, this factorization has a univer- sal property, and coincides with Stinespring’s dilation for normal maps into B(H ). The Stinespring Dilation Theorem[17] entails that every normal completely positive linear map (NCP- map) j : A ! B(H ) is of the form A p / B(K ) V ∗(·)V / B(H ) where V : H ! K is a bounded operator and p a normal unital ∗-homomorphism (NMIU-map). Stinespring’s theorem is fun- damental in the study of quantum information and quantum computing: it is used to prove entropy inequalities (e.g. [10]), bounds on optimal cloners (e.g. [20]), full completeness of quantum program- ming languages (e.g. [16]), security of quantum key distribution (e.g. [8]), analyze quantum alternation (e.g. [1]), to categorify quantum processes (e.g. [14]) and as an axiom to single out quantum theory among information processing theories.[2] A fair overview of all uses of Stinespring’s theorem and its consequences would warrant a separate article of its own. One wonders: is the Stinespring dilation categorical in some way? Can the Stinespring dilation theorem be generalized to arbitrary NCP-maps j : A ! B? In this paper we answer both questions in the affirmative. We use the dilation introduced by Paschke[11] for arbitrary NCP-maps, and we show that it coincides with Stinespring’s dilation (a fact not shown before) by introducing a universal property for Paschke’s dilation, which Stinespring’s dilation also satisfies. In the second part of this paper, we will study the class of maps that may appear on the right-hand side of a Paschke dilation, to prove the counter-intuitive fact that both maps in a Paschke dilation are extreme (among NCP maps with same value on 1). Let us give the universal property and examples right off the bat; proofs are further down. j Theorem 1. Every NCP-map j : A ! B has a Paschke dilation. A Paschke A / B ; O dilation of j is a pair of maps A r / P f / B , where P is a von r f Neumann algebra, r is an NMIU-map and f is an NCP-map with j = f ◦ r # such that for every other r0 / 0 f 0 / , where 0 is a von Neu- P A P B P 0 O 0 mann algebra, r0 is an NMIU-map, and f 0 is an NCP-map with j = f 0 ◦ r0, r s f 0 there is a unique NCP-map s : P ! P such that the diagram on the right ' P0 commutes. Example 2. A minimal Stinespring dilation A p / B(K ) V(·)V ∗ / B(H ) of an NCP-map is a Paschke dilation, see Theorem 14. Example 3. As a special case of the previous example, we see the GNS construction for a normal state j on a von Neumann algebra A , gives a Paschke dilation A p / B(H ) hx;(·)xi / C of j. 1 2 In particular, the Paschke dilation of (l; m) = 2 (l + m); C ! C is (l;m)7! l 0 a b 7! 1 (a+b+c+d) 2 0 m c d 2 C / M2 / C : © Westerbaan & Westerbaan R. Duncan and C. Heunen (Eds.): Quantum Physics and Logic (QPL) 2016 This work is licensed under the EPTCS 236, 2017, pp. 229–244, doi:10.4204/EPTCS.236.15 Creative Commons Attribution License. 230 Paschke Dilations This gives a universal property to the von Neumann algebra M2 of 2 × 2 complex matrices, (which is a model of the qubit.) The following examples can be proven using only the universal property of a Paschke dilation. Example 4. The Paschke dilation of an NMIU-map r : A ! B is A r / B id / B . Example 5. If A r / P f / B is a Paschke dilation, then P id / P f / B is a Paschke dilation of f . Example 6. Let j : A ! B1 ⊕ B2 be any NCP-map. A hr1;r2i / P1 ⊕ P2 f1⊕ f2 / B1 ⊕ B2 is a Paschke dilation of j if A ri / Pi fi / Bi is a Paschke dilation of pi ◦ j for i = 1;2. Example 7. Let j : A ! B be any NCP-map with Paschke dilation A r / P f / B and l > 0. Then A r / P l f / B is a Paschke dilation of lj. Example 8. Let A r / P f / B be a Paschke dilation for a map j : A ! B. If J : P ! P0 is 0 any