A Szeg\H{o} type theorem and distribution of symplectic eigenvalues

We study the properties of stationary G-chains in terms of their generating functions. In particular, we prove an analogue of the Szeg\H{o} limit theorem for symplectic eigenvalues, derive an expression for the entropy rate of stationary quantum Gaussian processes, and study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices. We also introduce a concept of symplectic numerical range, analogous to that of numerical range, and study some of its basic properties, mainly in the context of block Toeplitz operators.

The complete Heisenberg uncertainty principle for all the position and momentum observables assumes the form of the following matrix inequality: where J 2k = J 2 ⊕ · · · ⊕ J 2 k times with J 2 = 0 1 −1 0 .
Following the terminology in [19], we call a real 2k × 2k positive definite matrix A satisfying inequality (1.1) a G-matrix. A standard result in quantum theory states that a k-mode, mean zero, Gaussian quantum state is uniquely represented by its covariance matrix, which is a G-matrix. Conversely any 2k × 2k G-matrix is the covariance matrix of a unique (up to permutation) k-mode mean zero quantum Gaussian state in the Fock space Γ(C k ); [17,11]. Finite mode quantum Gaussian states and quantum Gaussian processes have been extensively studied in quantum optics, quantum probability, and quantum information -both in theory as well as in experiments. A comprehensive survey of Gaussian states and their properties can be found in the two books of Holevo [10,11]. For their applications to quantum information theory the reader is referred to the survey article by Weedbrook et al [29], Holevo's book [11], and the new book of Serafini [23].
In the present paper, our concern is with a stationary quantum Gaussian process. This is a chain of finite mode (k mode) quantum Gaussian states exhibiting stationarity. Let {ρ n } be a chain of quantum Gaussian states with covariance matrices {T n }. The stationarity property means that each T n is a positive definite block Toeplitz matrix such that T n is the leading principal sub-matrix of T n+1 . This sequence {T n } gives rise to an infinite block Toeplitz matrix Σ. We call this chain {ρ n } of quantum Gaussian states a stationary quantum Gaussian process and the infinite matrix Σ a G-chain [19]. Thus a G-chain Σ is an infinite block Toeplitz matrix. The classical version of such objects has been well studied in probability theory. (See for instance [12].) A study of the quantum version has been initiated in [19,20]. In order to study G-chains, we need to study properties of infinite block Toeplitz matrices with blocks of size 2k × 2k. Every leading n × n principal block sub-matrix gives a covariance matrix of an nk-mode quantum Gaussian state. Toeplitz matrices play an important part in the study of stationary processes in classical probability theory as well. See, e.g., Grenander and Szegő [8].
Among real positive definite matrices, G-matrices are characterised by a simple property of their symplectic eigenvalues. Williamson's theorem [32] tells us that for every 2k × 2k real positive definite matrix A, there exists a symplectic matrix M such that where d 1 (A) ≤ · · · ≤ d k (A) are positive numbers uniquely determined by A. These are uniquely determined by A. We call these numbers the symplectic eigenvalues of A. We can see that a matrix A is a G-matrix if and only if all its symplectic eigenvalues d j (A) ≥ 1 2 . There has recently been considerable interest in the study of various properties of symplectic eigenvalues (see for instance [1,4,6,9]), due to their close connection with quantum optics and thermodynamics [11,15].
Given a k-mode quantum Gaussian state with covariance matrix A, the von Neumann entropy of the state is given by where H is the Shannon entropy function given by [5,18], or [23] pages 61 -62. Let T n be the covariance matrix of a k-mode stationary quantum Gaussian process, truncated at level n. The entropy rate of the process is defined as n .
