An Effective Spectral Theorem for Bounded Self Adjoint Oper

Transcription

1 An Effective Spectral Theorem for Bounded Self Adjoint Operators Thomas TU Darmstadt jww M. Pape München, 5. Mai 2016

2 Happy 60th Birthday, Uli! Ois Guade zum Sech zga, Uli! I ll follow soon!

3 Spectral Theorem for Bounded Selfadjoint Operators John von Neumann s Spectral Theorem establishes a 1-1-correspondence between selfadjoint operators on separable Hilbert space H (e.g. l 2 ) and spectral measures on R. It restricts to a 1-1-correspondence between bounded selfadjoint operators on H and spectral measures on R with compact support. W.l.o.g. it suffices to establish a 1-1-correspondence between non-expansive selfadjoint operators A, i.e. 1 A 1, and spectral measures on I, the interval [ 1, 1] with Euclidean topology. We will show that this Spectral Theorem for non-expansive selfadjoint operators will be computable in the sense of Weihrauch s approach to Computable Analysis (TTE).

4 Recap of Weihrauch s TTE from an Abstract PoV (1) TTE stands for Type Two Effectivity meaning that data types (spaces) are represented by subspaces of B (Baire space) and maps between them are functions realized by effective/continuous partial maps on B (which may be considered as Turing machines with an infinite input and an infinite output tape). According to Brouwer such partial maps on B can be represented by elements of Baire space. This gives rise to K 2, the 2nd Kleene algebra, where we write α β or simply αβ for the application of the function coded by α to β. Like every pca (partial combinatory algebra) K 2 induces a topos, the so-called function realizability topos RT(K 2 ). We write K eff 2 for the sub-pca of K 2 on effective elements, i.e. total recursive functions. The Kleene-Vesley topos KV is the wide subcategory of RT(K 2 ) of maps with realizers in K eff 2.

5 Recap of Weihrauch s TTE from an Abstract PoV (2) RT(K 2 ) hosts a full reflective subcategory Mod(K 2 ) of so-called modest sets with a fairly concrete flavour. An object of Mod(K 2 ) is a pair X = ( X, ρ X ) where X is a set and ρx is a surjection from a subspace R X of B to X. A morphism from X to Y is a function f : X Y realized by some α B, i.e. x X. β ρ 1 X (x). αβ ρ 1(f (x)) which we write as α f. Modest sets in KV are the wide subcategory of Mod(K 2 ) consisting of morphisms realized by an element of K2 eff. For a modest set X we may endow X with the quotient topology induced by ρ X (whose domain R X carries the subspace topology inherited from R X B). Y

6 Admissible Representations à la Schröder and Simpson For which modest sets are realizable maps the continuous ones? The admissible representations, i.e. X where for every subspace S of B and every continuous f : S X there is a partial continuous φ : S R X with f = ρ X φ. More abstractly by a result of Schröder admissible representations are those X Mod(K 2 ) where η X : X Σ ΣX is a regular mono, i.e. a mono, or equivalently where X Σ I for some I Mod(K 2 ). We write AdmRep for the full subcategory of Mod(K 2 ) on admissible representations and AdmRep eff for the wide subcategory whose morphisms are those realized by some element of K eff 2.

7 QCB 0 (Schröder & Simpson) There is an obvious functor from AdmRep to Sp sending X to the space with underlying set X and the quotient topology induced by ρ X. It is full and faithful and its image is the full subcat of Sp on so-called QCB 0 spaces, i.e. T 0 quotients of subspaces of B. They have been characterized by Schröder as sequential T 0 spaces with a countable pseudo-base. We will employ a lazy guy s variant of TTE where we show some function to be computable by exhibiting it as a morphism in the effective part of AdmRep QCB 0 exploiting that it lives as a full subcat within the topos KV.

8 BQM in AdmRep AdmRep QCB 0 hosts complete separable metric spaces (csm s) and countably based domains. Thus, it hosts H, Σ H and the Hilbert lattice L which appears as a -subobject of Σ H. Our aim is to show that also states and observables in the sense of Basic Quantum Mechanics live within AdmRep. Eventually, we will show that a (variant of) the Spectral Theorem for non-expansive selfadjoint operators lives within AdmRep eff, i.e. that the 1-1-correspondence between non-expansive selfadjoint operators and spectral valuations on I is given by computable maps between spaces with admissible representations. Our use of the notions of state and observable is standard and we refer to the book by Pták and Pullmannová Orthomodular Structures as Quantum Logics (1991).

9 States in AdmRep (1) A (quantum) state is thought of as a probability measure on L. Since L has a non-discrete observation order and states preserve it we have to consider the unit interval [0, 1] topologized in such a way that its information ordering is the opposite of the usual order, i.e. with the lower topology, for which we write I. Thus, we define a state as a morphism s : L I such that (S1) s(0 L ) = 0 I and s(1 L ) = 1 I (S2) s(p Q) = s(p) + s(q) whenever P Q. Notice that preservation of suprema of countable chains is automatic for morphisms in AdmRep. The object Sta of states is closed under countable convex combinations and thus by Gleason s Theorem it suffices to show that all states induced by unit vectors ( pure states ) are in Sta.