isomorhism, then A J◦r / P f ◦J −1 / B is also a Paschke dilation of j. There is a converse to the last example: Lemma 9. If A ri / Pi fi / B (i = 1;2) are Paschke dilations for the same map j : A ! B, then there is a unique (NMIU) isomorphism J : P1 ! P2 such that J ◦ r1 = r2 and f2 ◦ J = f1. Proof. There are unique mediating maps s1 : P1 ! P2 and s2 : P2 ! P1. It is easy to see s1 ◦ s2 satisfies the same property as the unique mediating map id: P1 ! P1 and so s1 ◦s2 = id. Similarly s2 ◦ s1 = id. Define J = s1. We just saw J is an NCP-isomorphism. Note J(1) = J(r1(1)) = r2(1) = 1 and so J is unital. But then by [22, Corollary 47] J is an NMIU isomorphism. 1 Two universal properties for Stinespring’s dilation Let j : A ! B(H ) be a NCP-map where A is a von Neumann algebra and H is a Hilbert space. In this section, we prove that any minimal normal Stinespring dilation of j gives a Paschke dilation of j. Let us first recall the relevant definitions. Definition 10. A normal Stinespring dilation of j, is a triple (K ;p;V), where K is a Hilbert space, p : A ! B(K ) is an NMIU-map, and V : H ! K a bounded operator such that j = AdV ◦p, where ∗ 1 Adv : B(K ) ! B(H ) is the NCP-map given by AdV (A) = V AV for all A 2 B(K ). If the linear span of fp(a)Vx: a 2 A x 2 H g is dense in K , then (K ;p;V) is called minimal. It is a well-known fact that all minimal normal Stinespring dilations of j are unitarily equivalent (see e.g. [12, Prop. 4.2]). We will adapt its proof to show that a minimal Stinespring dilation admits a universal property (Prop. 13), which we will need later on. The adaptation is mostly straight-forward, except for the following lemma. p 0 A / B Lemma 11. Let p : A ! B, p : A ! C be NMIU-maps between von Neumann O 0 algebras, and let s : C ! B be an NCP-map such that s ◦ p = p. s 0 0 p0 Then s(p (a1)cp (a2)) = p(a1)s(c)p(a2) for any a1;a2 2 A and c 2 C . C 1 ∗ Be warned: many authors prefer to define AdV by AdV (A) = VAV instead. Westerbaan & Westerbaan 231 Proof. By Theorem 3.1 of [5], we know that, for all c;d 2 C , s(d∗d) = s(d)∗s(d) =) s(cd) = s(c)s(d): (1) Let a 2 A . We have s(p0(a)∗p0(a)) = s(p0(a∗a)) = p(a∗a) = p(a)∗p(a) = s(p0(a))∗s(p0(a)). By (1), we have s(cp0(a)) = s(c)s(p0(a)) ≡ s(c)p(a) for all c 2 C . Then also s(p0(a)c) = p(a)s(c) for 0 0 0 all c 2 C (by taking adjoints). Thus s(p (a1)cp (a2)) = p(a1)s(cp (a2)) = p(a1)s(c)p(a2) for all a1;a2 2 A and c 2 C . Lemma 12. Let K be a Hilbert space. If AdS = AdT for S;T 2 B(K ), then S = lT for some l 2 C. Proof. Let x 2 K be given, and let P be the projection onto flx: l 2 Cg. Then ∗ ∗ ∗ ∗ flS x: l 2 Cg = Ran( S PS ) = Ran( T PT ) = flT x: l 2 Cg: ∗ ∗ It follows that S x = aT x for some a 2 C with a 6= 0. While a might depend on x, there is a0 2 C ∗ ∗ ∗ with a0 6= 0 and S = a0T by Lemma 9 of [22]. Then S = a0 T. Proposition 13. Let (K ;p;V) and (K 0;p0;V 0) be normal Stinespring dilations of j. If (K ;p;V) is 0 0 0 minimal, then there is a unique isometry S: K ! K such that SV = V and p = AdS ◦p . Proof. Let us deal with a pathological case. If V = 0, then j = 0, V 0 = 0, K = f0g and p = 0, and so the unique linear map S: f0g ! K 0 satisfies the requirements. Assume V 6= 0. 0 0 0 (Uniqueness) Let S1;S2 : K ! K be isometries with SkV = V and AdSk ◦p = p. We must show that S1 = S2. We have, for all, a1;:::;an;a1;:::;an 2 A , x1;:::;xn;y1;:::;yn 2 H , and c 2 C , ∗ ∗ AdSk (c)∑p(ai)Vxi; ∑p(a j)Vy j = ∑ V p(a j ) AdSk (c)p(ai)Vxi; y j by rearranging i j i; j ∗ 0 ∗ 0 = ∑ V AdSk ( p (a j )cp (ai))Vxi; y j by Lemma 11 i; j 0 ∗ 0 ∗ 0 0 0 = ∑ (V ) p (a j )cp (ai)V xi; y j as SkV = V i; j Since the linear span of p(A )VH is dense in K , we get AdS1 = AdS2 .
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