# Contractions of Product Density Operators of Systems of Identical Fermions and Bosons

Authors: Wiktor Radzki
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Recurrence and explicit formulae for contractions (partial traces)
of antisymmetric and symmetric products of identical trace class operators
are derived. Contractions of product density operators of systems of identical
fermions and bosons are proved to be asymptotically equivalent to, respectively,
antisymmetric and symmetric products of density operators of a single
particle, multiplied by a normalization integer. The asymptotic equivalence
relation is defined in terms of the thermodynamic limit of expectation values
of observables in the states represented by given density operators. For
some weaker relation of asymptotic equivalence, concerning the thermodynamic
limit of expectation values of product observables, normalized antisymmetric
and symmetric products of density operators of a single particle are
shown to be equivalent to tensor products of density operators of a single
particle.


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## Body

## 1. Introduction

This paper (see also preprint [1]), presenting the results of a part of the author's thesis [2], deals with contractions (partial traces) of antisymmetric and symmetric product density operators representing mixed states of systems of identical noninteracting fermions and bosons, respectively.Ifℋ is a separable Hilbert space of a single fermion (boson), then the space of the n-fermion (resp. n-boson) system is the antisymmetric (resp. symmetric) subspace ℋ∧n (resp. ℋ∨n) of ℋ⊗n. Density operators of n-fermion (resp. n-boson) systems are identified with those defined on ℋ⊗n and concentrated on ℋ∧n (resp. ℋ∨n).Recall that the expectation value of an observable represented by a bounded self-adjoint operatorB on given Hilbert space in a state described by a density operator ρ equals TrBρ. If B is an unbounded self-adjoint operator on a dense subspace of given Hilbert space, instead of B one can consider its spectral measure EB(Δ) (which is a bounded operator) of a Borel subset Δ of the spectrum of B. Then TrEB(Δ)ρ is the probability that the result of the measurement of the observable in question belongs to Δ [3].k-particle observables of n-fermion and n-boson systems (k<n) are represented, respectively, by operators of the form(1.1)Γkn∧B=AH(n)(B⊗I⊗(n-k))AH(n),Γkn∨B=SH(n)(B⊗I⊗(n-k))SH(n)
(multiplied by (nk)), where Aℋ(n) and Sℋ(n) are projectors of ℋ⊗n onto ℋ∧n and ℋ∨n, respectively, I is the identity operator on ℋ and B is a self-adjoint operator on ℋ⊗k (see [4]). Operators (1.1) are called antisymmetric and symmetric expansions ofB. In view of the earlier remark it is assumed that B is bounded. The expectation values of observables represented by Γkn∧B and Γkn∨B in states represented by n-fermion and n-boson density operators K and G, respectively, can be expressed as(1.2)Tr⁡KΓkn∧B=Tr⁡BLnkK,Tr⁡GΓkn∨B=Tr⁡BLnkG
(see [4, equations (1.7), (3.19)]), where k-particle density operators LnkK and LnkG are (n,k)-contractions ofK and G (see Definition 2.1), called also reduced density operators. Such operators were investigated by Coleman [5], Garrod and Percus [6], and Kummer [4] (see also, e.g., [7–9] and references therein). A presentation of the basic ideas concerning reduced density operators and their applications can be found in [10].In the present paper particular interest is taken in the case whenK and G are product density operators, that is, they are of the form(1.3)K=1Tr⁡ρ∧nρ∧n,G=1Tr⁡ρ∨nρ∨n,
where ρ∧n=Aℋ(n)ρ⊗nAℋ(n), ρ∨n=Sℋ(n)ρ⊗nSℋ(n), and ρ is a density operator of a single fermion or boson, respectively. The first objective of this paper is to find the recurrence and explicit formulae for LnkK and LnkG for K and G being, respectively, antisymmetric and symmetric products of identical trace class operators, including operators (1.3). The explicit form of the operators LnkK and LnkG proves to be quite complex. However, they can be replaced by operators with simpler structure if only the limiting values of expectations (1.2), in the sense of the thermodynamic limit, are of interest. The second objective of this paper is to find that simpler forms of contractions LnkK and LnkG for product density operators (1.3), equivalent to the complete expressions in the thermodynamic limit.The problems described above have been solved fork=1,2 by Kossakowski and Maćkowiak [11], and Maćkowiak [12]. The formulae they derived were exploited in calculations of the free energy density of large interacting n-fermion and n-boson systems [11, 12], as well as in the perturbation expansion of the free energy density for the M-impurity Kondo Hamiltonian [13]. In the case of investigation of many-particle interactions of higher order [14–17], or higher order perturbation expansion terms of the free energy density, the expressions for (Trρ∧n)-1Lnkρ∧n and (Trρ∨n)-1Lnkρ∨n with k≥3 can be used in the canonical and grand canonical ensemble approach, which is the physical motivation for the present paper.The main results of this paper are Theorems3.1, 3.4, 4.9, and 4.14.


## 2. Preliminaries

In this section notation and terminology are set up.


### 2.1. Basic Notation

Let(ℋ,〈·,·〉) be a separable Hilbert space over ℂ or ℝ. The following notation is used in the sequel. I: the identity operator on ℋ,ℬ(ℋ): the space of bounded linear operators on ℋ with the operator norm ∥·∥,𝒯(ℋ): the space of trace class operators on ℋ with the trace norm Tr|·|,ℬ*(ℋ): the space of bounded self-adjoint operators on ℋ,ℬ≥0*(ℋ): the set of nonnegative definite bounded self-adjoint operators on ℋ,𝒟(ℋ): the set of density operators (matrices) on ℋ, that is,
(2.1)D(H)={D∈T(H)∣D=D*,D≥0,Tr⁡D=1}.Setℋ⊗n=ℋ⊗⋯⊗ℋ︸n and denote by Sn the group of permutations of the set {1,…,n}. Let Aℋ(n),Sℋ(n)∈ℬ(ℋ⊗n) be the projectors such that (2.2)AH(n)(ψ1⊗⋯⊗ψn)=1n!∑π∈Snsgn⁡πψπ(1)⊗⋯⊗ψπ(n),SH(n)(ψ1⊗⋯⊗ψn)=1n!∑π∈Snψπ(1)⊗⋯⊗ψπ(n)
for every ψ1,…,ψn∈ℋ. The closed linear subspaces ℋ∧n=Aℋ(n)ℋ⊗n and ℋ∨n=Sℋ(n)ℋ⊗n of ℋ⊗n are called the antisymmetric and symmetric subspace, respectively.Theantisymmetric and symmetric product of operators B∈ℬ(ℋ⊗k), C∈ℬ(ℋ⊗m) are defined as B∧C=Aℋ(k+m)(B⊗C)Aℋ(k+m) and B∨C=Sℋ(k+m)(B⊗C)Sℋ(k+m), respectively. It is assumed B∧n=B∧⋯∧B︸n, B∨n=B∨…∨B︸n, and B∧1=B∨1=B. Clearly, if B∈ℬ(ℋ) then B∧n=Aℋ(n)B⊗n=B⊗nAℋ(n), B∨n=Sℋ(n)B⊗n=B⊗nSℋ(n), and if B∈ℬ≥0*(ℋ) then B∧n,B∨n∈ℬ≥0*(ℋ⊗n).Setℝ+=[0,+∞) and ℝ¯+=ℝ+∪{+∞}. The product of measuresμ,μ1 is denoted by μ⊗μ1 and μ⊗n stands for μ⊗⋯⊗μ︸n. In subsequent sections use is made of product integral kernels, described in the appendix.


