Podsędkowska

[ Pobierz całość w formacie PDF ]
//-->Entropy2015,17,1181-1196; doi:10.3390/e17031181OPEN ACCESSentropyISSN 1099-4300www.mdpi.com/journal/entropyArticleEntropy of Quantum MeasurementHanna Podsedkowska¸Faculty of Mathematics and Computer Sciences, University of Łód´ , ul. S. Banacha 22, 90-238 Łód´ ,zzPoland; E-Mail: hpodsedk@math.uni.lodz.plAcademic Editor: Kevin H. KnuthReceived: 31 October 2014 / Accepted: 9 March 2015 / Published: 12 March 2015Abstract:A notion of entropy of a normal state on a finite von Neumann algebra inSegal’s sense is considered, and its superadditivity is proven together with a necessary andsufficient condition for its additivity. Bounds on the entropy of the state after measurementare obtained, and it is shown that a weakly repeatable measurement gives minimal entropyand that a minimal state entropy measurement satisfying some natural additional conditionsis repeatable.Keywords:entropy; von Neumann algebra; instrument1. IntroductionThe notion of the entropy of a state of a physical system was introduced by John von Neumann(see [1]) in the setup that is now classical for quantum mechanics. In this approach, the observables of aphysical system are identified with self-adjoint operators on a separable Hilbert space, and the states ofthe system, with the positive operators of trace one on this space. This setting has been generalized inmore modern theories, in particular in the so-called algebraic approach to quantum physics in which thebounded observables of a physical system form the self-adjoint part of a C*-, or von Neumann, algebra(see [2–5]). The origin of this approach goes back to I. Segal [6], who first indicated the basic features ofsuch an algebraic formalism. However, despite its obvious importance, the unique notion of the entropyof a state on an arbitrary C*-, or von Neumann, algebra has not been unambiguously established. On theother hand, a lot of work has been done in this field, and an interested reader may consult, e.g., [7–10].In our considerations, we adopt a definition of entropy due to I. Segal, which is similar to the classicalBoltzmann–Gibbs entropy and applies to normal states on a finite von Neumann algebra.In the paper, we show the superadditivity of the entropy considered, together with a necessary andsufficient condition of its additivity and give bounds on the entropy of the state after measurement.Entropy2015,171182Moreover, we show that a weakly repeatable measurement gives minimal entropy and that a minimalstate entropy measurement satisfying some natural additional conditions is repeatable.2. Preliminaries and NotationLetMbe a von Neumann algebra,i.e.,an algebra of bounded operators on a Hilbert spaceHwith identity1being the identity operator, closed in the weak operator topology given by the familyof seminorms:Mx→ |ξ|xη|,ξ, η∈ H,and such thatx∗∈Mwheneverx∈M.For a projectionp∈M,we setp⊥=1−p.ByM∗is denotedthe predual ofM,which is a Banach space of bounded linear functionals onM,such that(M∗)∗=M.The