classical and quantum propagation of chaos

8/3/2019 Classical and Quantum Propagation of Chaos

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arXiv:quant-ph/0108143v1

31Aug2001 Chapter 1

Classical and Quantum

Propagation of Chaos

1.1 Overview

The concept of molecular chaos dates back to Boltzmann [3], who derived thefundamental equation of the kinetic theory of gases under the hypothesis thatthe molecules of a nonequilibrium gas are in a state of molecular disorder.The concept of propagation of molecular chaos is due to Kac [8, 9], who calledit propagation of the Boltzmann property and used it to derive the homo-geneous Boltzmann equation in the infinite-particle limit of certain Markoviangas models (see also [5, 17]). McKean [12, 13] proved the propagation of chaos

for systems of interacting diffusions that yield diffusive Vlasov equations in themean-field limit. Spohn [16] used a quantum analog of the propagation of chaosto derive time-dependent Hartree equations for mean-field Hamiltonians, andhis work was extended in [1] to open quantum mean-field systems.

This article examines the relationship between classical and quantum prop-agation of chaos. The rest of this introduction reviews some ideas of quantumprobability and dynamics. Section 1.2 discusses the classical and quantum con-cepts of propagation of chaos. In Section 1.3, classical propagation of chaosis shown to occur when quantum systems that propagate quantum molecularchaos are suitably prepared, allowed to evolve without interference, and thenobserved. Our main result is Corollary 1.3.7, which may be paraphrased asfollows:

Let O be a complete observable of a single particle, taking its values in

a countable set J, and let Oi denote the observable O of particle i in a sys-tem of n distinguishable particles of the same species. Suppose we allow thatquantum n-particle system to evolve freely, except that we periodically measureO1, O2, . . . , On. The resulting time series of measurements is a Markov chain inJn. If the sequence ofn-particle dynamics propagates quantum molecular chaos,then these derived Markov chains propagate chaos in the classical sense.

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Results like this may be of interest to probabilists who already know someexamples of the propagation of chaos and who may be surprised to learn of

novel examples arising in quantum dynamics. An effort has been made hereto expound the propagation of quantum molecular chaos for such an audience,while the classical propagation of chaos per se is discussed only briefly in Sec-tion 1.2.1. The reader is referred to [18] and [14] for two definitive surveys ofthe classical propagation of chaos.

1.1.1 Quantum kinematics

In quantum theory, the state of a physical system is inherently statistical: thestate of a system S does not determine whether or not S has a given property,but rather, the state provides only the probability that S would be found tohave that property, if we were to check for it. The properties that S mightor might not have are represented by orthogonal projectors on some Hilbert

space. If a projector P represents a property P of S, then the complementaryproperty NOT P is represented by I P. The identity and zero operators Iand 0 represent the trivial properties TRUE and FALSE respectively. If P andQ are properties whose orthogonal projectors are P and Q, the properties (PAND Q) and (P OR Q) are defined if and only if P and Q commute, in whichcase P Q = QP represents (P AND Q) and P + Q P Q represents (P OR Q).A countable resolution of the identity is a countable family of projectors {Pj}such that

PjPj = PjPj = 0 ; j = j

and I =

j Pj . This represents a partition of the space of outcomes of a mea-surement on S into a countable set of elemental properties, which are mutuallyexclusive and collectively exhaustive. A state of the system S is a functionthat assigns probabilities to the properties of S, or their projections. Thus wesuppose that (I) = 1, (0) = 0 and also that

(I P) + (P) = (I) = 1.

Indeed, it is rational to suppose that any state must satisfy

1 = (I) =

j

Pj

=j

(Pj) (1.1)

for any countable resolution of the identity {Pj}.Having made these introductory comments, we revert to a more technical

description of the mathematical set-up. Suppose the properties of a quantumsystem S are represented by orthogonal projectors in B(H), the bounded oper-

ators on a Hilbert spaceH

. The statistical states (also called simply states) ofthat quantum system are identified with the normal positive linear functionalson B(H) that assign 1 to the identity operator. A positive linear functional on B(H) is normal if

aA

(Pa) = 1


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be defined as the predual of the Heisenberg dynamics; if is normal then therelation

Tr((D)A) = Tr(D(A)) A B(H)

implicitly defines a trace-preserving map known as the predual of . In theSchrodinger picture, the density operator D that describes the quantum stateundergoes the transformation D (D), where is a normal completelypositive unital endomorphism of B(H).