### 2.2. Contractions of Operators

The definition and basic properties of contractions of operators are now recalled for the reader's convenience. They were studied in [4–7]. A discussion of properties of reduced density operators can also be found in [10].Letℋ be a separable Hilbert space over the field 𝕂=ℂ or ℝ.Definition 2.1.
Letk,n∈ℕ, k<n, and K∈𝒯(ℋ⊗n). Then the (n,k)-contraction ofK is the operator LnkK∈𝒯(ℋ⊗k) such that
(2.3)∀C∈B(H⊗k):TrH⊗n(C⊗I⊗(n-k))K=TrH⊗kCLnkK.
It is also assumed LnnK=K.Remark 2.2.
The operatorLnkK always exists and is defined uniquely by (2.3). LnkK is a partial trace of K with respect to the component ℋ⊗(n-k) of ℋ⊗n=ℋ⊗k⊗ℋ⊗(n-k). If ℋ=ℋY:=L2(Y,μ), where the measure μis separable and σ-finite, and 𝒦 is a product integral kernel of K (see the appendix) then LnkK has an integral kernel 𝒦0 given by formula (A.4), according to Lemma A.5 and Corollary A.6.Under the assumptions of Definition2.1 one has Trℋ⊗kLnkK=Trℋ⊗nK, and if p∈ℕ, k<p<n, then Lpk(LnpK)=LnkK. Moreover, if K∈ℬ*(ℋ⊗n) then LnkK∈ℬ*(ℋ⊗k), and if K∈ℬ≥0*(ℋ⊗n) then LnkK∈ℬ≥0*(ℋ⊗k).Contractions of density operators are calledreduced density operators. Contractions preserve the Fermi and the Bose-Einstein statistics of the contracted operator, that is, for K∈Aℋ(n)𝒯(ℋ⊗n)Aℋ(n) and G∈Sℋ(n)𝒯(ℋ⊗n)Sℋ(n) one has LnkK∈Aℋ(k)𝒯(ℋ⊗k)Aℋ(k) and LnkG∈Sℋ(k)𝒯(ℋ⊗k)Sℋ(k). For such K and G (1.2) hold.The following theorem is a part of Coleman's theorem [4, 5].Theorem 2.3.
Letn∈ℕ, n≥2. For every (n-fermion) density operator D∈𝒟(ℋ⊗n), D=Aℋ(n)DAℋ(n), one has ∥Ln1D∥≤(1/n)∥D∥.


## 2.1. Basic Notation

Let(ℋ,〈·,·〉) be a separable Hilbert space over ℂ or ℝ. The following notation is used in the sequel. I: the identity operator on ℋ,ℬ(ℋ): the space of bounded linear operators on ℋ with the operator norm ∥·∥,𝒯(ℋ): the space of trace class operators on ℋ with the trace norm Tr|·|,ℬ*(ℋ): the space of bounded self-adjoint operators on ℋ,ℬ≥0*(ℋ): the set of nonnegative definite bounded self-adjoint operators on ℋ,𝒟(ℋ): the set of density operators (matrices) on ℋ, that is,
(2.1)D(H)={D∈T(H)∣D=D*,D≥0,Tr⁡D=1}.Setℋ⊗n=ℋ⊗⋯⊗ℋ︸n and denote by Sn the group of permutations of the set {1,…,n}. Let Aℋ(n),Sℋ(n)∈ℬ(ℋ⊗n) be the projectors such that (2.2)AH(n)(ψ1⊗⋯⊗ψn)=1n!∑π∈Snsgn⁡πψπ(1)⊗⋯⊗ψπ(n),SH(n)(ψ1⊗⋯⊗ψn)=1n!∑π∈Snψπ(1)⊗⋯⊗ψπ(n)
for every ψ1,…,ψn∈ℋ. The closed linear subspaces ℋ∧n=Aℋ(n)ℋ⊗n and ℋ∨n=Sℋ(n)ℋ⊗n of ℋ⊗n are called the antisymmetric and symmetric subspace, respectively.Theantisymmetric and symmetric product of operators B∈ℬ(ℋ⊗k), C∈ℬ(ℋ⊗m) are defined as B∧C=Aℋ(k+m)(B⊗C)Aℋ(k+m) and B∨C=Sℋ(k+m)(B⊗C)Sℋ(k+m), respectively. It is assumed B∧n=B∧⋯∧B︸n, B∨n=B∨…∨B︸n, and B∧1=B∨1=B. Clearly, if B∈ℬ(ℋ) then B∧n=Aℋ(n)B⊗n=B⊗nAℋ(n), B∨n=Sℋ(n)B⊗n=B⊗nSℋ(n), and if B∈ℬ≥0*(ℋ) then B∧n,B∨n∈ℬ≥0*(ℋ⊗n).Setℝ+=[0,+∞) and ℝ¯+=ℝ+∪{+∞}. The product of measuresμ,μ1 is denoted by μ⊗μ1 and μ⊗n stands for μ⊗⋯⊗μ︸n. In subsequent sections use is made of product integral kernels, described in the appendix.


## 2.2. Contractions of Operators

The definition and basic properties of contractions of operators are now recalled for the reader's convenience. They were studied in [4–7]. A discussion of properties of reduced density operators can also be found in [10].Letℋ be a separable Hilbert space over the field 𝕂=ℂ or ℝ.Definition 2.1.
Letk,n∈ℕ, k<n, and K∈𝒯(ℋ⊗n). Then the (n,k)-contraction ofK is the operator LnkK∈𝒯(ℋ⊗k) such that
(2.3)∀C∈B(H⊗k):TrH⊗n(C⊗I⊗(n-k))K=TrH⊗kCLnkK.
It is also assumed LnnK=K.Remark 2.2.
The operatorLnkK always exists and is defined uniquely by (2.3). LnkK is a partial trace of K with respect to the component ℋ⊗(n-k) of ℋ⊗n=ℋ⊗k⊗ℋ⊗(n-k). If ℋ=ℋY:=L2(Y,μ), where the measure μis separable and σ-finite, and 𝒦 is a product integral kernel of K (see the appendix) then LnkK has an integral kernel 𝒦0 given by formula (A.4), according to Lemma A.5 and Corollary A.6.Under the assumptions of Definition2.1 one has Trℋ⊗kLnkK=Trℋ⊗nK, and if p∈ℕ, k<p<n, then Lpk(LnpK)=LnkK. Moreover, if K∈ℬ*(ℋ⊗n) then LnkK∈ℬ*(ℋ⊗k), and if K∈ℬ≥0*(ℋ⊗n) then LnkK∈ℬ≥0*(ℋ⊗k).Contractions of density operators are calledreduced density operators. Contractions preserve the Fermi and the Bose-Einstein statistics of the contracted operator, that is, for K∈Aℋ(n)𝒯(ℋ⊗n)Aℋ(n) and G∈Sℋ(n)𝒯(ℋ⊗n)Sℋ(n) one has LnkK∈Aℋ(k)𝒯(ℋ⊗k)Aℋ(k) and LnkG∈Sℋ(k)𝒯(ℋ⊗k)Sℋ(k). For such K and G (1.2) hold.The following theorem is a part of Coleman's theorem [4, 5].Theorem 2.3.
Letn∈ℕ, n≥2. For every (n-fermion) density operator D∈𝒟(ℋ⊗n), D=Aℋ(n)DAℋ(n), one has ∥Ln1D∥≤(1/n)∥D∥.


