Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces Roderich Tumulka∗ November 19, 2019 Abstract The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, GAP (ρβ), for a ther- mal density operator ρβ at inverse temperature β. More generally, GAP (ρ) is a probability measure on the unit sphere in Hilbert space for any density operator ρ (i.e., a positive operator with trace 1). In this note, we collect the mathemat- ical details concerning the rigorous definition of GAP (ρ) in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorov’s theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Fi- nally, we give a proof that GAP (ρ) depends continuously on ρ in the sense that convergence of ρ in the trace norm implies weak convergence of GAP (ρ). Key words: Gaussian measure, GAP measure, Scrooge measure, canonical ensem- ble in quantum mechanics. 1 Introduction The Gaussian adjusted projected (GAP) measures are certain probability distributions over quantum-mechanical wave functions that arise as the typical distribution of the conditional wave function for a quantum system in thermal equilibrium with a heat bath arXiv:2004.14226v1 [math-ph] 29 Apr 2020 at a given temperature [4, 3]. The goal of this note is to provide a rigorous foundation of the measure GAP (ρ) for every density operator ρ in the case of an infinite-dimensional separable Hilbert space H . After we formulate the precise definition, the existence and uniqueness of GAP (ρ) will follow from corresponding theorems about Gaussian measures on such Hilbert spaces. We then provide an explicit construction (and thereby ∗Fachbereich Mathematik, Eberhard-Karls-Universit¨at, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany. E-mail: [email protected] 1 an alternative existence proof) of GAP (ρ) and prove the continuous dependence of GAP (ρ) on ρ. The family of GAP measures is a family of probability measures on Hilbert spaces, first considered in [5] under the name “Scrooge measure” as the most spread-out measure on the unit sphere with a given density matrix, then in [4] as playing the role of a thermal equilibrium distribution of the wave function of a quantum system. The GAP measures relevant to thermal equilibrium are those associated with the canonical density operators 1 −βH ρβ = Z e , (1) where Z = tr e−βH is the normalization constant, β the inverse temperature and H the Hamiltonian. A detailed discussion of GAP measures and their physical background can be found in [4, 10, 3]. See [7] for a discussion of conditional wave functions with spin in thermal equilibrium, and [11] for a rigorous study about the support of GAP measures (that took existence and uniqueness for granted), in particular about what GAP (ρβ)-typical wave functions look like. 2 Mean, Covariance, and Density Operator We begin by introducing some notation. Let H be a separable complex Hilbert space, and for any topological space X let B(X) denote its Borel σ-algebra. We denote the unit sphere in H by S(H )= ψ H : ψ =1 . (2) { ∈ k k } Note that S(H ) lies in B(H ) (because it is the pre-image of 1 under the continuous { } function ), and that a subset of S(H ) lies in B(S(H )) if and only if it lies in B(H ) (becausek·k the open sets in S(H ) are the intersections of open sets in H with S(H )). In particular, a measure on S(H ), B(S(H )) can be regarded as a measure on (H , B(H )) that is concentrated on S(H ). Definition 1. Let µ be a probability measure on (H , B(H )). The vector ψ0 H is called the mean of µ if and only if, for every φ H , ∈ ∈ φ ψ0 = µ(dψ) φ ψ . (3) h | i Z h | i The operator C : H H is called the covariance operator of µ if and only if, for µ → every φ, χ H , ∈ φ Cµ χ = µ(dψ) φ ψ ψ0 ψ ψ0 χ . (4) h | i Z h | − ih − | i The mean of µ need not exist (e.g., the function ψ φ ψ may fail to be µ- ′ 7→ h | i H integrable), but if it exists it is unique. (Indeed, if φ ψ0 = φ ψ0 for every φ ′ h | i h | i ∈ then ψ0 = ψ0.) The definition of the covariance