The description of a quantum system evolving continuously in time requiresa normal and completely positive endomorphism of B(H) for each t > 0, todescribe the change of observables (in the Heisenberg picture) from time 0 totime t. A quantum dynamical semigroup, or QDS, is a family {t}t0 of normalcompletely positive (and unital) endomorphisms of the bounded operators onsome Hilbert space H, which is a semigroup (i.e., 0 = id and t s = t+s for

s, t 0) and which has weak*-continuous trajectories: for any B B(H) andany trace class operator T

Tr(T t(B))

is continuous in t. (We will not need the continuity of trajectories in this article.)Quantum dynamical semigroups describe the continuous change of the stateof an open quantum system whose dynamics are autonomous and Markovian.Many models of open quantum systems are QDSs (but not all, viz. [10]). Wewill use the notation ()t for the whole QDS:

()t = {t}t0. (1.3)

1.2 Classical and Quantum Molecular Chaos

1.2.1 Classical molecular chaos

Molecular chaos is a type of stochastic independence of particles manifestingitself in an infinite-particle limit.

Let n be the n-fold Cartesian power of a measurable space . A probabilitymeasure p on n is called symmetric if

p(E1 E2 En) = p(E(1) E(2) E(n))

for all measurable sets E1, . . . , E n and all permutations of {1, 2, . . . , n}.For k n, the k-marginal of p, denoted p(k), is the probability measure on Sk

satisfying

p(k)(E1 E2 Ek) = p(E1 Ek )

for all measurable sets E1, . . . , E k . In the context of classical probabilitytheory, one defines molecular chaos as follows [18]:


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Propagation of Chaos in Classical and Quantum Kinetics 5

Definition 1.2.1. Let be a separable metric space. Let p be a probabilitymeasure on , and for each n N, let pn be a symmetric probability measure

on n.The sequence {pn} is p-chaotic if the k-marginals p

(k)n converge weakly to

pk as n , for each fixed k N.

A sequence, indexed by n, of n-particle dynamics propagates chaos if molec-ularly chaotic sequences of initial distributions remain molecularly chaotic forall time under the n-particle dynamical evolutions. In the classical contexts[9, 12, 14, 18] the dynamics are Markovian and the state spaces are usuallytaken to be separable and metrizable. Accordingly, in my dissertation [4] Idefined propagation of chaos in terms of Markov transition kernels, as follows:

Definition 1.2.2 (Classical Propagation of Chaos). Let be a separablemetric space. For each n N, let K

n: n (n) [0, 1] be a Markov

transition kernel which is invariant under permutations in the sense that

Kn(x, E) = Kn( x, E)

for all permutations of the n coordinates ofx and the points ofE n. Here,(n) denotes the Borel -field of n.

The sequence {Kn}n=1 propagates chaos if the molecular chaos of a se-quence {pn} entails the molecular chaos of the sequence

n

Kn(x, )pn(dx)

n=1

. (1.4)

The preceding formulation of the propagation of molecular chaos is techni-cally straightforward but not flexible enough to cover the weaker kinds of prop-agation of chaos phenomena that occur in several applications, most notably inthe landmark derivation of the Boltzmann equation due to Lanford and King[11]. Nonetheless, it is still worthwhile to make Definition 1.2.2. Those modelsthat exhibit weak propagation of chaos phenomena usually have less realisticregularizations that propagate molecular chaos in the sense of Definition 1.2.2.Moreover, this definition has the pleasant feature that it implies that {Kn Ln}propagates molecular chaos when {Kn} and {Ln} do.

1.2.2 Quantum molecular chaos

The Hilbert space of pure states of a collection of n distinguishable componentsis H1Hn, where Hi is the Hilbert space for the ith component. The Hilbertspace for n distinguishable components of the same species will be denoted Hn.If Dn is a density operator on Hn, then its k-marginal, or partial trace, is adensity operator on Hk that gives the statistical state of the first k particles.The k-marginal may be denoted Tr(nk)Dn and defined as follows: Let O be


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any orthonormal basis ofH. If x Hk with k < n then for any w, x Hk

Tr(nk)Dn(w), x

=

y1,... ,ynkO

Dn(w y1 ynk), x y1 ynk .

