A Common Parametrization for Finite Mode Gaussian States, their Symmetries and associated Contractions with some Applications
aa r X i v : . [ qu a n t - ph ] M a y Klauder-Bargmann Integral Representation ofGaussian Symmetries and Generating Functions ofGaussian States
Tiju Cherian John and K. R. Parthasarathy
Indian Statistical Institute, Delhi Centre, 7, SJSS Marg, New Delhi, India, 110059 [email protected], [email protected]
In memory of V. S. Varadarajan.
Abstract
Let H be a finite dimensional complex Hilbert space of dimension n andlet Γ( H ) be the boson Fock space over H . A unitary operator U in Γ( H ) is called a n -mode gaussian symmetry if, for every gaussian state ρ in Γ( H ) ,the transformed state U ρU † is also gaussian. It is shown that every gaus-sian symmetry admits a Klauder-Bargmann integral representation in termsof coherent states. This construction provides an explicit strongly continuous,irreducible, and projective unitary representation of the Lie group which is thesemidirect product of the additive group H and the group Sp ( H ) of all reallinear symplectic transformations of H .For any bounded operator Z in Γ( H ) , the notion of a generating function G Z ( u, v ) with u, v in H is introduced by using the matrix entries of Z inthe overcomplete basis of exponential vectors. An explicit computation of thegenerating functions of gaussian symmetries, gaussian states and second quan-tizations of contraction operators in H reveal that all these operators belongto an adjoint-closed semigroup E ( H ) of bounded operators in Γ( H ) . Everyelement in E ( H ) is completely determined by its matrix entries at the , and -particle vectors in a complete orthonormal basis of n -mode finite parti-cle vectors of Γ( H ) . This enables the parametrization of all elements of E ( H ) by a -tuple consisting of a scalar c , two vectors λ , µ in C n and three matrices A, Λ , B in M n ( C ) . This yields, in particular, a new parametrization of allgaussian states in Γ( H ) , which is a very fruitful alternative to the customaryparametrization by position-momentum mean vectors and covariance matri-ces. The alternative parametrization suggested here leads to a rich harvest ofcorollaries:(i) every positive element Z in E ( H ) (and, in particular, every gaussianstate) admits a factorization Z = Z † Z , where Z is an element of E ( H ) and has the form Z = √ c Γ( √ Λ) exp nP nr =1 λ r a r + P nr,s =1 α rs a r a s o on thedense linear manifold generated by all exponential vectors, Λ being a positiveoperator in H , a r , ≤ r ≤ n are the basic annihilation operators correspondingto the n different modes in Γ( H ) , λ r ∈ C and [ α rs ] is a symmetric matrix in M n ( C ) ; ii) an explicit particle basis expansion of an arbitrary mean zero puregaussian state vector along with a density matrix formula for a general gaussianstate in terms of its E ( H ) -parameters;(iii) an easy test for the entanglement of pure gaussian states and a classof examples of pure n -mode gaussian states which are completely entangled;(iv) Tomography of an unknown gaussian state in Γ( C n ) by the estimationof its E ( C n ) -parameters using O ( n ) measurements with a finite number ofoutcomes. Keywords:
Quantum gaussian states, coherent states, particle basis, com-pletely entangled states, tomography of gaussian states, nonlinear informationchannel, symplectic group, generating function, semigroup of operators.
Contents E ( H )
236 Gaussian States and the Uncertainty Relations in terms of E ( H ) -parameters 277 Positive operators in E ( H )
348 A Density Matrix Formula for Gaussian States 409 Completely Entangled Pure Gaussian States and a Criterion forTesting Entanglement 4310 Tomography of Gaussian States 50
The principal aim of this paper is an analysis of gaussian states and their symmetriesthrough a new scheme of parametrization. It replaces the customary mean valuesand covariances of position and momentum observables which assume all values onthe real line. To this end we consider the n -mode boson Fock space Γ( H ) over an n -dimensional complex Hilbert space H with a chosen and fixed orthonormal basis { e j , ≤ j ≤ n } , where the index j stands for the j -th mode and n for the totalnumber of modes. The study is based on three already well-known tools and afourth one which is not widely used. We shall repeatedly use the gaussian integral2ormula, properties of exponential vectors and coherent states, the Weyl displace-ment operator and the quantum Fourier transform and, finally, the elementary ideaof generating function of a bounded operator on Γ( H ) .Section 2 contains a brief summary of some well-known properties of exponentialvectors and coherent states as well as the definition of generating function of abounded operator on Γ( H ) . In order to make the exposition fairly self-contained, theKlauder-Bargmann isometry from the Hilbert space Γ( H ) into L ( C n ) is describedalong with a short proof. The Klauder-Bargmann formula for the resolution ofthe identity operator on Γ( H ) as an integral of coherent states with respect to asuitably normalized Lebesgue measure in Γ( C n ) is given. This is repeatedly used inour analysis. Also, we define the crucial notion of generating function of a boundedoperator on Γ( H ) .In Section 3, the Weyl displacement operators are presented as a projectiveunitary representation of the additive group H in the Hilbert space Γ( H ) and theassociated quantum Fourier transform of a state in Γ( H ) is defined. The Wignerisomorphism between the Hilbert space B (Γ( H )) of all Hilbert-Schmidt operatorson Γ( H ) and L ( C n ) is established.A unitary operator U in Γ( H ) is called a gaussian symmetry if U ρU † is a gaussianstate whenever ρ is a gaussian state. In Section 4, every gaussian symmetry U isrealized as a Klauder-Bargmann integral in terms of coherent states with respect tothe Lebesgue measure in C n . This construction yields a strongly continuous, projec-tive unitary and irreducible representation of the Lie group which is the semidirectproduct of the additive group H and the group Sp ( H ) of all symplectic linear trans-forms of H . It is also shown that the generating function of a gaussian symmetryadmits an exponential formula.In Section 5 we construct the central object of our paper, namely, the operatorsemigroup E ( H ) contained in the algebra B (Γ( H )) of all bounded operators on Γ( H ) by using the idea of generating function. To this end, we identify H with C n through the mode-basis mentioned at the beginning of Section 2. We say that abounded operator Z on Γ( H ) is in the class E ( H ) if, for all u , v in H , the followingholds: h e (¯ u ) | Z | e ( v ) i = c exp (cid:8) u T α + β T v + u T A u + u T Λ v + v T B v (cid:9) , for some ordered -tuple ( c, α , β , A, Λ , B ) consisting of a scalar c = 0 , vectors α , β in C n and n × n complex matrices A, Λ , B with A and B being symmetric. Here e (¯ u ) and e ( v ) are the exponential vectors in Γ( H ) associated with ¯ u and v respectively, barindicating complex conjugation. We say that this -tuple are the E ( H ) -parametersof the operator Z . By the properties of the exponential vectors in Γ( H ) summarisedin Section 2, this parametrization is unambiguous. If Z is a selfadjoint element of E ( H ) , c is real, β = ¯ α , B = ¯ A and Λ is hermitian. Thus the E ( H ) -parameters ofa selfadjoint element in E ( H ) reduce to a quadruple ( c, α , A, Λ) . The class E ( H ) is shown to enjoy the following properties:1. E ( H ) is a † -closed multiplicative semigroup.2. A unitary operator U is in E ( H ) if and only if it is a gaussian symmetry.3. A density operator ρ is in E ( H ) if and only if ρ is a gaussian state. The element Λ in the quadruple of E ( H ) -parameters of a gaussian state is a positive andcontractive matrix operator in H . 3. Every positive element Z in E ( H ) (and, in particular, every gaussian state)admits a factorization Z = Z † Z , where Z is an element of E ( H ) and hasthe form Z = √ c Γ( √ Λ) exp nP nr =1 λ r a r + P nr,s =1 α rs a r a s o on the dense linearmanifold generated by all exponential vectors, Λ being a positive operator in H , a r , ≤ r ≤ n are the basic annihilation operators corresponding to the n different modes in Γ( H ) , λ r ∈ C and [ α rs ] is a symmetric matrix in M n ( C ) .5. Mean zero pure gaussian states are parametrized by a complex symmetricmatrix A of order n and general mean zero gaussian states are parametrizedby a pair of n × n complex matrices ( A, Λ) , where A is symmetric and Λ is positive definite contraction. An explicit particle basis expansion of anarbitrary mean zero pure gaussian state vector along with a density matrixformula for a general mean zero gaussian state is obtained in terms of the E ( H ) -parameters.Some of the proofs spill over to other sections; items 1), 2) and 3) above are provedin Section 5 while 4) is proved in Section 7 and 5) is obtained in Section 8.Section 6 is devoted entirely to gaussian states. Suppose ρ is a gaussian statewith E ( H ) -parameters ( c, α , A, Λ) and position-momentum parameters m , S where m is the mean annihilation vector in H and S is the n × n real covariance matrix.Then formulae for m and S in terms of ( α , A, Λ) and vice versa are obtained. Thisalso shows that the scalar parameter c is a function of ( α , A, Λ) . As a corollaryof the results described above, we obtain the uncertainty relation S + i/ J ≥ interms of the E ( H ) -parameters. Furthermore, we prove the necessary and sufficientconditions on a pair ( A, Λ) consisting of a complex symmetric matrix A and apositive semidefinite matrix Λ to be the E ( H ) -parameters of a gaussian state. Let C : C n → C n denote the complex conjugation map z ¯ z , it is shown that a pair ( A, Λ) as described above are the E ( H ) -parameters of a gaussian state if and onlyif I − Λ − AC > , in the sense of positive definiteness of real linear operators on C n . Matrix versionof the inequality above is provided in Theorem 6.13. This is equivalent to theuncertainty relations in terms of the new parametrization.Section 7 is dedicated to the study of positive operators in E ( H ) . The mainresult here is item 4) above.In Section 8 we prove item 5) described above. Furthermore, the architectureof a gaussian state is described. If U is an arbitrary unitary matrix operatorin H and Γ( U ) its second quantization in Γ( H ) , then the transformed gaussianstate ρ ′ = Γ( U ) ρ Γ( U ) † has E ( H ) -parameters given by ( c, U α , U AU T , U Λ U † ) . Thisshows that a conjugation by a Weyl operator followed by a second quantized uni-tary operator transforms ρ to a mean zero gaussian state of the form ρ ( A, D λ ) with E ( H ) -parameters ( c, , A, D λ ) , where the symmetric matrix A can be differentfrom the one we started with and D λ is a diagonal matrix with diagonal entries λ ≥ λ ≥ · · · ≥ λ n ≥ . Furthermore, the pair ( A, is the E ( H ) -parameters of amean zero pure gaussian state ρ ( A,