An important problem in information theory has been the study of the entropy rate of any given stationary process. This can be very complicated [3,7]. The entropy rate for a certain type of stationary quantum Gaussian process was calculated in [19]. We compute the entropy rate for a more general class, namely, the class of bounded partially symmetric stationary quantum Gaussian processes. Let Σ = A i−j be a G-chain corresponding to a stationary quantum Gaussian process. We call this process bounded if Σ is a bounded operator on l 2 2k (the space of square summable sequences of elements of C 2k ). In this case Σ is a Toeplitz operator generated by a matrix symbol A in L ∞ 2k×2k . The process is partially symmetric if A −n = A n for all n ∈ N. We show that a stationary quantum Gaussian process is partially symmetric and bounded if and only if its corresponding G-chain is generated by an A in L ∞ 2k×2k such that A(θ) is a G-matrix for almost all θ. The computation of the entropy rate requires a study of the distribution of symplectic eigenvalues of block Toeplitz matrices. To achieve this we prove a symplectic analogue of a fundamental theorem for the distribution of eigenvalues of Toeplitz matrices, well-known as the Szegő limit theorem [8,14]. The classical Szegő theorem can be stated as follows: Suppose ϕ : (−π, π) → R is an essentially bounded function, and (T n ) is the sequence of Hermitian Toeplitz matrices generated by ϕ. Then for every function f, continuous on the interval [essinf ϕ, esssup ϕ], one has where λ j (T n ), j = 1, 2, · · · , n, are the eigenvalues of T n . Many different versions and proofs of this theorem are available in the literature [2,22,24,25,26,27,28,30,31]. We prove an analogue of this theorem for symplectic eigenvalues, and apply this to compute the entropy rate and to study the distribution of symplectic eigenvalues of block Toeplitz matrices. In particular we prove that the union of the set of all symplectic eigenvalues of truncated n×n block Toeplitz matrices T n ( A) is dense in the set of all symplectic eigenvalues of A(θ) where A(θ) varies over the essential range of A.
In classical operator theory, the numerical range is an important and useful concept. We introduce an analogous notion of the symplectic numerical range and study its basic properties. We show that the closure of the symplectic numerical range of an operator is convex and contains the symplectic spectrum. We give a relationship between the symplectic numerical ranges of truncated block Toeplitz matrices and their symbol. This, in turn, helps us to have a better understanding of the distribution of symplectic eigenvalues of the truncated block Toeplitz matrices.
The paper is organised as follows: We give some basic notations and results in Section 2, introduce the notion of symplectic numerical range in Section 3, and study some of its basic properties, especially in the context of block Toeplitz operators. In Section 4 we prove a symplectic analogue of Szegő limit theorem, and give its applications.

Preliminaries
We begin with some basic facts about Toeplitz operators. For proofs and other details, the reader may refer to the book of Böttcher and Silbermann [2].
Let L ∞ k×k denote the set of all functions A = a ij from [−π, π] to the set of all k × k complex matrices, with A(−π) = A(π) and a ij essentially bounded for all i, j = 1, . . . , k. For an A in where A(θ) denotes the operator norm of A(θ). It is easy to see that · is a norm on L ∞ k×k . The space L ∞ k×k is a C * -algebra with the usual operations. Let L 2 k be the set of all functions x = ( x 1 , . . . , x k ) from [−π, π] to C k with x(−π) = x(π) and x ∈ L 2 for all i = 1, . . . , k. The space L 2 k is a Hilbert space with the inner product wherex is here understood as a column vector. It can be verified that M A is a bounded linear The space l 2 k is a Hilbert space with the inner product given by Clearly this inner product induces the l 2 norm on l 2 k . We denote this norm by · 2 . In a similar way, l 2 k (Z) is the Hilbert space of all square summable doubly infinite sequencesx of vectors with the l 2 norm.
Let A ∈ L ∞ k×k . For each n ∈ Z, let A n be the nth Fourier coefficient of A given by . This is a principal submatrix of L( A). If for n ∈ N, P n is the projection operator on l 2 k (Z) defined as P n (. . . , x −n , x −(n−1) , . . . , x 0 , . . . , x n , . . .) = (. . . , 0, 0, x −(n−1) , . . . , x 0 , . . . , x n , . . .), then P n L( A)P n converges strongly to L( A), and for every n, P n L( A)P n = T ( A). For A in L ∞ k×k , we say T ( A) is the infinite block Toeplitz matrix generated by A and A is the symbol of the block Toeplitz operator T ( A).
x n e ınθ .
A Hermitian operator T on a Hilbert space H is said to be positive semidefinite if x, T x ≥ 0 for all x in H. If equality here holds only for the null vector, then T is said to be positive definite.