A linear functional on B(Hn) is symmetric if it satisfies

(A1 An) = (A(1) A(2) A(n))

for all permutations of {1, 2, . . . , n} and all A1, . . . , An B(H). For eachpermutation of {1, 2, . . . , n}, define the unitary operator U on Hn whoseaction on simple tensors is

U(x1 x2 xn) = x(1) x(2) x(n). (1.5)

A density operator Dn represents a symmetric functional on B(Hn) if and onlyDn commutes with each U. Two special types of symmetric density operatorsare Fermi-Dirac densities, which represent the statistical states of systems offermions, and Bose-Einstein densities, which represent the statistical states ofsystems of bosons. Bose-Einstein density operators are characterized by thecondition that DnU = Dn for all permutations , and Fermi-Dirac densitiesare characterized by the condition that DnU = sign()Dn for all .

To recapitulate, n-component states are given by density operators on Hn

and, in the Schodinger picture, the dynamics transforms an initial state A Tr(DA) into a state of the form A Tr(D(A)), where is a normal completelypositive unital endomorphism of B(Hn). This is the context of the followingtwo definitions:

Definition 1.2.3. Let D be a density operator on H, and for each n N, letDn be a symmetric density operator on H

n.The sequence {Dn} is D-chaotic in the quantum sense if, for each fixed

k N, the density operators Tr(nk)Dn converge in trace norm to Dk asn .

The sequence {Dn} is quantum molecularly chaotic if it is D-chaotic inthe quantum sense for some density operator D on H.

The definitions of classical and quantum molecular chaos are somewhatincongruous. This definition of quantum molecular chaos requires that themarginals converge in the trace norm, whereas the notion of classical molecularchaos used in Probability Theory requires weak convergence of the marginals. In

fact, Definition 1.2.1 of classical molecular chaos and the commutative versionof Definition 1.2.3 (obtained by extending that definition of quantum molecularchaos to commutative von Neumann algebras) are not equivalent! Nonethe-less, I have chosen Definition 1.2.3 because the attractive theory of quantummean-field kinetics presented in Section 1.2.3 favors a formulation of quantummolecular chaos in terms of the trace norm.


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Definition 1.2.4 (Propagation of Quantum Molecular Chaos). For eachn N, let n be a normal completely positive map from Hn to itself that fixes

the identity and which commutes with permutations, i.e., such that

n(UAU) = U

n(A)U (1.6)

for all A B(Hn) and all permutations of {1, 2, . . . , n}, where U is asdefined in (1.5).

The sequence {n} propagates quantum molecular chaos if the quan-tum molecular chaos of a sequence of density operators {Dn} entails the quan-tum molecular chaos of the sequence {n(Dn)}.

We shall soon find that there are interesting examples of quantum dynamicalsemigroups (n)t with the collective property that {n,t} propagates quantummolecular chaos for each fixed t > 0. When this happens, it is convenient to saythat the sequence {(n)t} of QDSs propagates chaos.

Definition 1.2.5. For each n let (n)t be a QDS on B(Hn) that satisfies thepermutation condition (1.6). The sequence {(n)t} propagates molecularchaos if {n,t} propagates quantum molecular chaos for every fixed t > 0.

1.2.3 Spohns quantum mean-field dynamics

There are several successful mathematical treatments of quantum mean-fielddynamics. One of them, due to H. Spohn, relies upon the concept of propaga-tion of quantum molecular chaos. Spohns theorem [16] constitutes a rigorousderivation of the time-dependent Hartree equation for bounded mean-field po-tentials.