0) = | ψ A ih ψ A | . Remarkbly, the density matrix of ρ ( A, D λ ) admits a canonical expansion in terms of | ψ A ih ψ A | and λ . This completesthe architecture of an arbitrary gaussian state.4n Section 9 we study some important examples of gaussian states using thenew parametrization proposed in this paper. A whole class of completely entangled n -mode pure gaussian states is constructed. This yields examples of such entangledstates which are also invariant under the action of the permutation group S n on theset of all the n modes.In Section 10, we show how the tomography of an unknown gaussian state ρ isessentially the tomography of the truncated finite level state (Tr ρ P ) − P ρ P , where P is the orthogonal projection onto the subspace C ⊕ H ⊕ H s (cid:13) in Γ( H ) .Finally, a note on the notational conventions used in this paper. We considerall vectors in C n or Z n + as column vectors and the notation with round bracket ( z , z , . . . , z n ) is often used to denote the column vector [ z , z , . . . , z n ] T . Bold letterslike x , y , z etc. are used to indicate vectors in euclidean spaces R n and C n . Similarly,when we use the multi-index notation, bold letters like k , t , s etc. denote vectors in Z n + , meaning, all entries in the vector are non-negative integers. In the multi-indexconvention, if k = ( k , k , . . . , k n ) , ℓ = ( l , l , . . . , l n ) ∈ Z n + and z = ( z , z , . . . , z n ) ∈ C n , then k ! = k ! k ! · · · k n ! , | k | = k + k + · · · + k n , z k = z k z k · · · z k n n , k ≤ ℓ , if k j ≤ l j , ∀ j, k ∧ ℓ = ( k ∧ l , k ∧ l , . . . , k n ∧ l n ) , where k j ∧ l j = min { k j , l j } , ≤ j ≤ n .Furthermore, for any Hilbert space H (finite or infinite dimensional, real orcomplex), in the space of all bounded selfadjoint opell as real symmetric or Hermitianmatrices introduce the partial ordering A ≤ B to imply that B − A is a positiveoperator or a nonnegative definite matrix. We say that A < B if A ≤ B and B − A has null space. Let Γ( H ) denote the boson (symmetric) Fock space over a finite dimensional complexHilbert space H of dimension n . Choose and fix an orthonormal basis, { e j | ≤ j ≤ n } in H and identify H with C n , such that z = P nj =1 z j e j ∈ H is identified with z =( z , z , . . . , z n ) ∈ C n . Then Γ( H ) = Γ( C n ) = ⊗ nj =1 Γ( C e j ) . Let k = ( k , k , . . . , k n ) ,with nonnegative integer entries k j , ≤ j ≤ n . Denote by | k , k , . . . , k n i , thenormalized symmetric tensor product of e r taken k r times, r = 1 , , . . . , n so that | k , k , . . . , k n i is a unit vector in the subspace H s (cid:13) | k | of Γ( H ) . Write | k i = | k , k , . . . , k n i . The set {| k i | k ∈ Z n + } is a complete orthonormal basis for Γ( H ) , called a particlebasis with reference to the choice {| e j i | ≤ j ≤ n } in H . When H = C n we chooseits canonical basis so that e j is the column vector with 1 in the position j and elsewhere. The Fock space Γ( H ) as well as Γ( C n ) is also called an n -mode Fock spacedescribing a quantum system of an arbitrary finite number of boson particles butin n different modes. Any unit vector φ as well as the corresponding -dimensionalprojection | φ ih φ | is called a pure state . The pure state | k i , thus defined is understood5o be a state in which there are k r particles in the r -th mode for r = 1 , , . . . , n . Forany z ∈ Γ( H ) , define | e ( z ) i , | ψ ( z ) i in Γ( H ) respectively by | e ( z ) i = ⊕ ∞ k =0 z ⊗ k √ k ! , | ψ ( z ) i = e −k z k / | e ( z ) i . Then h e ( z ) | e ( z ′ ) i = e h z | z ′ i , k ψ ( z ) k = 1 . We call | e ( z ) i and | ψ ( z ) i respectively the exponential vector and coherent state with parameter z ; | e (0) i is known as the vacuum state and is denoted by | Ω i . Then | ψ ( z ) i is a superposition of finite particle states {| k i | k ∈ Z n + } , | k i having probabilityamplitude Π r z k r r / √ Π r k r ! . Using the multi-index conventions described in Section 1,we have | e ( z ) i = X k ∈ Z n + z k √ k ! | k i = X ≤ j< ∞ X | k | = j z k √ k ! | k i (2.1)and for the particle basis measurement in the coherent state | ψ ( z ) i , the probabilityof observing k r particles in the r -th mode for ≤ r ≤ n is e −| z | (cid:12)(cid:12) z k (cid:12)(cid:12) k ! = Π nr =1 e −| z r | (cid:12)(cid:12) z k r r (cid:12)(cid:12) k r ! . In other words, number of particles in different modes have independent Poissondistributions with mean values | z r | , r = 1 , , . . . , n . Later, in our exposition, weshall meet | ψ ( z ) i as an example of a pure gaussian state.We now recall some of the well-known properties of exponential vectors andcoherent states:1. The map z
7→ | e ( z ) i from C n into Γ( C n ) is analytic where as z
7→ | ψ ( z ) i isreal analytic.2. For any nonempty finite set F ⊂ H , the set {| e ( z ) i | z ∈ F } is linearly inde-pendent and the linear span of all coherent states is dense in Γ( H ) . In otherwords, the coherent states form a linearly independent and total set.3. Let H = ⊕ j S j , where S j ’s are mutually orthogonal subspaces. Furthermore,let P j denote the orthogonal projection of H onto S j . Then the map | e ( z ) i 7→ ⊗ j | e ( P j z ) i extends to an isomorphism between Γ( H ) and ⊗ j Γ( S j ) . In particular, if { v , v , . . . , v n } is any orthonomal basis of H and z = P j z j v j , then | ψ ( z ) i = ⊗ j | ψ ( z j v j ) i . In other words, in any orthonormal basis of H , any coherentstate | ψ ( z ) i can be viewed as a product state.6. Suppose for some function f ∈ L ( R ) , R R f ( x ) e − x + ux d x = 0 , ∀ u ∈ C , thenthis implies, in particular, that the Fourier transform of the function ¯ f e − x vanishes and so does f . Hence the set of functions { e − x + ux | u ∈ C } is totalin L ( R ) . Putting g u ( x ) = π − / e − ( x + u )+ √ ux , ∀ x ∈ R , (2.2)we conclude that the set { g u | u ∈ C } is total in L ( R ) .When H is one dimesional, it may be identified with C and then the map e ( u ) g u (2.3)is scalar product preserving between total sets in Γ( H ) and L ( R ) . So itextends uniquely to an isomorphism between Γ( H ) and L ( R ) . In general,when H is n -dimesional, by Property 3 above, we see that Γ( H ) is isomorphicto L ( R n ) via the mapping e ( u )
7→ ⊗ j g u j (2.4)where u = ⊕ nj =1 u j e j .5. Klauder-Bargmann Isometry . The map φ π − n/ h ψ ( z ) | φ i , z ∈ C n is an isometry from Γ( H ) into L ( C n ) , where C n is equipped with the n -dimensional Lebesgue measure.We indicate a proof of Property 5: Proof.
Since Γ( C n ) is the n -fold tensor product of copies of Γ( C ) , which, by def-inition, is ℓ ( Z + ) and by Property 3, | ψ ( z ) i = ⊗ nj =1 | ψ ( z j ) i , it is enough to proveProperty 5 when n = 1 . In this case, ψ ( z ) = { e −| z | / z k √ k ! , k = 0 , , , . . . } , z ∈ C . Let φ and φ ′ be two sequences given by { a j } and { b j } , j ∈ Z + . Putting z = re iθ ,and using polar co-ordinates along with Parseval’s identity we get π Z C h ψ ( z ) | φ i h ψ ( z ) | φ ′ i d z = 2 ∞ Z e − r π Z ∞ X j =0 a j r j √ j ! e ijθ ! ∞ X j =0 b j r j √ j ! e ijθ ! d θ π r d r = ∞ Z ∞ X j =0 ¯ a j b j j ! r j e − r r d r = ∞ X j =0 ¯ a j b j = h φ | φ ′ i . The last line above is obtained by integrating term by term and using the fact that R ∞ x j e − x d x = j ! . 7e have a few important corollaries from the Klauder-Bargmann isometry. Be-fore that, let us recall the weak operator integrals. Let ( X, F , µ ) be a measurespace and K a complex separable Hilbert space. A : X → B ( K ) is said to be weaklymeasurable if the function A φ,ψ : X → C defined by A φ,ψ ( x ) := h φ | A ( x ) | ψ i is mea-surable for every φ, ψ ∈ K . A weakly measurable A is said to be weakly integrable (or just integrable when it is clear from the context) if for some M ≥ , Z X |h φ | A ( x ) | ψ i| µ (d x ) ≤ M k φ kk ψ k . (2.5)In this case we can make sense of R X A ( x ) µ (d x ) as an element of B ( K ) using thefollowing equation h φ | Z X A ( x ) µ (d x ) | ψ i = Z X h φ | A ( x ) | ψ i µ (d x ); ∀ φ, ψ ∈ K (2.6)Existence and uniqueness of R X A ( x ) µ (d x ) are given by the Riez-representation the-orem for linear functionals. Corollary 2.1. [Klauder-Bargmann formula] The coherent states yield a res-olution of identity I into -dimensional projections: π n Z C n | ψ ( z ) ih ψ ( z ) | d z = I, (2.7) where the left hand side integral is a weak operator integral with respect to the n -dimensional Lebesgue measure on C n . In particular, for any element | φ i in Γ( H ) the following holds: | φ i = 1 π n Z C n h ψ ( z ) | φ i | ψ ( z ) i d z , (2.8) which has the interpretation that {| ψ ( z ) i | z ∈ H} is an ’overcomplete basis’for Γ( H ) .2. Any bounded operator Z admits the representation Z = 1 π n Z C n Z | ψ ( z ) ih ψ ( z ) | d z . (2.9)
3. The positive operator valued measure (POVM) m defined by m ( E ) := 1 π n Z E | ψ ( z ) ih ψ ( z ) | d z , E a Borel set, yields a R n -valued continuous measurement in Γ( H ) . emark 2.2. The formula in (2.7) was first discovered in the present form byKlauder in Page 125-126 of [Kla60], with a heuristic proof. A rigourous proof ofthis first appeared in Page 194 of [Bar61], where he proved it for a slightly differ-ent version of exponential vectors called principal vectors in the Segal-Bargmannspace. Equation (2.7) later appeared separately in the works of Glauber (againwith a heuristic proof [Gla63b, Gla63a]) and Sudarshan (who refers to Bargmann[Sud63, KS68]) and was used by them to prove various results in quantum op-tics including the well-known Glauber-Sudarshan P representation. We call (2.7),