## 3. Recurrence and Explicit Formulae for Contractions of Products of Trace Class Operators

In this section recurrence and explicit formulae for contractions of antisymmetric and symmetric powers of single particle operators are derived.In the whole section use is made of the Hilbert spaceℋY:=L2(Y,μ) over the field 𝕂=ℂ or ℝ, where the measure μ is separable and σ-finite.The following theorem in the case ofk=1,2 was proved in [11, 12].Theorem 3.1 (Recurrence formulae).
Letρ∈𝒯(ℋY). If k,n∈ℕ, 1<k<n, then
(3.1)(nk)Lnkρ∧n=(n-1k-1)(Ln-1k-1ρ∧(n-1))∧ρ-(n-1k)(Ln-1kρ∧(n-1))(I⊗(k-1)⊗ρ)AHY(k),(3.2)(nk)Lnkρ∨n=(n-1k-1)(Ln-1k-1ρ∨(n-1))∨ρ+(n-1k)(Ln-1kρ∨(n-1))(I⊗(k-1)⊗ρ)SHY(k),
and if n∈ℕ, n≥2, then
(3.3)nLn1ρ∧n=(Tr⁡ρ∧(n-1))ρ-(n-1)(Ln-11ρ∧(n-1))ρ,(3.4)nLn1ρ∨n=(Tr⁡ρ∨(n-1))ρ+(n-1)(Ln-11ρ∨(n-1))ρ.Proof.
Letϱ:Y×Y→𝕂 be a product integral kernel of ρ. For every m∈ℕ define the mapping ϱ∧m:Ym×Ym→𝕂 by the formula
(3.5)ϱ∧m(x1,…,xmy1,…,ym)=det⁡[ϱ(x1,y1)⋯ϱ(x1,ym)⋮⋯⋮ϱ(xm,y1)⋯ϱ(xm,ym)].
Then the mapping 𝒦:Yn×Yn→𝕂 given by
(3.6)K(x1,…,xn,y1,…,yn)=1n!ϱ∧n(x1,…,xny1,…,yn)
is a product integral kernel of ρ∧n=AℋY(n)ρ⊗n.
Equation (3.1) will be first proved for n>k+1. In view of Remark 2.2, an integral kernel ℒ:Yk×Yk→𝕂 of (nk)Lnkρ∧n can be given by(3.7)L(x′,y′)=(nk)∫Yn-kK(x′,x′′,y′,x′′)dμ⊗(n-k)(x′′)
for μ⊗2k-a.a. (x′,y′)∈Yk×Yk. Performing k! permutations of the first k rows and k! permutations of the first k columns of the determinant defining 𝒦 and expanding that determinant with respect to the kth column one obtains
(3.8)L(x1,…,xk,y1,…,yk)=(nk)1n!1(k!)2∑π,σ∈Sksgn⁡πsgn⁡σ∑j=1k(-1)k+j×∫Yn-kϱ(xπ(j),yσ(k))⋅ϱ∧(n-1)(xπ(1),…,xπ(j-1),xπ(j+1),…,xπ(k),xk+1,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)×dμ⊗(n-k)(xk+1,…,xn)+(nk)1n!1(k!)2∑π,σ∈Sksgn⁡πsgn⁡σ∑j=k+1n(-1)k+j×∫Yn-kϱ(xj,yσ(k))⋅ϱ∧(n-1)(xπ(1),…,xπ(k),xk+1,…,xj-1,xj+1,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)×dμ⊗(n-k)(xk+1,…,xn).
Consider the first term on the r.h.s. of (3.8). In all summands of ∑j=1k except the last one the (k-1)th row of the determinant (containing the variable xπ(k)) can be shifted into the jth position, changing thereby the sign of the determinant by (-1)(k-2)-(j-1)=(-1)-k-j+1. Then the first term of sum (3.8) assumes the form(3.9)(nk)1n!1(k!)2∑π,σ∈Sksgn⁡πsgn⁡σ∑j=1k-1(-1)k+j(-1)-k-j+1×∫Yn-kϱ(xπ(j),yσ(k))⋅ϱ∧(n-1)(xπ(1),…,xπ(j-1),xπ(k),xπ(j+1),…,xπ(k-1),xk+1,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)×dμ⊗(n-k)(xk+1,…,xn)+(nk)1n!1(k!)2∑π,σ∈Sksgn⁡πsgn⁡σ(-1)k+k×∫Yn-kϱ(xπ(k),yσ(k))⋅ϱ∧(n-1)(xπ(1),…,xπ(k-1),xk+1,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)dμ⊗(n-k)(xk+1,…,xn).
Let Tjk∈Sk denote the transposition j↔k for j<k (then (-1)k+j(-1)-k-j+1=(-1)=sgnTjk) and the identity permutation for j=k (with sgnTkk=1). Expression (3.9) can be written as
(3.10)∑j=1k(nk)1n!1(k!)2∑π,σ∈Sk(sgn⁡πsgn⁡Tjk)sgn⁡σ×∫Yn-kϱ(x(π∘Tjk)(k),yσ(k))⋅ϱ∧(n-1)(x(π∘Tjk)(1),…,x(π∘Tjk)(k-1),xk+1,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)×dμ⊗(n-k)(xk+1,…,xn)=(n-1k-1)1(k!)2∑τ,σ∈Sksgn⁡τsgn⁡σϱ(xτ(k),yσ(k))×∫Yn-k1(n-1)!⋅ϱ∧(n-1)(xτ(1),…,xτ(k-1),xk+1,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)dμ⊗(n-k)(xk+1,…,xn).
The function 𝒫1:Yk×Yk→𝕂, such that 𝒫1(x1,…,xk,y1,…,yk) is μ⊗2k-a.e. equal to expression (3.10), is an integral kernel of the operator
(3.11)(n-1k-1)(Ln-1k-1ρ∧(n-1))∧ρ,
which appears on the r.h.s. of (3.1).
Consider now the second term of the sum on the r.h.s. of (3.8). One can change the indices of the integral variables xk+1,…,xj in all summands of ∑j=k+1n except the first one, according to the rule xj→xk+1→xk+2→⋯→xj for the jth summand, and simultaneously change the order of the columns of the determinant inversely (which changes the sign by (-1)(j-1)-k=(-1)(k+1)-j). The resulting sum ∑j=k+1n then contains n-k terms identical to the one with j=k+1. Thus the second term of sum (3.8) equals(3.12)-(n-k)(nk)1n!1(k!)2∑π,σ∈Sksgn⁡πsgn⁡σ×∫Yn-kϱ(xk+1,yσ(k))⋅ϱ∧(n-1)(xπ(1),…,xπ(k),xk+2,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)dμ⊗(n-k)(xk+1,…,xn).=-(n-1k)1k!∑σ∈Sksgn⁡σ×∫Yϱ(xk+1,yσ(k))(∫Yn-1-k1(n-1)!⋅ϱ∧(n-1)(x1,…,xk,xk+2,…,xnyσ(1),…,yσ(k-1),xk+1,…,xn)×dμ⊗(n-1-k)(xk+2,…,xn)∫Yn-1-k)dμ(xk+1).
The function 𝒫2:Yk×Yk→𝕂, such that 𝒫2(x1,…,xk,y1,…,yk) is μ⊗2k-a.e. equal to expression (3.12), is an integral kernel of the operator
(3.13)-(n-1k)(Ln-1kρ∧(n-1))(I⊗(k-1)⊗ρ)AHY(k),
which occurs on the r.h.s. of (3.1). One concludes that the kernel ℒ of the operator on the l.h.s. of (3.1) is μ⊗2k-a.e. equal to the kernel 𝒫1+𝒫2 of the operator on the r.h.s. of (3.1), which proves the equality of both operators.
The proof of (3.1) for n=k+1 and the proof of (3.3) proceed analogously.
Similarly, the proof of (3.2) and (3.4) is accomplished by changing the product ∧ into ∨ and replacing determinants in all formulae by pernaments, defined for every complex matrix A=[ai,j]i,j=1m as (3.14)perA=∑π∈Smaπ(1),1⋯aπ(m),m.