Let V be a bounded Hermitian operator on H H such that V U(12) =

U(12)V(y x), representing a symmetric two-body potential. Let V1,2 denotethe operator on Hn defined by

V1,2(x1 x2 xn) = V(x1 x2) x3 xn, (1.7)

and for each i, j n with i < j, define Vij similarly, so that it acts on theith and jth factors of each simple tensor. This may be accomplished by settingVij = UV1,2U, where = (2j)(1i) is a permutation that puts i in the firstplace and j in the second place, and U is as defined in (1.5). Define the n-particle Hamiltonians Hn as the sum of the pair potentials Vij , with commoncoupling constant 1/n:

Hn =1

n

i


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Theorem 1.2.6 (Spohn). Suppose D is a density operator onH and {Dn} isaD-chaotic sequence of symmetric density operators onHn. Then the sequence

of density operators {Dn(t)} defined in (1.8) and (1.9) is D(t)-chaotic, whereD(t) is the solution at time t of the following initial-value problem in the Banachspace of trace-class operators:

d

dtD(t) =

i

Tr(n1)[V, D(t) D(t)]

D(0) = D. (1.10)

In other words, if Hn is as in (1.8) and n(A) = eiHnt/ A eiHnt/ thenthe sequence {n} propagates quantum molecular chaos. See [16] for a shortproof.

It must be emphasized that the preceding approach does not apply to sys-tems of Fermions. In other words, Theorem 1.2.6 yields time-dependent Hartree

equations but not time-dependent Hartree-Fock equations (which include an ex-change term that enforces the Pauli Principle). The problem is that there existsno molecularly chaotic sequence of Fermi-Dirac states because the antisymme-try of Fermi-Dirac states is incompatible with the factorization of molecularlychaotic states. To derive time-dependent Hartree-Fock equations despite thisproblem is a goal of current research [15].

Spohns approach can be generalized to handle Hamiltonians which involvethe usual unbounded kinetic energy operators, and to handle open quantummean-field systems. In [1], Theorem 1.2.6 is extended to systems where the freemotion of the single particle may have an unbounded self-adjoint generator andthe two-particle generator may have the Lindblad form. The propagation ofquantum molecular chaos is called the mean-field property in that article.

1.3 Classical Manifestations of the Propagation

of Quantum Molecular Chaos

A sequence {n} that propagates quantum molecular chaos mediates a varietyof instances of the propagation of classical molecular chaos.

Given a D-chaotic sequence of density operators {Dn} one can produce avariety of molecularly chaotic sequences {qn} of probability measures. For eachsingle-particle measurement M, the joint probabilities qn of the outcome of ap-plying M to all of the particles form a molecularly chaotic sequence. Conversely,there are ways to convert a p-chaotic sequence {pn} of probability measuresinto a D-chaotic sequence of density operators {Dn}. Suppose we are presented

with a sequence of quantum dynamics {n} that propagates quantum molecularchaos. We can first encode a sequence of molecularly chaotic probabilities {pn}as a quantum molecularly chaotic sequence of density operators {Dn}, thenallow {Dn} to develop into a new sequence {n(Dn)} under the dynamics,and finally read the resulting quantum states by applying some single-particlemeasurement to each of the particles. This procedure converts one molecularly-


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chaotic sequence of probability measures into another, i.e., it propagates chaosin the classical sense.

The next three sections examine the encode/develop/read-procedure thatconverts quantum molecular chaos to classical molecular chaos. Finally, in Sec-tion 1.3.4, we show how to produce Markov chains that propagate chaos byobserving quantum processes that propagate chaos.

1.3.1 Generalized measurements and the reading proce-

dure

Let H be a Hilbert space and (, F) a measurable space. A positive operatorvalued measure, or POVM, is a function X(E) from F to the positive operatorson H which is countably additive with respect to the weak operator topology:

i=1

X(Ei) = X(E)

in the weak operator topology whenever the sets Ei F are disjoint andE = Ei. In the special case that the positive operators X(E) are self-adjointprojections such that X(E)X(F) = 0 when E F = , the POVM is just aresolution of identity on (, F). A POVM defines an affine map D pD,X fromdensity operators on H to probability measures on . For any density operatorD,

pD,X() = Tr [DX()] (1.11)

is a countably additive probability measure on (, F).POVMs correspond to generalized -valued measurements just as the spec-

tral decomposition of a self-adjoint operator corresponds to a simple measure-ment. A generalized measurement is realized by performing a simple measure-ment on a composite system consisting of the system of interest an ancilliarysystem that has been prepared to have state E, without allowing any inter-action between the system itself and the ancilliary system or the environment.Suppose that Ha is the Hilbert space of an ancilliary space that has been pre-pared in state E, and P(d), is the spectral measure belonging to anobservable O on the composite system H Ha. If the system of interest is instate D when O is measured, a random is produced, governed by theprobability law Tr[(D E) P(d)]. This probability law has the form (1.11),since

Tr[(D E) P(d)] = Tr[D Tr(1)((I E)P(d))] = Tr(DX(d))

where X(d) is the POVM

X(d) = T r(1)((I E1/2)P(d)(I E1/2)).