elements ofM∗are callednormal.The positive elementsϕofM∗having norm one,i.e.,such thatϕ(1) = 1,are callednormal states.M+will stand for the positive elements ofM∗; its elements, which∗are not states, bear sometimes the name ofnon-normalized states.Forϕ∈M+, we define itssupport,∗denoted bys(ϕ),as the smallest projectionpinM,such that:ϕ(p)=ϕ(1).The following formula holds true:s(ϕ)= sup{q∈M:q— projection,ϕ(q)= 0}⊥.A linear mapΦ :M→Mis said to benormalif it is continuous in theσ(M,M∗)topology.For a linear normal positive mapΦ,we define itssupports(Φ)in the same way as for normal positivefunctionals,i.e.,as the smallest projectionpinM,such that:Φ(p) = Φ(1).For the support, the following relation holds true:Φ(s(Φ)x) = Φ(xs(Φ))= Φ(x),moreover, if:Φ(s(Φ)xs(Φ)) = 0ands(Φ)xs(Φ)≥0,thens(Φ)xs(Φ)= 0.positive functionals.The same relations hold true for the normalx∈M;Lemma 1.LetΦ :M→Mbe a linear normal positive map, and let≤a≤1,a∈M,be such that:Φ(a) = Φ(1).Then:s(Φ)=s(Φ)a=as(Φ).Entropy2015,17Proof.We have1−a≥0,and:so:s(Φ)(1−a)s(Φ)= 0,which yields:s(Φ)(1−a)= 0,showing the claim.3. Instruments in Quantum Measurement Theory1183Φ(1−a)= 0,In this chapter, we briefly recall the theory of instruments by E. Davies and J. Lewis (see [11,12]),which serves as a mathematical tool for a description of the process of quantum measurement.Let(Ω,F)be a measurable space of values of an observable of a physical system,i.e.,Ωis an arbitraryset, andFis aσ-fieldof subsets ofΩ(usually, we have asΩthe setRof all real numbers, andFisthe Borel subsetsB(R)ofR).LetMbe a von Neumann algebra. An instrument on(Ω,F)is a mapE:F→L+(M∗)from theσ-fieldFinto the set of all positive linear transformations on the predualM∗,such that:(i)(EΩϕ)(1) =ϕ(1)for allϕ∈M∗,∞(ii)E∞n=1∆nϕ=n=1E∆nϕfor anyϕ∈M∗and pairwise disjoint sets∆nfromF,where the series on the right-hand side isconvergent in theσ(M∗,M)-topologyonM.In measurement theory,EΩϕrepresents the state of the system after measurement, provided that beforemeasurement, the system was in the stateϕ.The mapEΩsends states to states; thus, it is aquantumchannel(in the terminology of quantum information theory). Accordingly, the mapsE∆could be calleddeficient channels,since they send states to “almost states” in the sense thatE∆ϕare positive normalfunctional,s but there may be(E∆ϕ)(1) = 1.In particular, in von Neumann’s measurement theory, ifobservableTwith the spectral decomposition:T=iλieiis measured in a system being in the stateϕ,we have:EΩϕ=ieiϕei,(1)where(eiϕei)(a) =ϕ(eiaei).In the language of density matrices, equality Equation (1) reads:EΩ(Dϕ) =ieiDϕei,(2)whereDϕis the density matrix corresponding to the stateϕ,i.e.,ϕ(a)= traDϕ.Entropy2015,171184It is worth noting that channels of the form of Equation (2) are objects of intensive investigations; inthe theory of instruments, they constitute the class of so-calledLüders instruments(cf. the remarks afterTheorem3).