Klauder-Bargmann formula .Now we turn our attention to the description of an arbitrary bounded operator on Γ( H ) in the particle basis. Any bounded operator Z in Γ( H ) , admits the followingmatrix representation in the particle basis: Z = X r , s ∈ Z n + Z rs | r ih s | , (2.10)where Z rs = h r | Z | s i . Define the generating function of the operator Z , G Z ( u, v ) := h e (¯ u ) | Z | e ( v ) i , (2.11)where ¯ u is understood using the identification of H with C n . If u = ( u , u , . . . , u n ) and v = ( v , v , . . . , v n ) , then by (2.1), G Z ( u, v ) is a power series in the n variables u , u , . . . , u n , v , v , . . . , v n : G Z ( u, v ) = X ≤ k,ℓ< ∞ X | r | = k | s | = ℓ u r v s √ r ! s ! h r | Z | s i . (2.12)The constant term in the power series above is h | Z | i = h Ω | Z | Ω i . We shall adoptthe following notations to identify terms up to the second degree, | χ j i := | , · · · , , ↑ j -th , , · · · , i , | χ ij i := | , · · · , , ↑ i -th , , · · · , , ↑ j -th , , · · · , i , | χ jj i := | , · · · , , ↑ j -th , , · · · , i , ≤ i, j ≤ n, i = j. (2.13)The linear terms on the right side of (2.12) are n X j =1 u j h χ j | Z | Ω i + n X j =1 v j h Ω | Z | χ j i = u T λ + v T µ , (2.14)where λ and µ are vectors in C n with j -th coordinate h χ j | Z | Ω i and h Ω | Z | χ j i respectively, j = 1 , , . . . , n . We call λ and µ the - particle annihilation and creation amplitude vector respectively of Z . The quadratic terms in the powerseries are n X j =1 u j √ h χ jj | Z | Ω i + X j = k u j u k h χ jk | Z | Ω i + n X j,k =1 u j v k h χ j | Z | χ k i + n X j =1 v j √ h Ω | Z | χ jj i + X j = k v j v k h Ω | Z | χ jk i = u T A Z u + u T Λ Z v + v T B Z v , (2.15)9here A Z = [ α jk ] , Λ Z = [ λ jk ] and B Z = [ β jk ] are in M n ( C ) with, α jk = ( h χ jk | Z | Ω i , j = k, √ h χ jj | Z | Ω i , j = k , λ jk = h χ j | Z | χ k i , β jk = ( h Ω | Z | χ jk i , j = k, √ h Ω | Z | χ jj i , j = k . We call A Z , Λ Z and B Z the - particle annihilation, exchange and creation amplitudematrix respectively of Z . Notice that A Z and B Z are complex symmetric matricesby construction. The correspondence | ψ ( z ) i 7→ e − i Im h u | z i | ψ ( u + z ) i , z ∈ H is a scalar product pre-serving map for any fixed u ∈ H . Since the coherent states constitute a total set(Property 2, Section 2) in Γ( H ) , it follows that there exists a unique unitary operator W ( u ) on Γ( H ) satisfying the relation W ( u ) | ψ ( z ) i = exp {− i Im h u | z i} | ψ ( u + z ) i , z ∈ H . (3.1)We call W ( u ) the Weyl operator at u ∈ H . It is also known as the displacementoperator at u . The Weyl operators obey the multiplication relations W ( u ) W ( v ) = exp( − i Im h u | v i ) W ( u + v ) , ∀ u, v ∈ H ,W ( u ) W ( v ) = exp {− i Im h u | v i} W ( v ) W ( u ) , u, v ∈ H . (3.2)Equations in (3.2) are known as Weyl commutation relations or canonical commuta-tion relations (
CCR ) and the C ∗ -algebra generated by the Weyl operators denoted CCR ( H ) is called the CCR -algebra. We recall a few basic properties of the Weyloperators.1. The map u W ( u ) is a strongly continuous, projective, unitary and irre-ducible representation of the additive group H known as the Weyl represen-tation in Γ( H ) . Furthermore, it follows from the irreducibility that the vonNeumann algebra generated by Weyl operators is all of B (Γ( H )) , i.e., CCR ( H ) sot = B (Γ( H )) , (3.3)where sot indicates the closure in the strong operator topology.2. The Weyl representation enjoys the factorizability property: if H = H ⊕H ⊕ · · · ⊕ H k , u = u ⊕ u ⊕ · · · ⊕ u k , u j ∈ H j for each j , then W ( u ) = W ( u ) ⊗ W ( u ) ⊗ · · · ⊗ W ( u k ) .3. For every fixed u ∈ H , the set { W ( tu ) : t ∈ R } is a strongly continuous, oneparameter unitary group and hence has the form W ( tu ) = e − it √ p ( u ) , t ∈ R , u ∈ H , (3.4)10here p ( u ) is a self-adjoint operator in Γ( H ) . Define q ( u ) = p ( − iu ) ,a ( u ) = 1 √ q ( u ) + ip ( u )) ,a † ( u ) = 1 √ q ( u ) − ip ( u )) . Then a ( u ) and a † ( u ) are the well-known annihilation and creation operatorsat u . Observe that a ( u ) e ( v ) = h u | v i e ( v ) , (3.5) a † ( u ) e ( v ) = ∞ X n =1 √ n ! n − X r =0 v ⊗ r ⊗ u ⊗ v ⊗ n − r − . (3.6)Changing v to sv, s ∈ R , and identifying coefficients of s n on both sides ofequations above, we get a ( u ) v ⊗ n = √ n h u | v i v ⊗ n − , (3.7) a † ( u ) v ⊗ n = 1 √ n + 1 n X r =0 v ⊗ r ⊗ u ⊗ v ⊗ n − r (3.8)for all v ∈ H . Furthermore, it may be noted from (3.6) that a † ( u ) e ( v ) = dds | s =0 e ( v + su ) . (3.9)When H = C n = R n ⊕ i R n the families { q ( x ) | x ∈ R n } and { p ( x ) | x ∈ R n } arecommuting families of self-adjoint operators or observables and the CCR in(3.2) becomes [ q ( x ) , p ( y )] = i x T y , ∀ x , y ∈ R n . These are the well-known
Heisenberg commutaion relations , again called CCR.It is also expressed as [ a ( u ) , a ( v )] = 0 , [ a † ( u ) , a † ( v )] = 0 , [ a ( u ) , a † ( v )] = h u | v i , ∀ u, v ∈ H or C n . It may also be noted that the map u a ( u ) and u a † ( u ) are respectivelyantilinear and linear in the variable u . Going back to Weyl operators we have W ( u ) = e ( a † ( u ) − a ( u )) , ∀ u ∈ H or C n . In all these relations we are dealing with unbounded operators and we havebeen silent on matters concerning their domains. For details we refer to[Par92]. Write p j = p ( e j ) , q j = q ( e j ) = − p ( ie j ) (3.10) a j = a ( e j ) = 1 √ q j + ip j ) a † j = a † ( e j ) = 1 √ q j − ip j ) (3.11)11or each ≤ j ≤ n . The operators p j , q j , a j and a † j are respectively called the momentum , position , annihilation and creation operators of the j -th mode. Inparticular, the observable a † j a j is the number operator describing the numberof particles in the j -th mode.4. Stone-von Neumann Theorem . If K is a complex separable Hilbert spaceand u W ′ ( u ) is a strongly continuous, projective and unitary representationof H in K satisfying the relations (3.2) with W replaced by W ′ , then thereexists a Hilbert space k and a unitary isomorphism Γ :
K → Γ( H ) ⊗ k suchthat Γ W ′ ( u )Γ − = W ( u ) ⊗ I k , ∀ u ∈ H , where I k is the identity oprerator in k . In particular, if W ′ is also irreduciblethen k = C , the -dimensional Hilbert space and Γ is a unitary isomorphismfrom K to Γ( H ) .5. Let L be a real linear transformation of H satisfying Im h Lu | Lv i = Im h u | v i , ∀ u, v ∈ H . Such a transformation is said to be symplectic. Define W L ( u ) = W ( Lu ) , u ∈ H . The map u W L ( u ) is a strongly continuous, projective, unitary and irre-ducible representation of H in Γ( H ) obeying (3.2). Hence by the Stone-vonNeumann theorem in Property 4 there exists a unitary operator Γ( L ) in Γ( H ) satisfying Γ( L ) W ( u )Γ( L ) − = W ( Lu ) , ∀ u ∈ H . (3.12)Such a unitary operator Γ( L ) is unique upto multiplication by a scalar ofmodulus unity. The operator Γ( L ) is said to intertwine the representations W and W L .Let B j (Γ( H )) ⊂ B (Γ( H )) , for j = 1 and denote the ideal of trace class operatorsand Hilbert-Schmidt operators respectively on Γ( H ) . Then B (Γ( H )) is a Banachspace with k ρ k = Tr p ρ † ρ, ρ ∈ B (Γ( H )) , B (Γ( H )) is a Hilbert space with scalarproduct h ρ | ρ i = Tr ρ † ρ and B (Γ( H )) ⊂ B (Γ( H )) as a linear manifold. Definition 3.1. If ρ ∈ B (Γ( H )) , then the complex valued function ˆ ρ ( z ) := Tr ρW ( z ) , z ∈ H is called the quantum Fourier transform (or Wigner transform ) of ρ .We summarize a few properties of the quantum Fourier transform:1. The function ˆ ρ is bounded and continuous on H .Since B (Γ( H )) is the predual of B (Γ( H )) and W ( z ) is a unitary operator, | Tr ρW ( z ) | ≤ k ρ k . The continuity of ˆ ρ follows from the strong continuity of the Weyl representa-tion. 12. The correspondence ρ → ˆ ρ is injective .Let ρ , ρ ∈ B (Γ( H )) . The equation ˆ ρ = ˆ ρ implies that Tr( ρ − ρ ) W ( z ) = 0 for all z ∈ H and by (3.3) Tr( ρ − ρ ) X = 0 for any X ∈ B ( H ) .3. The quantum Fourier transform is factorizable .Indeed, for ρ j ∈ B (Γ( H j )) , j = 1 , , by property 2 of Weyl operators, ( ρ ⊗ ρ ) ∧ ( u ⊕ v ) = ˆ ρ ( u ) ˆ ρ ( v ) .
4. A positive operator ρ of unit trace in Γ( H ) is called an n -mode state . Forsuch a state ρ , by Property 3, of Weyl operators, the function ˆ ρ ( t z ) , t ∈ R is the characteristic function of the probability distribution of the observable −√ p ( z ) = i ( a ( z ) − a † ( z )) = √ q ( y ) − p ( x )) for any fixed z = x + i y .5. Quantum Bochner Theorem [SW75, Par10]. A complex valued function f defined on H is the quantum Fourier transform of an n -mode state if and onlyif the following are satisfied:(a) f (0) = 1 and f is continuous at .(b) The kernel K ( z, w ) = e i Im h z | w i f ( w − z ) is positive definite.We now state the gaussian integral in a form which we frequently use in the rest ofthis paper and refer to Appendix A of [Fol89] for a proof. Proposition 3.2 (Gaussian integral formula) . Let A be an n × n complex matrixsuch that A is symmetric ( A = A T ) and Re A is a positive definite matrix. Thenfor any m ∈ C n , Z R n exp (cid:8) − x T A x + m T x (cid:9) d x = r π n det A exp (cid:26) m T A − m (cid:27) , (3.13) where the branch of the square root is determined in such a way that det − / A > when A is real and positive definite. We defined the quantum Fourier transform on the trace class ideal. Now weproceed to extend this definition to the Hilbert-Schmidt class in the same spirit asin the classical theory of Fourier transforms. Let F = {| e ( u ) ih e ( v ) | : u, v ∈ H} , then F ⊂ B (Γ( H )) ⊂ B (Γ( H )) . Since exponential vectors form a total subset (Property2, Section 2) of Γ( H ) , F is a total set in B (Γ( H )) . The following example illustratesan important property of the quantum Fourier transform of elements of F . Example 3.3.