This shows that the outcome of a generalized measurement is governed by somePOVM as in (1.11). Conversely, it can be shown that any POVM arises in this


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way from some conceivable (but perhaps impracticable) generalized measure-ment of the type we have just described [7].

The following lemma has a straightforward proof, which we omit.

Lemma 1.3.1. Let be a separable metric space with Borel -field (), andlet X : () B (H) be a POVM on .

For each n, letDn be a symmetric density operator on B(Hn), and supposethat {Dn} is D-chaotic in the quantum sense.

Then the sequence of probability measures {pn} is p-chaotic, pn and p beingdefined by

p(A) = Tr(DA)

pn(A1 A2 An) = Tr (Dn X(A1) X(An)) .

1.3.2 Lemmas concerning the encoding procedure

The procedure for encoding probability measures as density operators dependson our choice of a density operator valued function D() on the single-particlespace , now assumed to be a separable metric space. Let us choose a functionD() from to the density operators on a Hilbert space H, and assume Dis continuous for technical convenience. Then we can convert a probabilitymeasure pn on n into the density operator

n

D(1) D(2) D(n) pn(d1d2 dn)

on Hn. If {pn} is a p-chaotic sequence of symmetric measures on n then the

corresponding sequence of density operators is quantum molecularly chaotic but only in a weak sense! Think of the density operators as a subset of theBanach space of trace-class operators. Each continuous linear functional on thisspace has the form

T Tr(T B)

where B is a bounded operator, for B(H) is the Banach dual of the space oftrace-class operators on H. A sequence {Tn} of trace-class operators on H isweakly convergent if

limn

Tr(TnB) = Tr(T B)

for all B B(H).

Lemma 1.3.2. Let be a separable metric space and suppose {pn} is a p-chaotic sequence of symmetric measures on n. LetD(s) be a continuous func-tion from to the density operators on a Hilbert space H.


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Define D and Dn by

D =

D()p(d)

Dn =

n

D(1) D(2) D(n) pn(d1d2 dn).

(1.12)

Then, for each k, the sequence of marginals {Tr(nk)Dn} converges weakly toDk as n .

Proof. The integrals in (1.12) may be defined as Bochner integrals in the Ba-nach space of trace-class operators (see [6] Theorem 3.7.4). The partial trace isa bounded operator from the Banach space of trace-class operators on Hn to theBanach space of trace-class operators on Hk, so Theorem 3.7.12 of [6] implies

that

Tr(nk)n

D(1) D(n) pn(d1d2 dn)

=

n

D(1) D(k) pn(d1d2 dn).

(1.13)

Since Bochner integration commutes with application of bounded linear func-tionals, the right hand side of (1.13) converges weakly to

k

D(1) D(2) D(k) pk(d1d2 dk) = D

k

as n .

The preceding lemma does not conclude that {Dn} is quantum molecularlychaotic in the sense of our Definition 1.2.3, which requires convergence of thepartial traces in trace norm. Some additional conditions seem necessary in orderto conclude that the encoding procedure produces quantum molecular chaos.The easiest thing to do is suppose that the quantum systems involved are finitedimensional. This affords a quick way to construct Markov transitions on anyspace . The opposite approach is to let the mediating quantum dynamicsoccur in any Hilbert space H but to suppose that is discrete. We follow thesetwo approaches in the next two lemmas:

Lemma 1.3.3. IfH is a finite dimensional Hilbert space Cd, and Dn and D

are as in the statement of Lemma 1.3.2, then the sequence of states {Dn} isD-chaotic.