Notice that signs of permutations are omitted in this case, similarly as the multipliers ±1 in the Laplace expansions.Lemma 3.2.
Letk,m∈ℕ, 1<k<m, ρ∈𝒯(ℋY), jk∈{k,…,m}, and
(3.15)R:=∑jk-1=k-1jk-1∑jk-2=k-2jk-1-1…∑j1=1j2-1ρj1⊗ρj2-j1⊗⋯⊗ρjk-jk-1
(for k=2 the only summation index is jk-1=j1). Then AℋY(k)R=RAℋY(k) and SℋY(k)R=RSℋY(k).The proof of the above lemma consists in demonstrating the invariance ofR under permutations of factors in the tensor products. To this end it suffices to observe that R is invariant under transpositions of neighbouring factors.Lemma 3.3.
Letρ∈𝒯(ℋY), ξs∧:=Trρ∧s, ξs∨:=Trρ∨s for s∈ℕ, ξ0∧:=1, ξ0∨:=1, and
(3.16)Πm∧p(ρ):=∑jp=pm∑jp-1=p-1jp-1…∑j1=1j2-1ξm-jp∧(-1)p+jpρj1∧ρj2-j1∧⋯∧ρjp-jp-1,Πm∨p(ρ):=∑jp=pm∑jp-1=p-1jp-1…∑j1=1j2-1ξm-jp∨ρj1∨ρj2-j1∨⋯∨ρjp-jp-1
for p,m∈ℕ, p≤m. (For p=1 the only summation index is j1 and the summation runs over the operators ρj1.) If 2≤p<m then
(3.17)Πm∧p(ρ)=(Πm-1∧(p-1)(ρ))∧ρ-(Πm-1∧p(ρ))(I⊗(p-1)⊗ρ)AHY(p),(3.18)Πm∨p(ρ)=(Πm-1∨(p-1)(ρ))∨ρ+(Πm-1∨p(ρ))(I⊗(p-1)⊗ρ)SHY(p).Proof.
Equation (3.17) will be first proved for p>2. One has
(3.19)Πm∧p(ρ)=ξm-p∧ρ∧⋯∧ρ+∑lp=pm-1∑lp-1=p-1lp∑lp-2=p-2lp-1-1…∑l1=1l2-1ξm-lp-1∧(-1)p+lp+1ρl1∧ρl2-l1∧⋯∧ρlp-1-lp-2∧ρlp-lp-1+1(3.20)=ξm-p∧ρ∧⋯∧ρ+∑lp=pm-1∑lp-1=p-1lp-1∑lp-2=p-2lp-1-1…∑l1=1l2-1ξm-lp-1∧(-1)p+lp+1ρl1∧ρl2-l1∧⋯∧ρlp-1-lp-2∧ρlp-lp-1+1+∑lp=pm-1∑lp-2=p-2lp-1∑lp-3=p-3lp-2-1…∑l1=1l2-1ξm-lp-1∧(-1)p+lp+1ρl1∧ρl2-l1∧⋯∧ρlp-2-lp-3∧ρlp-lp-2∧ρ.
The first and the third term after the last of equalities (3.20) yield
(3.21)∑jp-1=p-1m-1∑jp-2=p-2jp-1-1…∑j1=1j2-1(ξ(m-1)-jp-1∧(-1)(p-1)+jp-1ρj1∧ρj2-j1∧⋯∧ρjp-1-jp-2)∧ρ
for lp≡jp-1, lp-2≡jp-2,…, l1≡j1. By Lemma 3.2, the second term after the last of equalities (3.20) equals
(3.22)-(Πm-1∧p(ρ))(I⊗(p-1)⊗ρ)AHY(p).
The sum of expressions (3.21) and (3.22) is equal to the r.h.s. of (3.17) for p>2. After simplifications the proof also applies to the case of p=2.
The proof of (3.18) is analogous to that of (3.17).The next theorem provides the explicit form of(n,k)-contractions of product operators. The proof for k=1,2 was given in [11, 12]. The author of [12] emphasized that formula (3.23) for k=2 was derived by S. Pruski in 1978.Theorem 3.4 (Explicit formulae).
Letk,n∈ℕ, k<n, ρ∈𝒯(ℋY), ξs∧:=Trρ∧s, ξs∨:=Trρ∨s for s∈ℕ, and ξ0∧:=1, ξ0∨:=1. Then
(3.23)(nk)Lnkρ∧n=∑jk=kn∑jk-1=k-1jk-1…∑j1=1j2-1ξn-jk∧(-1)k+jkρj1∧ρj2-j1∧⋯∧ρjk-jk-1=∑i1=1n-(k-1)∑i2=1n-i1-(k-2)…∑ik-1=1n-i1-⋯-ik-2-1∑ik=1n-i1-⋯-ik-1ξn-i1-⋯-ik∧(-1)k+i1+⋯+ikρi1∧⋯∧ρik,(3.24)(nk)Lnkρ∨n=∑jk=kn∑jk-1=k-1jk-1…∑j1=1j2-1ξn-jk∨ρj1∨ρj2-j1∨⋯∨ρjk-jk-1=∑i1=1n-(k-1)∑i2=1n-i1-(k-2)…∑ik-1=1n-i1-⋯-ik-2-1∑ik=1n-i1-⋯-ik-1ξn-i1-⋯-ik∨ρi1∨⋯∨ρik.
(For k=1 the only summation indices are j1 and i1 and the summation runs over the operators ρj1 and ρi1, resp.)Proof.
For everyp,m∈ℕ, p≤m, let Πm∧p(ρ) be defined as in Lemma 3.3. Then the first of equalities (3.23) can be written as
(3.25)(nk)Lnkρ∧n=Πn∧k(ρ).
The proof of (3.25) will be carried out by (double) induction with respect to k and, for fixed k, with respect to n>k.(1∘)
(k=1) This part of the proof is by induction with respect to n>1.
(a)
(n=2) According to Theorem 3.1, 2L21ρ∧2=(Trρ)ρ-ρ2=Π2∧1(ρ).(b)
Assuming validity of formula (3.25) (with k=1) for n∈{2,…,m-1}, where m∈ℕ, m>2, its validity will be shown for n=m.
One has(3.26)Πm∧1(ρ)=ξm-1∧ρ+∑j1=2mξm-j1∧(-1)1+j1ρj1=ξm-1∧ρ-(Πm-1∧1(ρ))ρ.
Thus, according to the inductive hypothesis for n∈{2,…,m-1},
(3.27)Πm∧1(ρ)=ξm-1∧ρ-(m-1)(Lm-11ρ∧(m-1))ρ,
which, in view of Theorem 3.1, yields (m1)Lm1ρ∧m=Πm∧1(ρ).(2∘)
Assuming validity of formula (3.25) for k∈{1,…,p-1} (and every n>k), where p∈ℕ, p>1, its validity will be shown for k=p. For arbitrarily fixed p the proof will be carried out by induction with respect to n>p.
(a)
(n=p+1) By the inductive hypothesis with respect to k and Lemma 3.3,
(3.28)Πp+1∧p(ρ)=((p+1)-1p-1)(L(p+1)-1p-1ρ∧((p+1)-1))∧ρ-ρ∧p(I⊗(p-1)⊗ρ)AHY(p),
hence (p+1p)Lp+1pρ∧(p+1)=Πp+1∧p(ρ), according to Theorem 3.1.(b)
Assuming validity of formula (3.25) for n∈{p+1,…,m-1}, where k=p, m∈ℕ, m>p+1, its validity will be shown for n=m.
By the inductive hypothesis fork∈{1,…,p-1} and Lemma 3.3 one has(3.29)Πm∧p(ρ)=(m-1p-1)(Lm-1p-1ρ∧(m-1))∧ρ-(Πm-1∧p(ρ))(I⊗(p-1)⊗ρ)AHY(p).
According to the inductive hypothesis for n∈{p+1,…,m-1} one thus obtains
(3.30)Πm∧p(ρ)=(m-1p-1)(Lm-1p-1ρ∧(m-1))∧ρ-(m-1p)(Lm-1pρ∧(m-1))(I⊗(p-1)⊗ρ)AHY(p),
which, in view of Theorem 3.1, yields (mp)Lmpρ∧m=Πm∧p(ρ). This completes the inductive proof for (3.25) with respect to n>p and with respect to k.
Now turn to the second of equalities (3.23). For k=1 it is identity. Let k≥2. Setting j1=i1, j2=i1+i2,…, jk=i1+⋯+ik or, equivalently, i1=j1, i2=j2-j1, i3=j3-j2,…, ik=jk-jk-1, one checks that both sides of the equality in question are equal to(3.31)∑j1=1n-(k-1)∑j2=j1+1n-(k-2)…∑jk-1=jk-2+1n-1∑jk=jk-1+1nξn-jk∧(-1)k+jkρj1∧ρj2-j1∧⋯∧ρjk-jk-1.
The proof of (3.24) is analogous to that of (3.23).