For u , v ∈ C , consider ρ = | e ( u ) ih e ( v ) | ∈ B (Γ( C )) . Then ˆ ρ ( z ) = Tr | e ( u ) ih e ( v ) | W ( z )= h e ( v ) | W ( z ) | e ( u ) i = e ¯ vu e − | z | +¯ vz − ¯ zu . (3.14)Thus ˆ ρ ∈ L ( C ) ∩ L ( C ) . Since L ( C ) = L ( R ) ⊗ L ( R ) and (by Property 4 inSection 2) the set { e − x + ζx | ζ ∈ C } is total is L ( R ) itfollows that { ˆ ρ | ρ ∈ F } is atotal set in L ( C ) . 13 heorem 3.4 ( Wigner isomorphism ) . For ρ ∈ B (Γ( H )) , let F n ( ρ ) be the func-tion defined on C n such that F n ( ρ )( z ) = π − n/ ˆ ρ ( z ) , z ∈ C n . (3.15) Then F n extends uniquely to a Hilbert space isomorphism from B (Γ( H )) onto L ( C n ) .Proof. First we prove the theorem when n = 1 , i.e., H = C . Let ρ j = | e ( u j ) ih e ( v j ) | , j = 1 , . Using (3.14) and the gaussian integral formula (Proposition 3.2), we get π Z C ˆ ρ ( z ) ˆ ρ ( z )d z = e h u | v i + h v | u i π Z C exp (cid:8) −| z | + h v − u | z i + h z | v − u i (cid:9) d z = e h u | v i + h v | u i e h v − u | v − u i = e h u | u i + h v | v i = Tr ρ † ρ . Now Example 3.3, shows that F is a scalar product preserving map between totalsets and thus extends uniquely to a Hilbert space isomorphism. Hence F ⊗ n is anisomorphism from B (Γ( H )) onto L ( C n ) . Observe that F ⊗ n coincides with F n on B (Γ( H )) , hence F ⊗ n is the extension we sought of F n on B (Γ( H )) . Corollary 3.5.
1. The map e ( u ⊕ v )
7→ | e ( u ) ih e (¯ v ) | extends as an isomorphism η from Γ( H ⊕ H ) onto B (Γ( H )) .2. Let u = P j u j e j ∈ H define g u = ⊗ j g u j , where g u j ∈ L ( R ) is as defined byequation (2.2). The map | e ( u ) ih e (¯ v ) | 7→ g u ⊗ g v extends as an isomorphism η from B (Γ( H )) onto L ( C n ) = L ( R n ) ⊗ L ( R n ) .3. Let v = P j v j e j ∈ H , define by ¯ v := P j ¯ v j e j . The Wigner isomorphismsatisfies F n ( | e ( u ) ih e (¯ v ) | ) = g u ′ ⊗ g v ′ , where (cid:18) u ′ v ′ (cid:19) = S (cid:18) uv (cid:19) , S = 1 √ (cid:20) − I IiI iI (cid:21) , (3.16) Proof.
1. This follows from a direct computation showing that η is scalar productpreserving.2. We know from Property 4 in Section 2 that the map e ( u ) g u extends to aHilbert space isomorphism, the required result follows from 1.3. Again it is enough to prove this when H = C . We have F ( | e ( u ) ih e (¯ v ) | )( x, y ) = Tr | e ( u ) ih e (¯ v ) | W ( x + iy )= h e (¯ v ) | W ( x + iy ) | e ( u ) i = D e (¯ v ) (cid:12)(cid:12)(cid:12) e − ( x + y ) − ( x − iy ) u e ( u + x + iy ) E = e − ( x + y ) − ( x − iy ) u + v ( u + x + iy ) = e − ( x + y )+( v − u ) x + i ( v + u ) y + vu = e − x + (cid:16) v − u √ (cid:17) √ x − (cid:16) v − u √ (cid:17) e (cid:16) v − u √ (cid:17) × e − y + (cid:16) v + u √ (cid:17) √ y − (cid:16) − i v + u √ (cid:17) e − (cid:16) v + u √ (cid:17) × e vu = g u ′ ( x ) g v ′ ( y ) , ( H ⊕ H ) B (Γ( H )) L ( C n )Γ( H ⊕ H ) B (Γ( H )) L ( C n ) η Γ( S ) ˜Γ( S ) η F n ˆΓ( S ) η η Figure 1: The maps ˜Γ( S ) and ˆΓ( S ) are the corresponding compositions. e ( u ⊕ v ) | e ( u ) ih e (¯ v ) | g u ⊗ g v e ( u ′ ⊕ v ′ ) | e ( u ′ ) ih e (¯ v ′ ) | g u ′ ⊗ g v ′ η Γ( S ) ˜Γ( S ) η F n ˆΓ( S ) η η Figure 2: u ′ ⊕ v ′ = S ( u ⊕ v ) where S as in equation (3.16).where (cid:18) u ′ v ′ (cid:19) = 1 √ (cid:20) − i i (cid:21) (cid:18) uv (cid:19) . (3.17) Remarks 3.6.
1. All the isomorphisms above are described by the Figure 1 viathe mappings in Figure 2.2. The quantum Fourier transform on B (Γ( H )) can be viewed as the secondquantization Γ( S ) on Γ( H ⊕ H ) , where S is the unitary matrix given by (3.16).The eigenvalues of S are λ = ie − iπ/ and λ = e iπ/ with multiplicities n each. Then λ j = − , j = 1 , . Write F ′ n = ˜Γ( S ) = η − F n , then F ′ n : B (Γ( H )) → B (Γ( H )) is an isomorphism and satisfies the property ( F ′ n ) = − I. (3.18)In the classical theory, the Fourier transform defined on L ( R n ) extends to aunitary F on L ( R n ) , furthermore F = − I . Equation (3.18) can be viewed asa noncommutative analogue of this fact. Theorem 3.7 (Quantum Fourier Inversion) . If ρ ∈ B (Γ( H )) then, ρ = 1 π n Z H ˆ ρ ( z ) W ( − z )d z, (3.19) where the integral is a weak operator integral with respect to the n -dimensionalLebesgue measure (inherited from C n ) on H .Proof. If φ, ψ ∈ Γ( H ) , by Theorem 3.4 h φ | ρ | ψ i = Tr | φ ih ψ | † ρ = Z C n F n ( | φ ih ψ | )( z ) F n ( ρ )( z )d z = 1 π n Z H h φ | W ( − z ) | ψ i ˆ ρ ( z )d z = h φ | π n Z H ˆ ρ ( z ) W ( − z )d z | ψ i . This is same as (3.19). 15e conclude this section with two results connecting the notion of generatingfunction defined in Section 2 with the quantum Fourier transform.
Proposition 3.8.
Let ρ ∈ B (Γ( H )) , then the quantum Fourier transform of ρ canbe expressed in terms of the generating function (equation (2.11)) of ρ as ˆ ρ ( u ) = e − | u | π n Z H exp (cid:8) −| z | − h u | z i (cid:9) G ρ (¯ z, u + z )d z. (3.20) Proof.
By the Klauder-Bargmann formula, ˆ ρ ( u ) = Tr ρW ( u ) = 1 π n Z H h ψ ( z ) | ρe − i Im h u | z i | ψ ( u + z ) i d z = 1 π n Z H exp (cid:26) −| z | − | u | − h u | z i (cid:27) h e ( z ) | ρ | e ( u + z ) i d z, which is same as (3.20).The following lemma is a corollory of Klauder-Bargmann formula. Lemma 3.9. If ρ is a trace class operator then Tr ρ = 1 π n Z H Tr ρ | ψ ( z ) ih ψ ( z ) | d z. (3.21) Furthermore, a positive operator ρ is trace class if and only the quantity on the rightside of the equation above is finite.Proof. Assume that ρ is trace class. If ρ = | φ ih φ | then (3.21) is immediate fromthe Klauder-Bargmann isometry. If ρ is a positive trace class operator then (3.21)follows from the preceding case by an application of the spectral theorem. Nowthe required result follows from the fact that any trace class operator is a linearcombination of four positive trace class operators.Now suppose ρ ≥ . Let { f k } be any orthonormal basis of Γ( H ) , then we caninterchange the summation and integral in the following computation because allthe quantities involved are nonnegative, X k h f k | ρ | f k i = X k π n Z H h f k |√ ρ | ψ ( z ) ih ψ ( z ) | √ ρ | f k i d z = 1 π n Z H X k h f k |√ ρ | ψ ( z ) ih ψ ( z ) | √ ρ | f k i d z = 1 π n Z H Tr ρ | ψ ( z ) ih ψ ( z ) | d z. Hence ρ ∈ B ( H ) if and only if the integral above coverges.16 heorem 3.10. Let H = H ⊕ H with dim H = n and dim H = n so that Γ( H ) = h ⊗ h where h i = Γ( H i ) , i = 0 , . Let ρ be any state on Γ( H ) and let ρ − i := Tr i ρ ∈ B ( h − i ) denote the h − i marginal of ρ , i = 0 , . For any u , v ∈ H , π n Z H Tr ρ | ψ ( u ⊕ z ) ih ψ ( v ⊕ z ) | d z = G ρ (¯ v , u ) e − ( k u k + k v k ) . (3.22) In other words, G ρ ( u , v ) = 1 π n Z H G ρ ( u + ¯ z, v + ¯ z ) e −k z k d z. (3.23) Proof.