If the space H is finite-dimensional, a positive semidefinite operator is positive definite if and only if it is invertible. This is not the case when H is infinite-dimensional (consider, e.g., the operator T = diag(1, 1/2, 1/3, .....) on the space l 2 ). So, we will use the term positive invertible for an operator that is positive definite and invertible. Let T n ( A) be the truncated n × n block The essential range of A is given by the set of all k × k matrices B such that for every > 0, m({t : Here m(·) denotes the Lebesgue measure. We denote the essential range of A by R( A). Clearly the essential range of A is closed in the space of k × k matrices and is contained in the closure of the range of A. The following proposition gives an equivalent condition for T n ( A) to be positive definite for each n. See [16].  Here we point out that symplectic eigenvalues of T n ( A) and A(θ) are defined only when T ( A) is a partially symmetric operator on l 2 2k , and T n ( A) and A(θ) are positive definite. A stationary G-chain Σ = A i−j is bounded if it is bounded as a linear operator on l 2 2k , and is partially symmetric if it is a partially symmetric linear operator. The following theorem gives a characterisation of a partially symmetric bounded stationary G-chain in terms of its symbol.
Clearly Σ 0 is the infinite 2k × 2k block Toeplitz matrix corresponding to the sequence B n n∈Z , where

Symplectic numerical range
Let H be a real separable Hilbert space. We denote the direct sum H ⊕ H by H. It is easy to see that the space H is isomorphic to N K where K is a two dimensional real Hilbert space and the operator J = 0 I −I 0 on H is orthogonally equivalent to N J 2 . Henceforth we will identify H with N K and the operator J with N J 2 .
Definition 3.1. Let A be a positive definite operator on H. We define the symplectic numerical range of A to be the set This is a subset of (0, ∞). It is unbounded as the set of vectors (u, v) with u, Jv = 1 is unbounded. An infinite dimensional version of Williamson's theorem was proved in [21]: for any positive invertible operator A on H there exists a positive invertible operator P on H and a symplectic transformation L : H → H such that The symplectic spectrum of A is the spectrum of the positive invertible operator P. If A is a 2n × 2n real positive definite matrix, then its symplectic spectrum is the set of its symplectic eigenvalues {d 1 (A), . . . , d n (A)} ⊆ (0, ∞). We denote by σ s (A) the symplectic spectrum of A.
Clearly α(t) is continuous in t and α(1) = α. Since the left-hand side of (3.1) belongs to W s (P ) and the right-hand side to W (P ), it follows that inf W s (P ) ≥ inf W (P ).

(3.2)
Now let x be any unit vector in H, and let u = 1 √ 2 (x⊕x) and v = 1 √ 2 (−x⊕x). Then u, Jv = 1. We see that This implies that inf W (P ) ≥ inf W s (P ). When H is finite-dimensional, we have (See Theorem 5 of [1].) This gives part (iii).
Let A ∈ L ∞ 2k×2k be such that all matrices A(θ) in R( A) are real positive definite. Then the symplectic numerical range of A is the set We next give a relationship between W s ( A) and W s ( A(θ)) for A(θ) ∈ R( A). Clearly u n and v n are in L 2 2k , and u n , J 2k v n = 1. Let µ n = 1 2 ( u n , A u n + v n , A v n ). Then µ n ∈ W s ( A), and we have This proves µ n → µ. Hence W s (B) ⊆ W s ( A). Since W s ( A) is convex, the closed convex hull of To prove the reverse inclusion, we use the fact that every element of W s ( A) is a limit of finite sums of the form where A(θ j ) ∈ R( A), and α j ≥ 0 are such that j α j u j , Jv j = 1. Let β j = u j , Jv j . Without loss of generality we may assume that β j ≥ 0 for all j. Replacing u j by β j u j and v j by β j v j we can take u j , J 2k v j = 1 for every j, and α j = 1. This shows that every element of W s ( A) is a limit of convex combinations of elements of A(θ)∈R( A) W s ( A(θ)).
for every partially symmetric, bounded, positive invertible operator Proof. Let d be a symplectic eigenvalue of T n ( A). Then there exist vectorsû = (u 1 , · · · , u n ),v = (v 1 , · · · , v n ), u j , v j ∈ R 2k with û, Jv = 1 and Let u and v be the elements of L 2 2k given by u(θ) = u j e ıjθ , v(θ) = v j e ıjθ . Then Using (3.6), we know that the above integrand is nonnegative almost everywhere. Hence, we have d ≥ m A . Also d = m A if and only if for almost all θ. So the last statement of the theorem by using Proposition 3.1(iii) and (3.5).