∗Consider now for eachE∆its dual mapE∆:M→Mdefined by:∗ϕ(E∆(x)) = (E∆ϕ)(x),ϕ∈M∗, x∈M.The dual instrument is then defined as a mapE∗:F→L+(M)fromFinto the set of all positive normalnlinear transformations onM,such that:∗(i*)EΩ(1) =1,∞(ii*)E∗∞(x)n=1∆n=n=1∗E∆n(x)for anyx∈Mand pairwise disjoint sets∆nfromF,where the series on the right-hand side isconvergent in theσ(M,M∗)-topologyonM.For an instrumentE,its associated observable is defined as a mape:F→Mby the formula:∗e(∆)=E∆(1).(3)Thus,eis a positive operator valued measure (≡ POVM, semi-spectral measure). If for any∆,e(∆)isa projection, theneis a projection-valued measure (≡ PVM, spectral measure).Suppose that the measured system is in stateϕ.Then, for observablee(∆),we wantϕ(e(∆))to bethe probability that the observed value is in set∆,which should be equal to(E∆ϕ)(1).This leads tothe equality:∗ϕ(e(∆))= (E∆ϕ)(1) =ϕ(E∆(1)),which justifies the definition of observable adopted earlier.Among many important classes of instruments, there are weakly repeatable and repeatable ones,which express the celebrated von Neumann repeatability hypothesis:if the physical quantity is measuredtwice in succession in a system, then we get the same value each time(cf. [1,12]). Their definitions areas follows.Definition 1.An instrumentEassociated with observableeis called weakly repeatable if the followingcondition holds:(E∆1(E∆2ϕ))(1) = (E∆1∩∆2ϕ)(1)for all sets∆1,∆2∈Fand anyϕ∈M∗, or equivalently,∗∗∗E∆1(E∆2(1)) =E∆1∩∆2(1),∆1,∆2∈F,which in terms of observable reads:∗E∆1(e(∆2)) =e(∆1∩∆2).The weak repeatability of an instrument may be characterized in the following way.Entropy2015,17Lemma 2.LetEbe an instrument. The following are equivalent:(i)(ii)(iii)(iv)(v)(vi)Eis weakly repeatable,∗ ∗for any∆, Θ∈F,such that∆∩Θ =∅,we haveE∆EΘ= 0,∗ ∗∗2for any∆, Θ∈F,we haveE∆EΘ=E∆∩Θ,∗ ∗for any∆∈F,we haveE∆E∆= 0,where∆ = Ω\∆,∗∗∗for any∆∈F,we haveE∆E∆(1) =E∆(1),∗∗∗∗∗for any∆, Θ∈F,such that∆⊂Θ,we haveE∆EΘ(1) =EΘE∆(1) =E∆(1).1185Proof.First, we shall show the equivalence of Conditions (ii)–(iv).(ii)⇐⇒(iii): Suppose that (ii) holds. For any∆, Θ∈F,we have:∗ ∗∗ ∗∗ ∗∗ ∗∗∗∗∗2E∆EΘ=E∆E∆∩Θ+E∆E∆∩Θ=E∆E∆∩Θ=E∆∩Θ+E∆∩ΘE∆∩Θ=E∆∩Θ,showing the implication (ii)=⇒(iii). The converse implication is obvious.(iv)⇐⇒(v): For any∆∈F,we have:∗∗1=E∆(1) +E∆(1),hence:∗∗∗∗∗E∆(1) =E∆E∆(1) +E∆E∆(1).Thus:∗∗∗E∆(1) =E∆E∆(1)if and only if:∗∗E∆E∆(1) = 0,∗ ∗∗ ∗which, since the mapE∆E∆is positive, holds if and only ifE∆E∆= 0.∗ ∗∗ ∗(ii)=⇒(vi): For∆⊂Θ,we have∆∩Θ =∅,and thus,E∆EΘ=EΘE∆= 0.Consequently,∗∗∗∗∗∗∗∗∗E∆EΘ(1) =E∆EΘ(1) +E∆EΘ(1) =E∆EΩ(1) =E∆(1),∗∗∗and, analogously,EΘE∆(1) =E∆(1).(vi)=⇒(v): Obvious.(iv)=⇒(ii). Let∆∩Θ =∅.Then,Θ⊂∆, and from the additivity ofE∗, we get:∗∗∗∗E∆=EΘ+E∆∩Θ≥ EΘ.∗Consequently, for eachx∈M,x≥0,we obtain on account of the positivity ofE∆and the inequality:∗∗EΘ(x)≤ E∆(x),the relation:∗∗∗∗≤ E∆EΘ(x)≤ E∆E∆(x) = 0,∗ ∗showing thatE∆EΘ= 0.Thus, Conditions (ii)–(iv) are equivalent. Clearly, (i)=⇒(v). We shall show that:(ii) and (iii)=⇒(i). For arbitrary∆1,∆2∈F,we have:∗∗∗∗∗∗E∆1E∆2(1) =E∆1E∆1∩∆2(1) +E∆1∩∆2(1) =E∆1∩∆2(1),showing the weak repeatability ofE. [ Pobierz całość w formacie PDF ]