Take H = C n and H i = C n i , i = 0 , . Let I i denote the identity operatorin h i , i = 0 , . Using Lemma 3.9 and the general property that, Tr A ( I ⊗ B ) =Tr(Tr A ) B and Tr C ( D ⊗ I ) = Tr(Tr C ) D for operators B, D in B ( h ) , B ( h ) respectively, we have π n Z C n Tr ρ | ψ ( u ⊕ z ) ih ψ ( v ⊕ z ) | d z = 1 π n Z C n Tr ρ | ψ ( u ) ih ψ ( v ) | ⊗ | ψ ( z ) ih ψ ( z ) | d z = 1 π n Z C n Tr ρ ( | ψ ( u ) ih ψ ( v ) | ⊗ I )( I ⊗ | ψ ( z ) ih ψ ( z ) | )d z = 1 π n Z C n Tr(Tr ( ρ | ψ ( u ) ih ψ ( v ) | ⊗ I ) | ψ ( z ) ih ψ ( z ) | )d z = Tr ρ ( | ψ ( u ) ih ψ ( v ) | ⊗ I )= Tr(Tr ρ ) | ψ ( u ) ih ψ ( v ) | = Tr ρ | ψ ( u ) ih ψ ( v ) | = G ρ (¯ v , u ) e − ( | u | + | v | ) . We continue our discussions with a finite dimensional Hilbert space H with anorthonormal basis { e j | ≤ j ≤ n } and the identification of z = P nj =1 z j e j ∈ H with z = ( z , z , . . . , z n ) ∈ C n . Let H R be the real linear span of the orthonormalbasis, then H = H R + i H R , i.e, if z ∈ H , z = x + iy, x, y ∈ H R . Furthermore,let L R ( H ) be the real algebra of real linear operators on H . Then for L ∈ L R ( H ) , Lz = ( Ax + By ) + i ( Cx + Dy ) , where A, B, C, D are operators in H R with respectivereal matrices denoted again by A, B, C, D . Write L = (cid:20) A BC D (cid:21) , (4.1)17here L ∈ M n ( R ) , i.e., a n × n real matrix. We now have L z = (cid:2) I iI (cid:3) L (cid:20) xy (cid:21) , z ∈ C n , x , y ∈ R n . (4.2)where I is the identity matrix of order n . Lemma 4.1.
Let
L, M ∈ L R ( H ) and z = x + iy, z ′ = x ′ + iy ′ , where x, x ′ , y, y ′ ∈ H R .Then h Lz | M z ′ i = (cid:2) x T y T (cid:3) L T ( I + iJ ) M (cid:20) x ′ y ′ (cid:21) , (4.3) where J = (cid:20) I − I (cid:21) . In particular, | Lz | = (cid:2) x T y T (cid:3) L T L (cid:20) xy (cid:21) . (4.4) Proof.
By (4.2) h Lz | M z ′ i = (cid:2) x T y T (cid:3) L T (cid:20) I − iI (cid:21) (cid:2) I iI (cid:3) M (cid:20) x ′ y ′ (cid:21) = (cid:2) x T y T (cid:3) L T ( I + iJ ) M (cid:20) x ′ y ′ (cid:21) . Example 4.2.
Consider Λ ∈ M n ( C ) as a real linear transformation on C n then Λ = (cid:20) Re Λ − Im ΛIm Λ Re Λ (cid:21) . (4.5)In this case J commutes with Λ . Furthermore, Λ is selfadjoint if and only if Λ issymmetric. Thus J Λ is skew symmetric in this case. Now by Lemma 4.1 Λ ≥ ifand only if Λ ≥ . Definition 4.3.
A real linear operator L ∈ L R ( H ) is called a symplectic transfor-mation of H if Im h Lz | Lz ′ i = Im h z | z ′ i , ∀ z, z ′ ∈ H . By Lemma 4.1, this is equivalent to (cid:2) x T y T (cid:3) L T J L (cid:20) x ′ y ′ (cid:21) = (cid:2) x T y T (cid:3) J (cid:20) x ′ y ′ (cid:21) , ∀ x , y , x ′ , y ′ ∈ R n , or equivalently, L T J L = J. (4.6)Any L ∈ M n ( R ) satisfying (4.6) is called a symplectic matrix .Suppose L is any symplectic matrix. Since J is an orthogonal matrix, takingdeterminants on both sides of (4.6) we get (det L ) = 1 . Thus L is nonsingular.Furthermore, (4.6) shows that L and ( L − ) T are orthogonally equivalent through J .Thus a is an eigenvalue of L if and only if a − is so, hence det( L ) = 1 . Multiplying18y ( L − ) T on the left and by L − on the right on both sides of (4.6) shows that L − is symplectic. Thus symplectic matrices form a group under multiplication.Indeed, it is a unimodular Lie group, denoted Sp (2 n, R ) and known as the symplecticreal matrix group of order n . From our discussions it is clear that all symplectictransformations of H constitute a group, denoted Sp ( H ) , isomorphic to the Liegroup Sp (2 n, R ) .We now make a detailed analysis of the unitary operators Γ( L ) , L ∈ Sp ( H ) occuring in Property 5 of Weyl operators in Section 3. Proposition 4.4.
Let L ∈ Sp ( H ) . Then | h Ω | Γ( L ) | Ω i| = α ( L ) − / where α ( L ) = det 12 ( I + L T L ) . Proof.
Let ρ = | Ω ih Ω | . Its quantum Fourier transform (Definition 3.1) is given by, ˆ ρ ( z ) = h Ω | W ( z ) | Ω i = e − | z | , z ∈ H . (4.7)By quantum Fourier inversion formula (3.19), | Ω ih Ω | = 1 π n Z H e − | z | W ( − z )d z. Conjugation by Γ( L ) gives Γ( L ) | Ω ih Ω | Γ( L ) − = 1 π n Z H e − | z | W ( − Lz )d z. Multiplying by h Ω | on the left and | Ω i on the right in both sides of this equation,using the unitarity of Γ( L ) , Lemma 4.1, and gaussian integral formula (Proposition3.2), we get | h Ω | Γ( L ) | Ω i| = 1 π n Z C n exp (cid:26) −
12 ( | z | + | L z | ) (cid:27) d z = 1 π n Z R n exp (cid:26) − (cid:2) x T y T (cid:3) ( I + L T L ) (cid:20) xy (cid:21)(cid:27) d x d y = (cid:18) det 12 ( I + L T L ) (cid:19) − / . Theorem 4.5.
For L ∈ Sp ( H ) , there exists a unique unitary operator Γ ( L ) in Γ( H ) satisfying the following:1. h Ω | Γ ( L ) | Ω i = α ( L ) − / , where α ( L ) = det 12 ( I + L T L ) . (4.8)19 . Γ ( L ) W ( u )Γ ( L ) − = W ( Lu ) , ∀ u ∈ H . Γ ( L ) | e ( v ) i = α ( L ) / π n R C n exp (cid:8) − ( | z | + | L z | ) + h z | v i (cid:9) | e ( L z ) i d z , ∀ v ∈ H . Proof.
Let Γ( L ) be any unitary operator satisfying Γ( L ) W ( u )Γ( L ) − = W ( Lu ) .Define Γ ( L ) = | h Ω | Γ( L ) | Ω i|h Ω | Γ( L ) | Ω i Γ( L ) . By Proposition 4.4, Γ ( L ) is a well defined unitary operator differing from Γ( L ) by a scalar multiple of modulus unity and satisfying properties (1) and (2) of thetheorem. To prove (3) we look at the rank one operator | e ( v ) ih Ω | as the inverse ofits quantum Fourier transform, | e ( v ) ih Ω | = 1 π n Z C n exp (cid:26) − | z | − h z | v i (cid:27) W ( − z )d z. Conjugation by Γ ( L ) on both sides yields Γ ( L ) | e ( v ) ih Ω | Γ ( L ) − = 1 π n Z C n exp (cid:26) − | z | − h z | v i (cid:27) W ( − L z )d z. Now right multiplication by | Ω i on both sides followed by a change of variable z
7→ − z in the integral on the right hand side completes the proof. Corollary 4.6.
The generating function (equation (2.11)) of Γ ( L ) is given by G Γ ( L ) ( u, v ) = α ( L ) − / exp (cid:8) u T Au + u T Λ v + v T Bv (cid:9) , where α ( L ) is as in (4.8), A, Λ , B are respectively the -particle annihilation, ex-change and creation amplitude matrices of the operator Γ ( L ) so that A = A Γ ( L ) = 12 (cid:2) I iI (cid:3) L ( I + L T L ) − L T (cid:20) IiI (cid:21) , Λ = Λ Γ ( L ) = (cid:2) I iI (cid:3) L ( I + L T L ) − (cid:20) I − iI (cid:21) , (4.9) B = B Γ ( L ) = 12 (cid:2) I − iI (cid:3) ( I + L T L ) − (cid:20) I − iI (cid:21) . Proof.
Left multiplying by h e (¯ u ) | on both sides of the identity (3) in Theorem 4.5we get G Γ ( L ) ( u, v ) = α ( L ) / π n Z C n exp (cid:26) −
12 ( | z | + | L z | ) + h z | v i + h ¯ u | L z i (cid:27) d z (4.10) = α ( L ) / π n Z R n exp ( − (cid:20) xy (cid:21) T (cid:18) I + L T L (cid:19) (cid:20) xy (cid:21) + m T (cid:20) xy (cid:21)) d x d y , m T = v T (cid:2) I − iI (cid:3) + u T (cid:2) I iI (cid:3) L . By gaussian integral formula now itfollows that G Γ ( L ) ( u, v ) = α ( L ) − / exp (cid:26) m T ( I + L T L ) m (cid:27) . Expanding the exponent in the right side after substituting the expression for m interms of u , v we get the required result. Theorem 4.7 (Klauder-Bargmann representation for Γ ( L ) ) . Let | ψ ( z ) i denote thecoherent state at z ∈ H . Then the unitary operator Γ ( L ) has the following weakoperator integral representation Γ ( L ) = α ( L ) / π n Z H | ψ ( Lz ) ih ψ ( z ) | d z. (4.11) In particular, the map L Γ ( L ) from Sp ( H ) into U ( H ) is strongly continuous.Proof. First we show that the right side of (4.11) defines a bounded operator on Γ( H ) . It is a consequence of Klauder-Bargmann isometry (in Section 2) that forany L ∈ Sp ( H ) , the map φ K L π − n/ h ψ ( Lz ) | φ i is an isometry of Γ( H ) into L ( H ) .Therefore for any φ, φ ′ ∈ Γ( H ) the map z
7→ h φ | ψ ( Lz ) i h ψ ( z ) | φ ′ i is measurable andby Cauchy-Schwartz inequality π n Z H |h φ | ψ ( Lz ) i h ψ ( z ) | φ ′ i| d z = h | K L ( φ ) | | | K I ( φ ′ ) | i L ≤ k φ kk φ ′ k . Thus the right side of (4.11) defines a bounded operator on Γ( H ) . For H = C n wehave h e (¯ u ) | ψ ( L z ) i h ψ ( z ) | e ( v ) i = exp (cid:26) −
12 ( | z | + | L z | ) + h ¯ u | L z i + h z | v i (cid:27) . Now equation (4.10) implies that G Γ ( L ) ( u, v ) = α ( L ) / π n Z C n h e (¯ u ) | ψ ( L z ) i h ψ ( z ) | e ( v ) i d z , ∀ u , v ∈ H which yields (4.11). Remarks 4.8.