## 4. Asymptotic Form for Contractions of Product States

The explicit forms of the contractions of product states given by Theorem3.4 are quite complex. In the present section they are replaced by simpler operators, equivalent in the thermodynamic limit. The main results in this section are Theorems 4.9 and 4.14.In what follows use is made of the Hilbert spaceℋΩ:=L2(Ω,μ) (over ℂ or ℝ), where the measureμ is separable, σ-finite, and satisfies the condition μ(Ω)=+∞. For everyμ-measurable subset Y⊂Ω it is assumed ℋY:=L2(Y,μ).Letℳ(Ω) be a fixed family of measurable subsets of Ω such that 0<μ(Y)<+∞ for every Y∈ℳ(Ω) (it can be the family of all such subsets). Fix d∈ℝ, d>0, and assume that there exists a sequence {Yn}n∈ℕ⊂ℳ(Ω) such that n/μ(Yn)→d as n→∞.Definition 4.1.
Fixd∈ℝ, d>0, and let {bY,n}(Y,n)∈ℳ(Ω)×ℕ be a family of complex numbers. A complex number b is said to be the thermodynamic limit of this family if for every sequence {Yn}n∈ℕ⊂ℳ(Ω) such that limn→∞n/μ(Yn)=d the condition limn→∞bYn,n=b is fulfilled. In such a case b is denoted by d-limn,μ(Y)→∞bY,n.Special attention will be given to the families of complex numbers of the formTr(LnkKY,n)CY, where k,n∈ℕ, n>k, KY,n∈𝒯(ℋY⊗n), and CY∈ℬ(ℋY⊗k).Definition4.1 does not guarantee the convergence of families {bY,n} of interest in physics. To obtain such a convergence, additional conditions (such as conditions of uniform growth [18]) are usually imposed on the sequence {Yn}n∈ℕ in question. However, those additional conditions do not affect considerations in this paper.Expression of expectation values of observables in mixed states by using trace, mentioned in Introduction, is the motivation for the following definition.Definition 4.2.
Fixk∈ℕ and d∈ℝ, d>0. Families {AY,n}(Y,n)∈ℳ(Ω)×ℕ and {BY,n}(Y,n)∈ℳ(Ω)×ℕ of operators AY,n,BY,n∈𝒯(ℋY⊗k) are said to be asymptotically equivalent (symbolically: AY,n≈BY,n), if for every family {CY,n}(Y,n)∈ℳ(Ω)×ℕ of operators CY,n∈ℬ(ℋY⊗k) with uniformly bounded operator norms one has
(4.1)d-lim⁡n,μ(Y)→∞Tr⁡(AY,n-BY,n)CY,n=0.Condition (4.1) is required to hold in particular for families {CY,n}(Y,n)∈ℳ(Ω)×ℕ such that CY,n=CY,m for all Y∈ℳ(Ω), n,m∈ℕ.Remark 4.3.
The authors of [11, 12] used some different definition of asymptotic equivalence of families of operators, closer to Definition 4.10 in this paper.Remark 4.4.
For fixedk∈ℕ and d∈ℝ, d>0, the relation ≈ is an equivalence relation. If AY,n≈BY,n then for every family of operators CY,n as in Definition 4.2 the limit d-limn,μ(Y)→∞TrAY,nCY,n exists if and only if the limit d-limn,μ(Y)→∞TrBY,nCY,n exists, in which case both limits are equal. Notice also that if AY,n≈BY,n then AY,n+DY,n≈BY,n+DY,n and aAY,n≈aBY,n for every family {DY,n}(Y,n)∈ℳ(Ω)×ℕ⊂𝒯(ℋY⊗k) and a∈ℂ. Furthermore, for every family {AY,n}(Y,n)∈ℳ(Ω)×ℕ⊂𝒯(ℋY⊗k) with uniformly bounded trace norms Tr|AY,n| and for every sequence {an}n∈ℕ⊂ℂ convergent to a∈ℂ one has anAY,n≈aAY,n.Lemma 4.5.
Let{AY,n}(Y,n)∈ℳ(Ω)×ℕ and {BY,n}(Y,n)∈ℳ(Ω)×ℕ be as in Definition 4.2. Then
(4.2)d-lim⁡n,μ(Y)→∞Tr⁡|AY,n-BY,n|=0⟹AY,n≈BY,n.
Moreover, if the operators AY,n, BY,n are self-adjoint then
(4.3)AY,n≈BY,n⟹d-lim⁡n,μ(Y)→∞Tr⁡|AY,n-BY,n|=0.Proof.
Implication (4.2) follows from Definition 4.2 and the estimate
(4.4)|Tr⁡(AY,n-BY,n)CY,n|≤‖CY,n‖Tr⁡|AY,n-BY,n|.
Now assume thatAY,n≈BY,n, which is equivalent to the condition(4.5)DY,n≈0,
where DY,n:=AY,n-BY,n. The operators DY,n have the spectral representations
(4.6)DY,n=∑i=1∞λi(Y,n)Pφi(Y,n),
where Pφi(Y,n) are the projectors onto orthogonal one dimensional subspaces of eigenvectors φi(Y,n) of DY,n, corresponding to eigenvalues λi(Y,n)∈ℝ. Since ∑i=1∞|λi(Y,n)|=Tr|DY,n|<+∞, for every (Y,n)∈ℳ(Ω)×ℕ there exists m(Y,n)∈ℕ such that ∑i=m(Y,n)+1∞|λi(Y,n)|<1/n. Thus the operators
(4.7)FY,n=∑i=1m(Y,n)λi(Y,n)Pφi(Y,n)
satisfy the condition
(4.8)d-lim⁡n,μ(Y)→∞Tr⁡|DY,n-FY,n|=d-lim⁡n,μ(Y)→∞∑i=m(Y,n)+1∞|λi(Y,n)|=0,
which, in view of implication (4.2) proved and condition (4.5), yields FY,n≈DY,n≈0. In particular,
(4.9)d-lim⁡n,μ(Y)→∞Tr⁡FY,nCY,n=0,
where
(4.10)CY,n=∑i=1m(Y,n)sgn⁡(λi(Y,n))Pφi(Y,n),‖CY,n‖=1.
Observe that TrFY,nCY,n=Tr|FY,n|, hence condition (4.9) gives
(4.11)d-lim⁡n,μ(Y)→∞Tr⁡|FY,n|=0.
Since Tr|DY,n|≤Tr|DY,n-FY,n|+Tr|FY,n|, conditions (4.8) and (4.11) yield
(4.12)d-lim⁡n,μ(Y)→∞Tr⁡|AY,n-BY,n|≡d-lim⁡n,μ(Y)→∞Tr⁡|DY,n|=0,