Proof. From Lemma 1.3.2 we know that {Tr(nk)Dn} converges weakly to Dk

as n . But a sequence of trace-class operators on Cd converges in tracenorm if it converges weakly, since the Banach space of trace class operators on


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on Cd is finite-dimensional. Hence, limn

Tr(nk)(Dn) = Dk in trace norm, as

required. Now we consider a set J with a discrete topology and -field. We note that if

the word separable were removed from Definition 1.2.1 of classical molecularchaos then the following lemma would hold even for uncountable sets J:

Lemma 1.3.4. LetJ be a countable set equipped with the discrete topology andits Borel -field (so that every subset of J is measurable), and let {D(j)}jJ bea family of density operators onH indexed by J.

Suppose p is a probability measure on J and {pn} is a p-chaotic sequence ofprobability measures. Define

D =jJ

p(j)D(j)

Dn =

(j1,... ,jn)Jnpn(j1, . . . , jn)D(j1) D(j2) D(jn).

(1.14)

Then {Dn} is D-chaotic.

Proof. Since the series defining Dn converges in trace norm and the partialtrace operator Tr(nk) is a bounded operator with respect to the trace norm, itfollows that

Tr(nk)Dn =Jn

pn(j1, . . . , jn)D(j1) D(j2) D(jk)

=Jk

p(k)n (j1, . . . , jk)D(j1) D(j2) D(jk).

Now {p(k)n } converges weakly to pk as n tends to infinity, for {pn} is p-chaotic.

Since a sequence of elements of 1 converges in norm if and only if it converges

weakly [2], the sequence {p(k)n } converges in the 1 norm to pk as n tends to

infinity. Hence, in trace norm,

limn

Tr(nk)Dn =Jk

p(j1)p(j2) p(jk) D(j1) D(j2) D(jk)

= Dk,

proving that {Dn} is D-chaotic.

1.3.3 Putting it together: encoding, developing, and read-ing

We now formulate two abstract propositions that follow from the above twolemmas on the encoding procedure. Proposition 1.3.5 relies on Lemma 1.3.3and is therefore limited by the hypothesis that the mediating quantum system is


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finite-dimensional. Proposition 1.3.6 is derived from Lemma 1.3.4. It allows thequantum dynamics to take place in an arbitrary Hilbert space but requires the

measurable spaces involved to be discrete. Despite these technical restrictionsthere remains a rich variety of classical examples of the propagation of chaosresiding within each instance of the propagation of quantum molecular chaos.

Proposition 1.3.5. Let be a separable metric space with Borel -field (),let D(s) be a continuous function from to the density operators onCd, andlet X : () B(Cd) be a POVM on . For each n, let n be a normalcompletely positive unital endomorphism of B((Cd)n) that satisfies (1.6).

Define the Markov transition kernel Kn on n by

Kn((1, 2, . . . , n), A1 A2 An)

= Tr [(D(1) D(n)) n(X(A1) X(An))] .

If {n} propagates quantum molecular chaos then {Kn} propagates molecularchaos in the classical sense.

The proof of this proposition is omitted because it follows directly fromLemma 1.3.1 and Lemma 1.3.3. Likewise, the following proposition followsfrom Lemma 1.3.1 and Lemma 1.3.4:

Proposition 1.3.6. LetH be a Hilbert space and, for each n, let n be a com-pletely positive endomorphism of B(Hn) that satisfies (1.6). Let J be a count-able set equipped with the discrete topology and-field, and let X : J B(H)be a POVM on J.

Define the Markov transition matrices Kn on Jn by

Kn((j1, . . . , jn), (j1, . . . , jn))

= Tr [(D(j1) D(jn))n(X(j1) X(j

n))]

If {n} propagates quantum molecular chaos then the sequence of Markov tran-sition kernels {Kn} propagates classical molecular chaos.

1.3.4 Periodic measurement of complete observables

The periodic measurement of a complete observable of a quantum system pro-duces a Markov chain of measurement values.