operator applies only to those µ that have a mean. Even in that case, the covariance operator may not exist, but if it exists 2 it is unique and positive. (Indeed, an operator Cµ is uniquely determined by the sesqui- linear form (φ, χ) φ Cµ χ ; positivity is immediate from (4).) Let D(H ) be7→ the h set| ofi all density operators, i.e., of all positive operators in the trace class with trace 1. Lemma 1. Every probability measure µ on S(H ), B(S(H )) possesses a mean and a covariance, as well as a unique operator ρµ :H H such that → φ ρµ χ = µ(dψ) φ ψ ψ χ (5) h | i Z h | ih | i for all φ, χ H ; ρµ, called the density operator of µ, lies in D(H ). If the mean of µ is zero then∈ the density operator coincides with the covariance operator. Proof. We first consider the mean. For any φ H , the function ∈ f : S(H ) C , f(ψ)= φ ψ (6) → h | i is continuous and thus measurable; it is also bounded by φ and thus, in particular, µ-integrable; therefore, k k L(φ) := µ(dψ) f(ψ) (7) Z is a bounded conjugate-linear form on H with L 1. By the Riesz lemma, there is a unique ψ H such that L(φ)= φ ψ , whichk isk ≤equivalent to (3). 0 ∈ h | 0i We now turn to the density operator. For any φ, χ H , the function ∈ f : S(H ) C , f(ψ)= φ ψ ψ χ (8) → h | ih | i is measurable and bounded, f(ψ) φ χ . (9) | |≤k kk k Therefore, f is µ-integrable, and B(φ, χ) := µ(dψ) f(ψ) (10) Z is a bounded sesqui-linear form on H with B 1. By standard arguments using the k k ≤ Riesz lemma, there is a unique operator ρµ : H H such that B(φ, χ) = φ ρµ χ , which is equivalent to (5). → h | i Since, for φ = χ, f(ψ) is always real and non-negative, ρµ is positive (and, in particular, self-adjoint). In order to compute tr ρµ, choose any orthonormal basis ϕn : n N of H and use the monotone convergence theorem to interchange summation{ ∈ } and integration in 2 tr ρµ = ϕn ρµ ϕn = µ(dψ) ϕn ψ (11) h | i Z h | i Xn Xn 2 2 = µ(dψ) ϕn ψ = µ(dψ) ψ =1 . (12) Z h | i Z k k Xn 3 In order to prove the existence of the covariance operator, repeat the above argument for the density operator starting from (8) with ψ replaced by ψ ψ0; (9) gets replaced by − f(ψ) φ 1+ ψ 2 χ , (13) | |≤k k k 0k k k which leads to B (1 + ψ )2, and to a unique operator C such that B(φ, χ) = k k ≤ k 0k µ φ Cµ χ , which is equivalent to (4). h | Thei last statement of Lemma 1 is obvious. 3 Definition of the GAP Measure in Finite Dimen- sion The measure GAP (ρ) on (the Borel σ-algebra of) S(Cd), d N, is built starting from the measure G(ρ), the Gaussian measure on Cd with mean zero∈ and covariance matrix ρ, which can be defined as follows: Let S be the subspace of Cd on which ρ is supported, i.e., its positive spectral subspace, or equivalently the orthogonal complement of its kernel, or equivalently its range; let k = dim S and ρ+ the restriction of ρ to S; then G(ρ) is a measure on Cd supported on S with the following density relative to the Lebesgue measure λ on S: dG(ρ) 1 −1 (ψ)= k exp( ψ ρ+ ψ ) . (14) dλ π det ρ+ −h | | i Now we define the adjusted Gaussian measure GA(ρ) on Cd as: GA(ρ)(dψ)= ψ 2G(ρ)(dψ) . (15) k k If ΨGA is a GA(ρ)-distributed vector, then GAP (ρ) is the distribution of the vector projected to the unit sphere, i.e., of ΨGA ΨGAP := (16) ΨGA k k Like G(ρ), but not like GA(ρ), GAP (ρ) has covariance matrix ρ (see [4] or Lemma 2 below). 4 Definition of the GAP Measure in Infinite Dimen- sion One can define for any measure µ on H , B(H ) the “adjust-and-project” procedure. We denote by Aµ the adjusted measure Aµ(dψ)= ψ 2 µ(dψ) (17) k k 4 and the projection on the unit sphere is defined as: ψ P : H 0 S(H ) , P (ψ)= . (18) \{ }→ ψ k k −1 Then the adjusted-and-projected measure is P∗Aµ = Aµ P , where P∗ denotes the action of P on measures. Since the function P is measurable◦ relative to B(H ) and B(S(H )), P∗A is a mapping from the measures on H to the measures on S(H ). If µ(dψ) ψ 2 =1 (19) Z k k then the adjusted projected measure P∗Aµ is a probability measure on S(H ).