which proves implication (4.3).The following lemma follows from Lemma4.5.Lemma 4.6.
Fixk,m∈ℕ. Let {AY,n}(Y,n)∈ℳ(Ω)×ℕ and {BY,n}(Y,n)∈ℳ(Ω)×ℕ be families of self-adjoint operators AY,n,BY,n∈𝒯(ℋY⊗k) such that AY,n≈BY,n, and let {DY,n}(Y,n)∈ℳ(Ω)×ℕ be a family of operators DY,n∈𝒯(ℋY⊗m) with uniformly bounded trace norms Tr|DY,n|. Then
(4.13)AY,n⊗DY,n≈BY,n⊗DY,n,DY,n⊗AY,n≈DY,n⊗BY,n,AY,n∧DY,n≈BY,n∧DY,n,DY,n∧AY,n≈DY,n∧BY,n,AY,n∨DY,n≈BY,n∨DY,n,DY,n∨AY,n≈DY,n∨BY,n.In the sequel{ρY}Y∈ℳ(Ω) denotes a family of nonnegative definite self-adjoint operators ρY∈𝒯(ℋY), and for every (Y,n)∈ℳ(Ω)×ℕ it is assumed that (4.14)ξY,0∧:=1,ξY,0∨:=1,ρY∧1:=ρY,ρY∨1:=ρY,ξY,n∧:=Tr⁡ρY∧n>0,ξY,n∨:=Tr⁡ρY∨n>0,sY,n∧:=ξY,n-1∧ξY,n∧,sY,n∨:=ξY,n-1∨ξY,n∨.The objective of this section is to find density operators of the most simple form which are asymptotically equivalent to the operators(4.15)σ∧Y,n(k):=Lnk(1ξY,n∧ρY∧n),σ∨Y,n(k):=Lnk(1ξY,n∨ρY∨n),
defined for fixed k∈ℕ and every (Y,n)∈ℳ(Ω)×ℕ, n>k.Remark 4.7.
For every(Y,n)∈ℳ(Ω)×ℕ the operator I+sY,n+1∧ρY is invertible and ∥(I+sY,n+1∧ρY)-1∥=1. Furthermore, if sY,n+1∨∥ρY∥<1 then I-sY,n+1∨ρY is invertible and ∥(I-sY,n+1∨ρY)-1∥=(1-sY,n+1∨∥ρY∥)-1.The next theorem is a version of a theorem studied in [11, 12] (see Remark 4.3).Theorem 4.8.
Ifσ∧Y,n(1)≈σ∧Y,n+1(1) and the reals sY,n+1∧∥ρY∥, (Y,n)∈ℳ(Ω)×ℕ, are uniformly bounded then
(4.16)σ∧Y,n(1)(I+sY,n+1∧ρY)≈(n+1)-1sY,n+1∧ρY,(4.17)σ∧Y,n(1)≈(n+1)-1sY,n+1∧ρY(I+sY,n+1∧ρY)-1.
If σ∨Y,n(1)≈σ∨Y,n+1(1) and the reals sY,n∨∥ρY∥, (Y,n)∈ℳ(Ω)×ℕ, are uniformly bounded then
(4.18)σ∨Y,n(1)(I-sY,n+1∨ρY)≈(n+1)-1sY,n+1∨ρY.
If, additionally, sY,n∨∥ρY∥≤ϵ for some ϵ<1 and every (Y,n)∈ℳ(Ω)×ℕ then
(4.19)σ∨Y,n(1)≈(n+1)-1sY,n+1∨ρY(I-sY,n+1∨ρY)-1.Proof.
By Theorem3.1 and the assumption σ∧Y,n(1)≈σ∧Y,n+1(1) one has
(4.20)σ∧Y,n(1)-(n+1)-1sY,n+1∧ρY≈-(n+1)-1nσ∧Y,n(1)(sY,n+1∧ρY).
Since Tr|σ∧Y,n(1)(sY,n+1∧ρY)|≤sY,n+1∧∥ρY∥Tr|σ∧Y,n(1)|=sY,n+1∧∥ρY∥, relation (4.20) yields (4.16), in view of Remark 4.4.
Now turn to the proof of relation (4.17). According to Remark 4.7,(4.21)Tr⁡|σ∧Y,n(1)-(n+1)-1sY,n+1∧ρY(I+sY,n+1∧ρY)-1|≤‖(I+sY,n+1∧ρY)-1‖Tr⁡|σ∧Y,n(1)(I+sY,n+1∧ρY)-(n+1)-1sY,n+1∧ρY|=Tr⁡|σ∧Y,n(1)(I+sY,n+1∧ρY)-(n+1)-1sY,n+1∧ρY|.
The explicit form of σ∧Y,n(1) given by Theorem 3.4 shows that σ∧Y,n(1) commutes with I+sY,n+1∧ρY, and since both operators are self-adjoint, σ∧Y,n(1)(I+sY,n+1∧ρY) is also self-adjoint. Thus conditions (4.16), (4.21), and Lemma 4.5 yield (4.17).
The proof of relations (4.18), (4.19) runs parallel to that of (4.16), (4.17). Notice that in this case the expression ∥(I+sY,n+1∧ρY)-1∥=1 from estimate (4.21) is replaced by ∥(I-sY,n+1∧ρY)-1∥=(1-sY,n+1∧∥ρY∥)-1≤(1-ϵ)-1 (see Remark 4.7).The following theorem fork=2 (with the reservation of Remark 4.3) was obtained in [11, 12]. The author of [12] gave also arguments that can be used to check the assumptions of this theorem.Theorem 4.9 (Asymptotic formulae I).
Ifσ∧Y,n(k)≈σ∧Y,n+1(k) for every k∈ℕ and
(4.22)sY,n∧‖ρY‖≤2forevery(Y,n)∈M(Ω)×N
then, for every k∈ℕ, k≥2,
(4.23)σ∧Y,n(k)≈k!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)︸k.
Ifσ∨Y,n(k)≈σ∨Y,n+1(k) for every k∈ℕ and(4.24)sY,n∨‖ρY‖≤ϵforsomeϵ<1andevery(Y,n)∈M(Ω)×N
then, for every k∈ℕ, k≥2,
(4.25)σ∨Y,n(k)≈k!σ∨Y,n(1)∨⋯∨σ∨Y,n(1)︸k.Proof.
First equivalence (4.23) will be proved. Observe that
(4.26)2Tr⁡|σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)|=Tr⁡|(σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1))(I⊗(q-1)⊗(I+sY,n+1∧ρY))AHY(q)+(σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1))(I⊗(q-1)⊗(I-sY,n+1∧ρY))AHY(q)|≤Tr⁡|(σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1))(I⊗(q-1)⊗(I+sY,n+1∧ρY))AHY(q)|+‖I-sY,n+1∧ρY‖Tr⁡|σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)|,
hence
(4.27)(2-‖I-sY,n+1∧ρY‖)Tr⁡|σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)|≤Tr⁡|(σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1))(I⊗(q-1)⊗(I+sY,n+1∧ρY))AHY(q)|.
Since the operators ρY are trace class, infφ∈ℋY;∥φ∥=1〈φ,ρYφ〉=0. Thus, by assumption (4.22) and the self-adjointness of the operators I-sY,n+1∧ρY, one obtains
(4.28)‖I-sY,n+1∧ρY‖=sup⁡φ∈HY‖φ‖=1⁡|〈φ,(I-sY,n+1∧ρY)φ〉|=max⁡{1-sY,n+1∧inf⁡φ∈HY‖φ‖=1⁡〈φ,ρYφ〉,sY,n+1∧sup⁡φ∈HY‖φ‖=1⁡〈φ,ρYφ〉-1}=1.
The rest of the proof of (4.23) is by induction with respect to k≥2.