10 States in AdmRep (2) We write S(H) for the -subobject of H consisting of unit vectors modulo the relation of being elements of the same closed linear subspaces. We have shown that Theorem 1 The map π : L I S(H) : P x x, Px is a QCB 0 morphism and thus lives within AdmRep eff. Moreover, π is a -mono. Here we identify closed linear subspaces with their projectors. For every x S(H) the map π x = λp:l.π(p)(x) is a morphism in AdmRep, thus continuous, and as it validates (S1) and (S2) we have π x Sta. Thus, the underlying set of Sta contains all (quantum) state.

11 Observables and von Neumann s Spectral Theorem A (quantum) observable on a csm X is a spectral measure on X, i.e. a function µ : B(X ) L such that for all x S(H), the map µ x = λb:b(x ).π(µ(b))(x) is a measure on X. For X = R the famous von Neumann Spectral Theorem establishes a 1-1-correspondence between (unbounded) selfadjoint operators on H and observables on R which restricts to a 1-1-correspondence between bounded selfadjoint operators on H and observables µ with bounded support, i.e. µ([ c, c]) = 1 L for some c > 0. We restrict attention to observables on I = [ 1, 1] which by the Spectral Theorem correspond to selfadjoint operators between 1 and 1. The correspondence is given by x, Ax = 1 1 λ dµ x (λ)

12 Observables on I in AdmRep Even for csm s X the set B(X ) of Borel subsets of X does not form an object in AdmRep. For this reason we replace measures on X by their restrictions to Cl(X ), the closed subsets of X. Such gadgets are called valuations and can be characterized as Scott continuous maps ν : Cl(X ) I with ν( ) = 0, ν(x ) = 1 and ν(a B) + ν(a B) = ν(a) + ν(b) for A, B Cl(X ). Since Scott continuity is automatic in QCB 0 for every X QCB 0 there is an object Val(X ) QCB 0 whose underlying set are the valuations on X. We define spectral valuations (in analogy with spectral measures) as maps ν : Cl(I) L such that for every x S(H) the map ν x = λc:cl(i).π(ν(c))(x) Val(I). Let SVal be the corresponding object of spectral valuations in QCB 0 AdmRep.

13 Spectral Theorem for Observables on I in AdmRep eff establishes an isomorphism in AdmRep eff between SVal and the object Obs(I) H H of selfadjoint operators between 1 and 1. Since integration w.r.t. valuations exists within RT(K 2 ) with every ν SVal we associate the map λx:s(h). 1 1 λ dν x(λ) from which we can construct the unique linear operator A with x, Ax = 1 1 λ dν x(λ) for all x S(H). On the other hand for every A Obs(I) the corresponding ν SVal is given by ν(c)(x) = inf{ x, f (A)x χ C f C(I)} where χ C : I {0, 1} I is the characteristic function of C and f (A) is obtained via spectral calculus. These back-and-forth maps are effective since their definition can be interpreted in KV.

14 Natural Topology on Obs(I) Separable Banach spaces E 1 and E 2 are csm s and thus objects of AdmRep. Let L(E 1, E 2 ) be the -closed subobject of E E 1 2 on (bounded) linear operators. Using the Banach-Steinhaus theorem one can show that convergence in L(E 1, E 2 ) coincides with convergence in the strong operator topology (pointwise convergence w.r.t. norm topologies). This applies in particular to Obs(I) H H. One can further show that A n converges to A in Obs(I) iff lim x, f (A n)x = x, Ax for all f C(I) and x H iff n lim inf π(ν n(c))(x) = π(ν(c))(x) for all C Cl(I) where ν n and ν n correspond to A n and A via the Spectral Theorem. Remark This is in accordance with M. Schröder s result characterizing the natural topology on Val(I) as lim n ν n = ν iff lim inf n ν n(c) = ν(c) for all C Cl(I).

15 Towards the General von Neumann Spectral Theorem First one proves the Spectral Theorem for Unitary Operators on H, i.e. that every unitary U : H H can be expressed as for a unique A Obs([ π, π]). U = cos(a) + i sin(a) The General Spectral Theorem of von Neumann establishes a 1-1-correspondence between spectral measures/valuations on R and self adjoint operators A on H with dense domain of definition and closed graph. It goes by applying the Spectral Theorem for Unitary Operators to the Cayley transform of A. Remark If D is a dense subspace of H this gives rise to a -subobject of H which again is in AdmRep. The graph of linear operator A : D H being a closed subspace of H H is equivalent to A preserving converging sequences and thus to being a morphism in AdmRep.