For any L ∈ Sp ( H ) , a unitary operator Γ in Γ( H ) is said to inter-twine W ( u ) and W ( Lu ) for all u ∈ H if Γ W ( u )Γ − = W ( Lu ) . Let G L denote the set of all such intertwiners. We know that there exists a uniqueelement Γ ( L ) which satisfies the condition h Ω | Γ ( L ) | Ω i > .1. If U is a unitary operator in H , i.e., U ∈ U ( H ) and Γ( U ) is the associ-ated second quantization operator satisfying Γ( U ) e ( u ) = e ( U u ) , ∀ u ∈ H , then h Ω | Γ( U ) | Ω i = 1 and hence Γ ( U ) = Γ( U ) .21. For any U, V in U ( H ) and L ∈ Sp ( H ) , Γ ( U )Γ ( L )Γ ( V ) and Γ ( U LV ) ∈ G ULV and h Ω | Γ ( U )Γ ( L )Γ ( V ) | Ω i = h Ω | Γ ( U LV ) | Ω i > . Hence Γ ( U )Γ ( L )Γ ( V ) = Γ ( U LV ) .
3. For any L ∈ Sp ( H ) , h Ω | Γ ( L ) − | Ω i = h Ω | Γ ( L ) | Ω i > , Γ ( L ) − and Γ ( L − ) lie in G L − and hence Γ ( L − ) = Γ ( L ) − . Definition 4.9. An n -mode state ρ ∈ B (Γ( H )) is called a gaussian state if thereexists an m ∈ C n and a n × n real, symmetric matrix S such that ˆ ρ ( z ) = exp (cid:26) − i Im h z | m i − (cid:2) x T y T (cid:3) S (cid:20) xy (cid:21)(cid:27) , (4.12)where z z = x + i y is an identification of H with C n . In this case, we write ρ = ρ m ,S , m is the mean annihilation vector , simply called the mean of the stateand S is the position-momentum covariance matrix . Remarks 4.10.
1. A n × n real, symmetric matrix S is theposition-momentum covariance of a gaussian state if and only if the followingmatrix inequality holds S + i J ≥ . (4.13)2. We say that a unitary operator U on Γ( H ) is a gaussian symmetry if, for anygaussian state ρ in Γ( H ) , the state U ρU † is also gaussian. It is a theorem thatany such gaussian symmetry U is equal to λW ( u )Γ ( L ) , where λ is a scalar ofmodulus unity, u ∈ H , L ∈ Sp ( H ) (See [Par13, BJS19]). Now Corollary 4.11below says that every gaussian symmetry has a Klauder-Bargmann integralrepresentation (4.14). Corollary 4.11 (Klauder-Bargmann representation for symmetries of gaussianstates) . For u ∈ H , L ∈ Sp ( H ) , W ( u )Γ ( L ) = α ( L ) / π n Z H exp {− i Im h u | Lz i} | ψ ( u + Lz ) ih ψ ( z ) | d z. (4.14) Proof.
This is immediate from (4.11).
Theorem 4.12.
Denote by H s (cid:13) Sp ( H ) , the Lie group which is the semidirect productof the additive group H and the Lie group Sp ( H ) acting on H so that the multipli-cation in H s (cid:13) Sp ( H ) is defined by ( u, L )( v, M ) = ( u + Lv, LM ) for all u, v ∈ H , L, M ∈ Sp ( H ) . Then the map ( u, L ) W ( u )Γ ( L ) is a strongly continuous, projective unitaryrepresentation of H s (cid:13) Sp ( H ) in Γ( H ) .Proof. Only the strong continuity of the map remains to be proved. To this end weconsider the Lebesgue measure preserving group action ( u, L ) : z → u + Lz, ( u, L ) ∈H s (cid:13) Sp ( H ) . This yields a strongly continuous unitary representation ( U ( u,L ) f )( z ) = f (( u, L ) − z ) , f ∈ L ( H ) H by identification of H with C n . This implies that themap φ
7→ h φ | ψ ( u + Lz ) i , z ∈ H , φ ∈ Γ( H ) is continuous as a map from Γ( H ) into L ( H ) , thanks to the Klauder-Bargmannisometry. Now an application of Klauder-Bargmann representation implies that ( u, L )
7→ h φ | W ( u )Γ ( L ) | φ ′ i is continuous in ( u, L ) for any φ, φ ′ in Γ( H ) . In other words, ( u, L ) W ( u )Γ ( L ) is weakly continuous. Unitarity implies strong continuity. Remark 4.13.
Writing W ( u, L ) = W ( u )Γ ( L ) we conclude from Theorem 4.12 that the map ( u, L ) W ( u, L ) is a strongly contin-uous projective irreducible unitary representation of the semidirect product group H s (cid:13) Sp ( H ) in Γ( H ) and W ( u, L ) W ( v, M ) = e − i Im h u | Lv i σ ( L, M ) W ( u + Lv, LM ) ∀ u, v ∈ H , L, M ∈ Sp ( H ) where σ ( L, M ) is a continuous function of ( L, M ) . E ( H ) Recall the definition of the generating function of a bounded operator in Γ( H ) fromSection 2. Definition 5.1.
An operator Z on Γ( H ) is said to be in the class E ( H ) if thereexists c ∈ C , α , β ∈ C n , A, B, Λ ∈ M n ( C ) , with A and B symmetric, such that thegenerating function of Z is of the form G Z ( u, v ) = c exp (cid:8) u T α + β T v + u T A u + u T Λ v + v T B v (cid:9) , ∀ u , v ∈ C n . (5.1)The ordered -tuple ( c, α , β , A, Λ , B ) completely characterizes Z ∈ E ( H ) and wecall them the E ( H ) -parameters of Z . Examples 5.2.
1. Let K be any contraction operator on H , then the secondquantization contraction Γ( K ) satisfies Γ( K ) | e ( v ) i = | e ( Kv ) i , v ∈ H . So, G Γ( K ) )( u, v ) = exp (cid:8) u T K v (cid:9) . (5.2)Hence Γ( K ) ∈ E ( H ) with parameters (1 , , , , K, .2. Let z ∈ H , then the associated Weyl displacement operator satisfies W ( z ) e ( v ) = exp (cid:26) − | z | − h z | v i (cid:27) e ( z + v ) . So G W ( z ) )( u, v ) = exp (cid:26) − | z | − h z | v i + h ¯ u | z + v i (cid:27) . (5.3)Hence W ( z ) ∈ E ( H ) with parameters ( e − | z | , z , − ¯ z , , I, .
23. Let L ∈ Sp ( H ) . By Corollary 4.6, Γ ( L ) ∈ E ( H ) , with parameters ( α ( L ) − / , , , A Γ ( L ) , Λ Γ ( L ) , B Γ ( L ) ) , where α ( L ) is as in (4.8) and A Γ ( L ) , Λ Γ ( L ) , B Γ ( L ) are as in (4.9).4. Let Z j ∈ E ( H j ) , with parameters ( c j , α j , β j , A j , Λ j , B j ) , j = 1 , then Z ⊗ Z ∈ E ( H ⊕H ) with parameters ( c c , α ⊕ α , β ⊕ β , A ⊕ A , Λ ⊕ Λ , B ⊕ B ) using the identification of Γ( H ⊕ H ) with Γ( H ) ⊗ Γ( H ) described inSection 2.Suppose Z ∈ E ( H ) with parameters ( c, α , β , A, Λ , B ) , by (5.1) we have, G Z ( u, v ) = c { u T α + β T v + u T A u + u T Λ v + v T B v )+ 12! ( u T α + β T v + u T A u + u T Λ v + v T B v ) + · · · } . (5.4)From the discussion in Section 2 and comparing with the definition of 1-particlecreation, annihilation vectors of equation (2.14) and the 2-particle creation, annihi-lation and exchange matrices of equation (2.15) we see that c = h Ω | Z | Ω i , c α = λ , c β = µ ,c ( A + αα T A Z , c ( B + ββ T B Z , c (Λ + αβ T ) = Λ Z . (5.5) Remark 5.3.
The most notable feature of an operator Z ∈ E ( H ) is the propertythat all the matrix entries of Z in the particle basis are completely determined bythe entries { h k | Z | ℓ i || k | + | ℓ | ≤ } which is a finite set of cardinality n + 3 n + 1 .If Z is selfadjoint then it is determined by ( n + 1) entries out of which n are real, ( n + 1) − n may be complex entries. Later we shall prove that a state ρ in Γ( H ) isgaussian if and only if it is in E ( H ) . Thus tomography of a gaussian state in Γ( H ) requires the estimation of atmost n + 1) − n events which are one dimensionalprojections in the subspace spanned by , and -particle vectors. Proposition 5.4. If ρ is an n -mode gaussian state then ρ ∈ E ( H ) .Proof. Take H = C n . Let ρ = ρ m ,S be a gaussian state on Γ( C n ) . By (4.12), (5.3)and (4.3), ˆ ρ ( z ) = exp (cid:26)(cid:2) x T y T (cid:3) (cid:20) ( ¯ m − m ) i ( ¯ m + m ) (cid:21) − (cid:2) x T y T (cid:3) S (cid:20) xy (cid:21)(cid:27) (5.6) G W ( z ) ( u, v ) = e u T v exp (cid:26) − (cid:2) x T y T (cid:3) I (cid:20) xy (cid:21) + (cid:2) x T y T (cid:3) (cid:18)(cid:20) IiI (cid:21) u − (cid:20) I − iI (cid:21) v (cid:19)(cid:27) . (5.7)24ow by the Wigner isomorphism Theorem 3.4, and equations (5.6) and (5.7), G ρ ( u , v ) = Tr ρ | e ( v ) ih e (¯ u ) | = h ρ || e ( v ) ih e (¯ u ) |i B (Γ( H )) = 1 π n Z C n ˆ ρ ( z ) h e (¯ u ) | W ( z ) | e ( v ) i d z = 1 π n Z C n ˆ ρ ( z ) G W ( z ) ( u, v )d z , = e u T v π n Z R n exp (cid:26) − (cid:2) x T y T (cid:3) ( 12 + S ) (cid:20) xy (cid:21) + q T (cid:20) xy (cid:21)(cid:27) , (5.8)where q = (cid:20) IiI (cid:21) u − (cid:20) I − iI (cid:21) v + (cid:20) ( ¯ m − m ) i ( ¯ m + m ) (cid:21) . The right hand side of (5.8) being agaussian integral in R n implies that ρ m ,S ∈ E ( H ) . Proposition 5.5.
Suppose Z ∈ E ( H ) with parameters ( c, α , β , A, Λ , B ) . If Z = Z † ,then c ∈ R , α = ¯ β , A = ¯ B , and Λ = Λ † . Furthermore, if Z ≥ and Z = 0 , then c > and Λ ≥ .Proof.