(1∘) (k=2) By Theorem 3.1 for n≥2 one has(4.29)1(n+1)2(n+12)σ∧Y,n+1(2)=nn+1σ∧Y,n(1)∧((n+1)-1sY,n+1∧ρY)-1(n+1)2(n2)σ∧Y,n(2)(I⊗(sY,n+1∧ρY))AHY(2).
Assumption (4.22) gives Tr|σ∧Y,n(2)(I⊗(sY,n+1∧ρY))AℋY(2)|≤sY,n+1∧∥ρY∥Tr|σ∧Y,n(2)|≤2, hence, by (4.29), Remark 4.4, and the assumption σ∧Y,n(2)≈σ∧Y,n+1(2), one obtains
(4.30)σ∧Y,n(2)+σ∧Y,n(2)(I⊗(sY,n+1∧ρY))AHY(2)≈2nn+1σ∧Y,n(1)∧((n+1)-1sY,n+1∧ρY).
Thus, in view of equivalence (4.16) from Theorem 4.8 and Lemma 4.6, one has
(4.31)σ∧Y,n(2)+σ∧Y,n(2)(I⊗(sY,n+1∧ρY))AHY(2)≈2nn+1σ∧Y,n(1)∧(σ∧Y,n(1)(I+sY,n+1∧ρY)).
Furthermore, assumption (4.22) implies that the trace norms of the operators on the r.h.s of (4.31) are uniformly bounded. Therefore, according to Remark 4.4,
(4.32)(σ∧Y,n(2)-2σ∧Y,n(1)∧σ∧Y,n(1))(I⊗(I+sY,n+1∧ρY))AHY(2)≈0.
The explicit form of σ∧Y,n(2), σ∧Y,n(1)∧σ∧Y,n(1) given by Theorem 3.4 implies that σ∧Y,n(2)-2σ∧Y,n(1)∧σ∧Y,n(1) and (I⊗(I+sY,n+1∧ρY))AℋY(2) commute, which proves the self-adjointness of the operator on the l.h.s of (4.32). Thus conditions (4.32), (4.27) for q=2, (4.28), and Lemma 4.5 yield relation (4.23) for k=2.
(2∘) Assuming validity of equivalence (4.23) for k∈{2,…,q-1}, where q∈ℕ, q>2, its validity will be proved for k=q.
By Theorem3.1 for n≥q one has(4.33)1(n+1)q(n+1q)σ∧Y,n+1(q)=1(n+1)q-1(nq-1)σ∧Y,n(q-1)∧((n+1)-1sY,n+1∧ρY)-1(n+1)q(nq)σ∧Y,n(q)(I⊗(q-1)⊗(sY,n+1∧ρY))AHY(q).
Assumption (4.22) implies
(4.34)Tr⁡|σ∧Y,n(q)(I⊗(q-1)⊗(sY,n+1∧ρY))AHY(q)|≤sY,n+1∧‖ρY‖Tr⁡|σ∧Y,n(q)|≤2,
hence, in view of (4.33), Remark 4.4, and the assumption σ∧Y,n(q)≈σ∧Y,n+1(q),
(4.35)σ∧Y,n(q)+σ∧Y,n(q)(I⊗(q-1)⊗(sY,n+1∧ρY))AHY(q)≈q!(n+1)q-1(nq-1)σ∧Y,n(q-1)∧((n+1)-1sY,n+1∧ρY).
Thus, by relation (4.16) from Theorem 4.8, Lemma 4.6, and Remark 4.4, one has
(4.36)σ∧Y,n(q)+σ∧Y,n(q)(I⊗(q-1)⊗(sY,n+1∧ρY))AHY(q)≈q!(q-1)!σ∧Y,n(q-1)∧(σ∧Y,n(1)(I+sY,n+1∧ρY)),
since the trace norms of the operators on the r.h.s. of (4.36) are uniformly bounded, by assumption (4.22). Furthermore, in view of Lemma 4.6 and the inductive hypothesis σ∧Y,n(q-1)≈(q-1)!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)︸q-1, condition (4.36) yields
(4.37)σ∧Y,n(q)+σ∧Y,n(q)(I⊗(q-1)⊗(sY,n+1∧ρY))AHY(q)≈q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)︸q-1∧(σ∧Y,n(1)(I+sY,n+1∧ρY)),
hence
(4.38)(σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)︸q)(I⊗(q-1)⊗(I+sY,n+1∧ρY))AHY(q)≈0.
From the explicit form of σ∧Y,n(q), σ∧Y,n(1)∧⋯∧σ∧Y,n(1) given by Theorem 3.4 one finds that σ∧Y,n(q)-q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1) and (I⊗(q-1)⊗(I+sY,n+1∧ρY))AℋY(q) commute, which proves the self-adjointness of the operator on the l.h.s of (4.38). Thus conditions (4.38), (4.27), (4.28), and Lemma 4.5 yield σ∧Y,n(q)≈q!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)︸q. Validity of relation (4.23) has been proved.
Now turn to equivalence (4.25). Similarly to (4.27) one has (4.39)(2-‖I+sY,n+1∨ρY‖)Tr⁡|σ∨Y,n(q)-q!σ∨Y,n(1)∨⋯∨σ∨Y,n(1)|≤Tr⁡|(σ∨Y,n(q)-q!σ∨Y,n(1)∨⋯∨σ∨Y,n(1))(I⊗(q-1)⊗(I-sY,n+1∨ρY))SHY(q)|.
Furthermore, according to assumption (4.24),
(4.40)2-‖I+sY,n+1∨ρY‖≥2-(1+sY,n+1∨‖ρY‖)≥1-ϵ>0.
The rest of the proof of (4.25) is by induction with respect to k≥2 and proceeds analogously to the proof of (4.23) with condition (4.28) replaced by (4.40) and the operators I∓sY,n+1∧ρY replaced by I±sY,n+1∨ρY (inversion of signs).Theorem4.9 allows to replace (n,k)-contractions of antisymmetric and symmetric product density operators by antisymmetric and symmetric products of 1-particle contractions, respectively, if the number n of particles in the system is large. Further simplification, consisting in replacement of antisymmetric and symmetric products by tensor products, will be now proved possible. To this end weaker conditions on the asymptotic equivalence relation will be imposed.Definition 4.10.
Fixk∈ℕ and d∈ℝ, d>0. Families {AY,n}(Y,n)∈ℳ(Ω)×ℕ, {BY,n}(Y,n)∈ℳ(Ω)×ℕ of operators AY,n,BY,n∈𝒯(ℋY⊗k) are called weakly asymptotically equivalent (symbolically: AY,n~BY,n), if d-limn,μ(Y)→∞Tr(AY,n-BY,n)CY,n=0 for every family {CY,n}(Y,n)∈ℳ(Ω)×ℕ of operators of the form CY,n=⨂i=1kCY,n(i), where CY,n(i)∈ℬ(ℋY) (i∈{1,…,k}, (Y,n)∈ℳ(Ω)×ℕ) are operators with uniformly bounded operator norms.The relation~ has the properties analogous to the properties of the relation ≈ from Remark 4.4.Definition 4.11.