Since d j ( A(θ)) ≤ A(θ) and A(θ) ≤ A for almost all θ, Now let f be any continuous function on 0, A . Then by using the Weierstrass approximation theorem and arguing as above, we can show that (4.1) holds for f. Remark 4.1. We know that d is a symplectic eigenvalue of a positive definite matrix A if and only if ±d are eigenvalues of the non-Hermitian matrix iJA. In [26] Tilli has proved a very general version of Szegő's limit theorem for non-Hermitian block Toeplitz matrices. It is possible to derive Theorem 4.1 from Tilli's general results. The proofs of the general version are, naturally, more intricate. We have given a short self-contained presentation for the special case we need.
The entropy rate of a stationary quantum Gaussian process with the associated G-chain T ( A) is given by the formula where S(T n ( A)) denotes the entropy of the quantum Gaussian state with the corresponding Gmatrix T n ( A). As a consequence of Theorem 4.1, we obtain a closed expression for the entropy rate of a partially symmetric, bounded stationary quantum Gaussian process in terms of the entropies of the Gaussian states with G-matrices A(θ).
Using (1.2)we can see that the entropy of any G-matrix B can be written as (4.9) Hence the entropy rate S T ( A) of T ( A) is given by Since f is continuous and A ∈ L ∞ 2k×2k , we can apply Theorem 4.1 to get Using the formula (4.9) for the sum inside this integral, we obtain (4.8).
The entropy rate of a special kind of stationary quantum Gaussian process has been computed in the paper [19]. There the authors considered a block Toeplitz matrix T n ( A) given by where A and B are 2k × 2k real symmetric matrices such that A + tB is a G-matrix for each t ∈ [−2, 2], and {p 1 , p 2 , . . .} is a probability distribution over {1, 2, . . .}. In this case A(θ) takes the form A + j∈Z\{0} p |j| Be ıjθ , θ ∈ [−π, π]. Our Corollary 4.2 gives a much more general result.
In the rest of the paper, we use Theorem 4.1 to study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices. Henceforth T ( A) is a partially symmetric, bounded, positive invertible operator on l 2 2k generated by A. Recall the definition of m A given in (3.5).  Proof. Let g be the distance function defined as For any > 0 define the function f : . Clearly f is continuous and f (x) = 1 if and only if x ∈ K. We can see that as → 0, f converges to the characteristic function χ K in the L 1 norm. Hence Applying Theorem 4.1 with f = f and taking → 0 we get (4.12).
We know that the map d j that takes a positive definite matrix B to its jth minimum symplectic eigenvalue d j (B) is continuous [1]. Since θ → A(θ) is measurable on [−π, π], the composite map θ → d j (θ) = d j ( A(θ)) is also measurable.
Let R j denote the essential range of the map d j (θ) and let R = k j=1 R j . Since A ∈ L ∞ 2k×2k , the set R is compact.
Lemma 4.5. For every A(θ) in R( A) and 1 ≤ j ≤ k, d j (A(θ))is in R j .
Proof. Let B = A(θ) be any element of R( A). We show that d j (B) ∈ R j . Let > 0. Since the map d j is continuous on positive definite matrices, we can find a δ > 0 such that For any subset X of R let B(X, δ) be its δ-neighbourhood: B(X, δ) = {x ∈ R : |x − s| < δ for some s ∈ X}.
Let D n be the set of symplectic eigenvalues of T n ( A). Let D = n D n Theorem 4.6. The set D is dense in R. Further for each δ > 0 let X δ be the set X δ = [m A , A ]\B(R, δ). (4.14) Then lim n→∞ c n (X δ ) n = 0.  This shows that the right hand side of (4.16) is strictly greater than the right hand side of (4.17). But since D ∩ B(x, ) = ∅, c n (Y ) = c n (R) for all n. So, the left hand sides of (4.16) and (4.17) are equal. This is a contradiction. Hence D must be dense in R.