1. Take H = C n without loss of generality. By definition, G Z † ( u , v ) = G Z (¯ v , ¯ u ) . Therefore self-adjointness of Z implies, G Z ( u , v ) = G Z (¯ v , ¯ u ) . Furthermore, if Z ∈ E ( H ) , c exp (cid:8) u T α + β T v + u T A u + u T Λ v + v T B v (cid:9) = ¯ c exp (cid:8) v T ¯ α + ¯ β T u + v T ¯ A v + v T ¯Λ u + u T ¯ B u (cid:9) , ∀ u , v ∈ C n . (5.9)This is possible only if c = ¯ c, α = ¯ β , A = ¯ B and Λ = Λ † . If Z ≥ and Z = 0 , then c = 0 and c = G Z (0 , > . Furthermore the kernel ( u , v ) G Z (¯ u , v ) is positivedefinite and is of the form G Z (¯ u , v ) = cf ( u ) f ( v ) e h u | Λ | v i , where f ( v ) = exp (cid:8) v T ¯ A v + ¯ β T v (cid:9) . Since cf ( u ) f ( v ) is already a positive definitekernel, e h u | Λ | v i has to be positive definite. Hence h u | Λ | v i is positive semidefinite, inother words Λ ≥ . Theorem 5.6.
The class E ( H ) is a semigroup.Proof. Let Z j ∈ E ( H ) , with parameters ( a j , α j , β j , A j , Λ j , B j ) , j = 1 , . Assumewithout loss of generality that H = C n . By the Klauder-Bargmann isometry andformula, G Z Z ( u , v ) = h e (¯ u ) | Z Z | e ( v ) i = 1 π n Z H h e (¯ u ) | Z | ψ ( z ) i h ψ ( z ) | Z | e ( v ) i = 1 π n Z H G Z ( u , z ) G Z (¯ z , v ) e −| z | d z , (5.10)25here the integrand is an L function of z of the form f ( u , v ) g ( u , v , z ) , with f ( u , v ) = a a π n exp (cid:8) u T α + β T v + u T A u + v T B v (cid:9) and g ( u , v , z ) = exp (cid:8) −| z | + z T B z + ¯ z T A ¯ z + ( β T + u T Λ ) z + ( α T + v T Λ T )¯ z (cid:9) which is an L function of z . If we write z = x + i y then g ( u , v , z ) assumes the form g ( u , v , x + i y ) = exp ( − (cid:2) x T y T (cid:3) R (cid:20) xy (cid:21) + (cid:20) uv (cid:21) T S + (cid:20) µν (cid:21) T ! (cid:20) xy (cid:21)) where R is a symmetric n × n matrix, S is a n × n complex matrix and µ , ν arevectors in C n . Because g ( u , v , z ) is integrable, the real part of R is strictly positivedefinite. Now an application of gaussian integral formula shows that G Z Z ( u , v ) asa function of ( u , v ) has the structure of the generating function of an operator in E ( H ) . This completes the proof. Notation.
Proposition 5.5 allows a reduction in the number of parameters requiredto describe a positive operator ρ belonging to E ( H ) . We now parametrize such a ρ by the quadruple ( c, α , A, Λ) , where c = h Ω | ρ | Ω i > , α ∈ C n , A is a complexsymmetric matrix and Λ is a positive matrix, so that the generating function of ρ takes the form G ρ ( u, v ) = c exp (cid:8) u T α + ¯ α T v + u T A u + u T Λ v + v T ¯ A v (cid:9) . (5.11) Theorem 5.7.
A state ρ in Γ( H ) is gaussian if and only if ρ ∈ E ( H ) .Proof. If ρ is a gaussian state then by Proposition 5.4 ρ ∈ E ( H ) . Conversely, let ρ ∈ E ( H ) be a state with parameters ( c, α , A, Λ) . Now by Proposition 3.8, ˆ ρ ( u ) takes the form ˆ ρ ( u ) = e − | u | + u T ¯ A u + α T u cπ n Z R n exp (cid:26) − (cid:2) x T y T (cid:3) M (cid:20) xy (cid:21) + ℓ T (cid:20) xy (cid:21)(cid:27) d x d y , (5.12)where M is an n × n complex symmetric matrix and ℓ ∈ C n is of the form ℓ T = u T M + ¯ u T M + p T , with M and M being n × n constant complex matrices and p ∈ C n is a constantcomplex vector. Since the function under the integral sign is integrable, M hasstrictly positive real part. If u = ξ + i η , an application of the gaussian integralformula in R n shows that ˆ ρ ( ξ + i η ) = c ′ exp (cid:8) − Q ( ξ , η ) + q T ξ + q T η (cid:9) , ∀ ξ , η ∈ R n , (5.13)where c ′ is a constant scalar, Q is a quadratic form in n real variables with complexcoefficients and q , q are some elements in C n . Furthermore, by Property 4 ofquantum Fourier transform, the map t ˆ ρ ( t ( ξ + i η )) , t ∈ R is the characteristicfunction of a probability distribution µ ξ , η on the real line for any fixed ξ , η . Hence µ ξ , η is a normal distribution on R , Q ( ξ , η ) ≥ , ∀ ξ , η and q j = i γ j , j = 1 , for somereal vectors γ , γ ∈ R n . Thus ˆ ρ is the quantum Fourier transform of a gaussianstate ρ in Γ( H ) . 26 orollary 5.8. Let ρ be a gaussian state, Z ∈ E ( H ) . Then for any t > , (Tr ρ t Z † Z ) − Zρ t Z † is a gaussian state.Proof. By the structure theorem for gaussian states (Theorem 4, [Par92]), (Tr ρ t ) − ρ t is again a gaussian state, and hence ρ t ∈ E ( H ) . Since E ( H ) is a semigroup, theCorollary follows immediately. Remark 5.9.
For any Z ∈ E ( H ) the map ρ Zρ t Z † Tr ρ t Z † Z on the set of all statesyields a nonlinear gaussian state preserving information channel. Proposition 5.10.
1. Any unitary operator in E ( H ) is a gaussian symmetry.2. Any finite rank projection operator in E ( H ) is conjugate to the vacuum pro-jection | Ω ih Ω | by a gaussian symmetry.Proof.
1. Let U ∈ E ( H ) be unitary. If ρ is any gaussian state, then ρ, U ρU † ∈E ( H ) . Thus U ρU † is a gaussian state. Hence U is a gaussian symmetry.2. Let P be a finite rank projection operator in E ( H ) . Then P is a constantmultiple of a gaussian state. By the structure theorem for gaussian states (Theorem4, [Par92]), it is a one dimensional projection onto the span of W ( z )Γ ( L ) | Ω i forsome z ∈ H , L ∈ Sp ( H ) . Lemma 5.11.
Let Z ≥ and Z ∈ E ( H ) with parameters ( c, α , A, Λ) , K be anycontraction operator in H and z ∈ H .1. The E ( H ) -parameters of the operator Γ( K ) Z Γ( K ) † are given by the -tuple ( c, K α , KAK T , K Λ K † ) .2. The E ( H ) -parameters of the operator W ( − z ) ZW ( z ) are are given by the -tuple ( h ψ ( z ) | Z | ψ ( z ) i , α − ( I − Λ − AC ) z , A, Λ) , where C is the complexconjugation map x + i y x − i y on C n .Proof. Both the results follow from a direct computation of the generating function.
Remarks 5.12.
Let Z be as in Lemma 5.11.1. If U is a unitary matrix which diagonalizes Λ then the E ( H ) -parameters of Γ( U ) Z Γ( U ) † are ( c, U α , U AU T , D ) , where D is a diagonal matrix consistingof the eigenvalues of Λ .2. The parameter α of Z can be brought to by conjugating Z with a Weyloperator if there exists m ∈ C n such that ( I − Λ − AC ) m = α . We will seein Proposition 6.5 that this can be done whenever Z is a trace class operator. E ( H ) -parameters Our investigations in the previous section show that there are two different ways ofparametrizing the set of all gaussian states in Γ( H ) , one obtained from the quantumFourier transform and the other from generating functions. In the first approach, a27aussian state ρ is described by the pair ( m , S ) , where m is the mean annihilationvector and S is the position-momentum covariance matrix. Such a description in-volves n + n (2 n + 1) = 2 n + 3 n real parameters. The parametrization ( c, µ , A, Λ) arising from the generating function has n + n ( n + 1) + n = 2 n + 3 n + 1 realparameters. It is natural to explore the relationship between the two parametriza-tions, particularly, in the context of tomography of gaussian states as well as quan-tum information theory in infinite dimensions. We will see in this section that theparameter c is a normalization factor which is a function of the other parameters µ , A and Λ . Furthermore, it is shown that µ is completely determined by the meanannihilation vector m of the state. We begin the section with two propositions thatdescribe the exact relationship between the two parametrizations. Also, these re-sults show that a mean zero gaussian state is completely determined by a pair ( A, Λ) of complex matrices with A being symmetric and Λ positive semidefinite. We knowthat a n -mode, mean zero gaussian state is determined by a covariance matrix, i.e.,a n × n real symmetric matrix S satisfying the uncertainity relations expressed bythe matrix inequality, S + i/ J ≥ . Theorem 6.13 shows that a pair ( A, Λ) deter-mines a mean zero gaussian state if and only if the real linear operator I − Λ − AC ispositive definite. This condition on the parameters ( A, Λ) expressed as a n × n realmatrix inequality can be viewed as the E ( H ) -version of the uncertainity relations. Proposition 6.1.
Let ρ = ρ m ,S be a gaussian state with quantum Fourier transformgiven by (4.12). Then the E ( H ) -parameters, ( c, µ , A, Λ) of ρ satisfy the following: c = det (cid:18) I + S (cid:19) − / exp (cid:26)(cid:2) Re { m } T Im { m } T (cid:3) J ( 12 I + S ) − J (cid:20) Re { m } Im { m } (cid:21)(cid:27) , µ = i (cid:2) I iI (cid:3) ( 12 I + S ) − J (cid:20) Re { m } Im { m } (cid:21) ,A = 14 (cid:2) I iI (cid:3) ( 12 I + S ) − (cid:20) IiI (cid:21) , Λ = I − (cid:2) I iI (cid:3) ( 12 I + S ) − (cid:20) I − iI (cid:21) . (6.1) Proof.
The expressions in (6.1) are obtained by applying the gaussian integral for-mula to the last integral in equation (5.8).
Notation.
Let C denote the complex conjugation map on C n , ie. C ( z ) = ¯ z . Let C denote the n × n real matrix corresponding to the real linear map C as in (4.1),then C = (cid:20) I − I (cid:21) . (6.2)Given a symmetric matrix A and a positive matrix Λ in M n ( C ) , define the n × n matrix M ( A, Λ) := I − (cid:20) Re { Λ } − Im { Λ } Im { Λ } Re { Λ } (cid:21) − (cid:20) Re { A } Im { A } Im { A } − Re { A } (cid:21) = I − Λ − A C , (6.3)where I denotes the n × n identity matrix. Then M ( A, Λ) is a real symmetricmatrix. If M ( A, Λ) ≥ , then define c ( A, Λ) := p det { M ( A, Λ) } . (6.4)28 emma 6.2. Let A be a symmetric matrix and Λ be a positive matrix in M n ( C ) ,then J T M ( − A, Λ) J = M ( A, Λ) . (6.5) where the matrix M ( A, Λ) is defined by (6.3). In particular, M ( A, Λ) invertible ifand only if M ( − A, Λ) is invertible.Proof. A direct computation shows that A C J = − J A C , since Λ J = J Λ , wehave Proposition 6.3.