Letk∈ℕ, k≥2. Fix π∈Sk. A set X⊂{1,…,k} is called a cyclic set of the permutationπ, if X={l1,…,lq} for some l1,…,lq∈{1,…,k}, q∈{2,…,k}, such that π(ls)=ls+1 for every s∈{1,…,q-1}, and π(lq)=l1. A singleton {l}⊂{1,…,k} such that π(l)=l is also called a cyclic set of the permutation π.Note that the set{1,…,k} from the above definition can be represented as the union of disjoint cyclic sets of π.Lemma 4.12.
Letk∈ℕ, k≥2. If B(1),…,B(k)∈𝒯(ℋY) then
(4.41)k!Tr⁡(BY,n(1)⊗⋯⊗BY,n(k))AHY(k)=∑π∈Sksgn⁡π∏j=1p(π)Tr⁡∏s=1qjBY,n(lj,s),
where p(π)∈{1,…,k} is the number of disjoint cyclic sets of π, indexed by j, and qj denotes the number of elements of the jth cyclic set of π, which is {lj,1,…,lj,qj}, where
(4.42)π(lj,qj)=lj,1,forqj≥2,π(lj,s)=lj,s+1,s=1,…,qj-1.
Clearly, ∑j=1p(π)qj=k and ⋃j=1p(π)⋃s=1qj{lj,s}={1,…,k}.Proof.
Let{φi}i∈ℕ be an orthonormal basis of ℋY. One has
(4.43)k!Tr⁡(BY,n(1)⊗⋯⊗BY,n(k))AHY(k)=∑π∈Sksgn⁡π∏j=1p(π)Mj,
where
(4.44)Mj:=∑ilj,1=1∞…∑ilj,qj=1∞〈φilj,1,B(lj,1)φiπ(lj,1)〉⋯〈φilj,qj,B(lj,qj)φiπ(lj,qj)〉.
If qj>2 for some j∈{1,…,p(π)} then, by condition (4.42) and Parseval's formula,
(4.45)Mj=∑ilj,1=1∞…∑ilj,qj=1∞〈φilj,1,B(lj,1)φilj,2〉〈φilj,2,B(lj,2)φilj,3〉⋯〈φilj,qj,B(lj,qj)φilj,1〉=∑ilj,1=1∞∑ilj,3=1∞…∑ilj,qj=1∞〈φilj,1,B(lj,1)B(lj,2)φilj,3〉⋯〈φilj,qj,B(lj,qj)φilj,1〉.
Performing successive summations one then obtains
(4.46)Mj=∑ilj,1=1∞〈φilj,1,(∏s=1qjB(lj,s))φilj,1〉=Tr⁡∏s=1qjB(lj,s).
The derivation of the above formula for qj=1,2, after simplifications, proceeds analogously. This completes the proof of (4.41), in view of (4.43).Lemma 4.13.
One has(4.47)d-lim⁡n,μ(Y)→∞‖σ∧Y,n(1)‖=0,
and if σ∨Y,n(2)≈2σ∨Y,n(1)∨σ∨Y,n(1) (see Theorem 4.9) then
(4.48)d-lim⁡n,μ(Y)→∞‖σ∨Y,n(1)‖=0.Proof.
To prove (4.47) it suffices to observe that, according to Theorem 2.3,
(4.49)‖σ∧Y,n(1)‖=‖Ln1(1ξY,n∧ρY∧n)‖≤1n1ξY,n∧‖ρY∧n‖≤1n1ξY,n∧Tr⁡ρY∧n=1n.
Now (4.48) will be proved. Let {φi}i∈ℕ be an orthonormal basis of ℋY for fixed Y∈ℳ(Ω). Then(4.50)Tr⁡2σ∨Y,n(1)∨σ∨Y,n(1)=2Tr⁡(σ∨Y,n(1)⊗σ∨Y,n(1))SHY(2)=∑π∈S2∑i1,i2=1∞〈φi1,σ∨Y,n(1)φiπ(1)〉〈φi2,σ∨Y,n(1)φiπ(2)〉=(Tr⁡σ∨Y,n(1))2+Tr⁡(σ∨Y,n(1)σ∨Y,n(1)).
Taking into account (4.50), the relation σ∨Y,n(2)≈2σ∨Y,n(1)∨σ∨Y,n(1), Definition 4.2 for CY,n=I⊗2, and the equality Trσ∨Y,n(2)=Trσ∨Y,n(1)=1, one obtains
(4.51)d-lim⁡n,μ(Y)→∞Tr⁡(σ∨Y,n(1)σ∨Y,n(1))=0.
Furthermore,
(4.52)‖σ∨Y,n(1)φ‖2=〈φ,σ∨Y,n(1)σ∨Y,n(1)φ〉≤Tr⁡(σ∨Y,n(1)σ∨Y,n(1))
for every φ∈ℋY such that ∥φ∥=1, hence (4.51) yields (4.48).Notice that (4.47) can be also proved analogously to (4.48) under the additional assumption σ∧Y,n(2)≈2σ∧Y,n(1)∧σ∧Y,n(1).The proof of the next theorem fork=2 was given in [11, 12].Theorem 4.14 (Asymptotic formulae II).
Letk∈ℕ, k≥2. One has
(4.53)k!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)︸k~σ∧Y,n(1)⊗⋯⊗σ∧Y,n(1)︸k,
and if d-limn,μ(Y)→∞∥σ∨Y,n(1)∥=0 (see Lemma 4.13) then
(4.54)k!σ∨Y,n(1)∨⋯∨σ∨Y,n(1)︸k~σ∨Y,n(1)⊗⋯⊗σ∨Y,n(1)︸k.Proof.
First (4.53) will be proved. Fix a family {CY,n}(Y,n)∈ℳ(Ω)×ℕ of operators such as in Definition 4.10 and set
(4.55)BY,n(r):=σ∧Y,n(1)CY,n(r),r=1,…,k.
Then, by Lemma 4.12, one has
(4.56)Tr⁡k!(σ∧Y,n(1)∧⋯∧σ∧Y,n(1))(CY,n(1)⊗⋯⊗CY,n(k))=k!Tr⁡(BY,n(1)⊗⋯⊗BY,n(k))AHY(k)=∑π∈Sksgn⁡π∏j=1p(π)Tr⁡∏s=1qjBY,n(lj,s)=Tr⁡(σ∧Y,n(1)⊗⋯⊗σ∧Y,n(1))(CY,n(1)⊗⋯⊗CY,n(k))+∑π∈Skπ≠Id⁡sgn⁡π∏j=1p(π)Tr⁡∏s=1qjBY,n(lj,s).
Thus
(4.57)Tr⁡(k!σ∧Y,n(1)∧⋯∧σ∧Y,n(1)-σ∧Y,n(1)⊗⋯⊗σ∧Y,n(1))(CY,n(1)⊗⋯⊗CY,n(k))=∑π∈Skπ≠Id⁡sgn⁡π∏j=1p(π)Tr⁡∏s=1qjBY,n(lj,s).
Now, let π∈Sk, π≠Id, be fixed. If qj=1 for some j∈{1,…,p(π)} then
(4.58)|Tr⁡∏s=1qjBY,n(lj,s)|≡|Tr⁡BY,n(lj,1)|≤‖CY,n(lj,1)‖Tr⁡|σ∧Y,n(1)|=‖CY,n(lj,1)‖,
whereas if qj≥2 then
(4.59)|Tr⁡∏s=1qjBY,n(lj,s)|≤‖∏s=1qj-1BY,n(lj,s)‖Tr⁡|BY,n(lj,qj)|≤‖σ∧Y,n(1)‖qj-1(∏s=1qj-1‖CY,n(lj,s)‖)‖CY,n(lj,qj)‖Tr⁡|σ∧Y,n(1)|≤‖σ∧Y,n(1)‖qj-1∏s=1qj‖CY,n(lj,s)‖.
Since π≠Id, there exists at least one j∈{1,…,p(π)} such that qj≥2, hence
(4.60)|∏j=1p(π)Tr⁡∏s=1qjBY,n(lj,s)|≤(∏j=1p(π)‖σ∧Y,n(1)‖qj-1)∏j=1p(π)∏s=1qj‖CY,n(lj,s)‖=‖CY,n‖∏j=1p(π)‖σ∧Y,n(1)‖qj-1
and at least one exponent qj-1 is nonzero. Thus, by the uniform boundedness of the norms ∥CY,n∥ and Lemma 4.13, the termodynamic limit of the l.h.s of (4.57) equals 0, which proves the validity of relation (4.53).
The proof of relation (4.54), after discarding the permutation signs and replacing ∧ by ∨, proceeds analogously.



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*Source: 890523-2010-12-30.xml*