Let O be a complete observable of a system, represented by a resolution of

the identity {|ejej |}jJ for some orthonormal basis {ej}jJ ofH. Considera (possibly open) quantum system whose evolution is governed by a QDS ( )t.If the natural quantum evolution of this system is interrupted periodically bythe measurement of O, then the resulting random sequence of measurementvalues is a Markov chain on J. To be specific, suppose the measurements of Oare performed at times 0, T, 2T, 3T , . . . but there is no other interference with


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the evolution ()t. The first measurement ofO results in a random outcome,namely, the pure state

Pej = |ejej |

into which the system has collapsed. (We use Dirac notation |ee| for projectiononto the span ofe.) In effect, measuring the observable O prepares a pure statePej and informs us of the index j J of that pure state. Having been preparedin the state Pej , the system is allowed to evolve T time units under ()t. Bytime T, the state of the system is T(Pej ). When O is measured at time T, themeasurement produces another random j J, and the system collapses intothe corresponding pure state Pej . The probability of the transition j j

is

Tr(T(Pej )Pej ) = Tr(PejT(Pej )) K(j,j),

defining a Markov transition K(, ) from J to itself. At time T the system hasbeen prepared in some pure state P

ej, and a new experiment begins: the system

undergoes T time units of the evolution ()t, transforming its state from Pejto T(Pej ), and then O is measured at time 2T, instantaneously forcing thesystem into the random state indexed by j with probability K(j, j). Thisis repeated, producing a random record of the indices j0, j1, j2, . . . of the purestates the system was in upon measurement at times 0 , 1, 2, . . . of O. Ideally,these successive measurement/experiments would be independent, both physi-cally and stochastically, and it is evident that the performance of those measure-ments in succession would produce a random sequence j0, j1, j2, . . . governed bythe (one-step) Markov transition kernel K(, ).

It will be convenient to denote by K[()t, O, T] the Markov transition pro-duced in this way, so that

K[()t, O, T](j,j) = Tr(PejT(Pej )). (1.15)

Imagine an n-component system whose (distinguishable) components areeach quantum systems with the (same) Hilbert space H, and which is governedby a QDS (n)t that satisfies the permutation condition (1.6). Let {ej} bean orthonormal basis for H indexed by J, and let O be the observable thatreturns the value j if the (component) system is in the pure state ej . Theobservable O determines a resolution of the identity {|ejej |}. The Hilbertspace for the n-component system is Hn and the state of the system is adensity operator on Hn. Let Oi denote the observable that returns the value jif the ith component in the pure state ej . We can imagine measuring Oi of eachof the components because the components are distinguishable. Simultaneousmeasurement of O1, . . . , On on the n-component system results in a randomvector (j1, . . . , j

n) J

n of measurement values, and forces the system into the

pure state ej1 ejn . Let On

denote the joint measurement (O1, . . . , On).Periodic measurement ofOn results in a Markov chain of values in Jn, since On

is a complete observable of the n-component system. The one-step transitionkernel for this Markov chain is

K[(n)t, On, T](j, j) = Tr(Pej1ejn T(Pej

1

ejn)). (1.16)


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by formula (1.15).

Corollary 1.3.7. Let (n)t and On

be as above for each n, and suppose that{(n)t} propagates quantum molecular chaos. Then, for each T 0, the se-quence of Markov transitions {K[(n)t, On, T]}nN propagates chaos in the clas-sical sense.

Proof. Consider the special case of Proposition 1.3.6 where

D(j) = X(j) = |ejej |.

In that case the sequence {Ln} of Markov transitions

Ln((j1, . . . , jn), (j1, . . . , j

n)) = Tr

(Qj1,...,jn)n(Qj1,...,jn)

(1.17)

with Qj1,...,jn = |ej1ej1 | |ejnejn | propagates molecular chaos. But

comparing (1.17) to (1.16) and noting that

Pej1ejn =ej1 ejnej1 ejn

= |ej1ej1 | |ej2ej2 | |ejnejn | = Qj1,...,jn

shows that

Ln = K[(n)t, On, T].

1.4 Acknowledgements

I am indebted to William Arveson, Sante Gnerre, Lucien Le Cam, Marc Rieffel,and Geoffrey Sewell for their advice and for their interest in this research. Iwould also like to thank Rolando Rebolledo for his collaboration and support.The author is supported by the Austrian START pro ject Nonlinear Schrodingerand quantum Boltzmann equations.


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