Let ( c, µ , A, Λ) be the E ( H ) -parameters of a gaussian state ρ .Then the covariance matrix S and the mean annihilation vector m of ρ satisfy thefollowing:1. The matrices S, A and Λ are related as follows, ( 12 I + S ) − = M ( − A, Λ) . (6.6)
2. The vectors µ and m are related by the equation ( I − Λ − AC ) m = µ (6.7) where C is the complex conjugation map described above. In other words, (cid:20) Re { m } Im { m } (cid:21) = M ( A, Λ) − (cid:20) Re { µ } Im { µ } (cid:21) . (6.8) In particular, µ = 0 if and only if m = 0 .Proof.
1. Write ( I + S ) − = (cid:20) P QQ T R (cid:21) as a block matrix where P, Q, R are of order n × n . We now solve for P, Q and R from (6.1). From the expression for A we get A = ( P − R ) + i ( Q + Q T ) . Hence P − R = 2( A + ¯ A ) . (6.9)From the expression for Λ we get − I ) = − ( P + R ) + i ( Q − Q T ) . Hence P + R = 2 I − (Λ + Λ T ) . (6.10)We get P and R from equations (6.9) and (6.10). The matrix Q is obtained bysubstitution. Finally we have P = I + ( A + ¯ A ) − Λ + Λ T R = I − ( A + ¯ A ) − Λ + Λ T Q = − i (cid:20) ( A − ¯ A ) + Λ − Λ T (cid:21) .
2. By comparing the real and imaginary parts on both sides of the expressionfor µ in (6.1), we get J (cid:20) Re { µ } Im { µ } (cid:21) = ( 12 I + S ) − J (cid:20) Re { m } Im { m } (cid:21) . By part 1) of the proposition we have ( I + S ) − = M ( − A, Λ) . Equation (6.5)completes the proof. 29 orollary 6.4. Let ( c, µ , A, Λ) be the E ( H ) -parameters of a gaussian state ρ . Then M ( e iθ A, Λ) > , ∀ θ ∈ R . (6.11) Proof.
By Lemma 5.11, ( c, e iθ/ µ , e iθ A, Λ) are the the E ( H ) -parameters of the state Γ( e iθ/ · I ) ρ Γ( e − iθ/ · I ) for any θ ∈ R . Proposition 6.3 provides the necessaryconclusion.If ρ is a nonzero, positive and trace class operator in E ( H ) then (Tr ρ ) − ρ is agaussian state (Theorem 5.7). In this case, M ( A, Λ) is positive definite (Corollary6.4). The following proposition proves a converse and also provides the value of Tr ρ in terms of the E ( H ) -parameters. Proposition 6.5.
Let ρ be a nonzero positive element of E ( H ) with parameters ( c, µ , A, Λ) , where µ = µ + i µ , µ , µ ∈ R n . Then ρ is trace class if and only if M ( A, Λ) defined by (6.3) is positive definite. In this case, Tr ρ = cc ( A, Λ) exp (cid:26) [ µ T , µ T ] M ( A, Λ) − (cid:20) µ µ (cid:21)(cid:27) . (6.12) Proof.
Recall the identification of z ∈ H with z = x + i y , x , y ∈ R n fixed in Section2. By Lemma 3.9, the positive operator ρ is trace class if and only if π n Z H h ψ ( z ) | ρ | ψ ( z ) i d z < ∞ , and in this case, Tr ρ is the value of the integral above. By Lemma 4.1, Z H h ψ ( z ) | ρ | ψ ( z ) i d zπ n = 1 π n Z H exp (cid:8) −| z | (cid:9) G ρ (¯ z, z )d z = cπ n Z C n exp (cid:8) −| z | + ¯ z T µ + ¯ µ T z + ¯ z T A ¯ z + ¯ z T Λ z + z T ¯ A z (cid:9) d z = cπ n Z R n exp (cid:26) − [ x T , y T ] M ( A, Λ) (cid:20) xy (cid:21) + 2[ µ T , µ T ] (cid:20) xy (cid:21)(cid:27) d x d y , (6.13)Hence Tr ρ is finite if and only if M ( A, Λ) > . Equation (6.12) is obtained byapplying the gaussian integral formula (3.13) to (6.13). Remarks 6.6.
1. Proposition 6.5 shows that the E ( H ) -parameter c of a gaus-sian state is purely a function of the other three parameters µ , A, Λ and c = c ( A, Λ) exp (cid:26) − [ µ T , µ T ] M ( A, Λ) − (cid:20) µ µ (cid:21)(cid:27) . (6.14)In particular, if µ = 0 then c = c ( A, Λ) .2. If ρ = ρ m ,S is a gaussian state, then W ( − m ) ρW ( − m ) † = ρ ,S . If ( c, µ , A, Λ) are the E ( H ) -parameters of ρ m ,S then that of the transformed state W ( − m ) ρW ( − m ) † are ( c ( A, Λ) , , A, Λ) .30. If M ( A, Λ) > then we have M ( A, Λ) + M ( − A, Λ) = 2 M (0 , Λ) > by (6.5).Hence I − Λ > , and by Example 4.2, this is equivalent to the positivedefiniteness of I − Λ . Thus the positive semidefinite matrix Λ is a strictcontraction in this case.4. By (6.6), if a pair ( A, Λ) determines a gaussian state ρ then M ( − A, Λ) =(1 / I + S ) − , where S is the covariace matrix of the state ρ . Since S ± i/ J ≥ ,we have M ( − A, Λ) − − / I ± iJ ) ≥ on C n . It may also be noted that theprojections / I + iJ ) and / I − iJ ) are orthogonal to each other in C n .Now we prove a restatement of the uncertainty relation S + i/ J ≥ satisfiedby the covaraince matrix of a gaussian state, in terms of the E ( H ) -parameters. Proposition 6.7.
A pair ( A, Λ) of complex matrices with A being symmetric and Λ positive semidefinite, determines the E ( H ) -parameters ( c ( A, Λ) , , A, Λ) of a gaus-sian state if and only if M ( − A, Λ) − −
12 ( I − iJ ) ≥ . (6.15) Proof.
Assume first that (6.15) holds. Then there exists a gaussian state ρ with co-variance matrix S = M ( − A, Λ) − − I . Now we get the desired result by Proposition6.1. Converse part follows from item 4 in Remarks 6.6. Lemma 6.8.
Let ρ = | ψ ih ψ | be a mean zero pure gaussian state. Then there exists L ∈ Sp ( H ) such that the E ( H ) -parameters of ρ are given by ( α ( L ) − / , , A Γ ( L ) , ,where α ( L ) and A Γ ( L ) are as in Corollary 4.6.Proof. By the structure theorem for gaussian states (Theorem 4, [Par92]), thereexists an L ∈ Sp ( H ) such that | ψ i = Γ ( L ) | ψ i . Now G ρ ( u, v ) = h e (¯ u ) | Γ ( L ) | Ω i h Ω | Γ ( L ) † | e ( v ) i = G Γ ( L ) ( u, G Γ ( L ) † (0 , v )= G Γ ( L ) ( u, G Γ ( L ) (¯ v, α ( L ) − / exp (cid:8) u T A Γ ( L ) u + v T ¯ A Γ ( L ) v (cid:9) where the last line follows from Corollary 4.6. Theorem 6.9.
Let ρ be a gaussian state with covariance matrix S and E ( H ) -parameters ( c, µ , A, Λ) . Then ρ is a pure state if and only if one of the followingholds:1. The matrix Λ = 0 .2. The covariance matrix S satisfies the relation ( 12 I + S ) − = (cid:20) P QQ I − P (cid:21) (6.16) for some real symmetric matrices P, Q of order n . roof. Write ( + S ) − = (cid:20) P QQ T R (cid:21) as a × block matrix. Then the expressionfor Λ in (6.1) shows that Λ vanishes if and only if condition 2 of the theorem holds.The necessity of condition 1 follows from Lemma 6.8 and item 2 in Remarks 6.6. Toprove sufficiency, take H = C n and note that the condition Λ = 0 implies G ρ (¯ u , v ) = F ( u ) F ( v ) , (6.17)where F ( x ) = √ ce x T ¯ A x , x ∈ C n . Expanding the left side of (6.17) in the particlebasis and comparing coefficients we get the martix elements in the particle basis as h k | ρ | ℓ i = β ( k ) β ( ℓ ) for some function β with P k ∈ Z + | β ( k ) | = Tr ρ = 1 . Hence ρ isa rank one operator and thus a pure state.It is interesting to note a corollary of the theorem above despite the fact that itdoes not play a role in the later part of this article. Corollary 6.10.
Let S be the covariance matrix of a pure gaussian state, i.e., S = L T L for some symplectic matrix L ∈ Sp (2 n, R ) . Let P = 1 / I + iJ ) , P ⊥ =1 / I − iJ ) . Then P and P ⊥ are mutually orthogonal projections with the propertythat in the direct sum decomposition C n = Ran P ⊕
Ran P ⊥ , the positive operator (1 / S ) − admits the block representation (cid:20) I P QQ ∗ I P ⊥ (cid:21) , (6.18) where I P and I P ⊥ are identity operators on Ran P and Ran P ⊥ respectively and Q : Ran P ⊥ → Ran P is the operator P (1 / S ) − | Ran P ⊥ .Proof. Since S is the covariance operator of a pure gaussian state, the exchangematrix Λ = 0 in its E ( H ) representation. Hence from the expression for Λ in (6.1), (cid:2) I iI (cid:3) ( 12 + S ) − (cid:20) I − iI (cid:21) = I (6.19) ⇒ (cid:20) I − iI (cid:21) (cid:2) I iI (cid:3) ( 12 + S ) − (cid:20) I − iI (cid:21) (cid:2) I iI (cid:3) = (cid:20) I − iI (cid:21) (cid:2) I iI (cid:3) . Since (cid:20) I − iI (cid:21) (cid:2) I iI (cid:3) = I + iJ, we get P ( 12 + S ) − P = P . (6.20)By doing a similar computation after taking transpose on both sides of (6.19) weget
12 ( I − iJ )( 12 + S ) −
12 ( I − iJ ) = 12 ( I − iJ ) . (6.21)But P ⊥ = I − P = ( I − iJ ) . Hence (6.21) is same as P ⊥ ( 12 + S ) − P ⊥ = P ⊥ . This together with (6.20) completes the proof.32ow we turn to the characterization of the E ( H ) -parameters of gaussian states.In Proposition 6.7, we saw uncertainity relations written in terms of the E ( H ) -parameters. That was a necessary and sufficient condition on the parameters ( A, Λ) to determine a mean zero gaussian state as an E ( H ) -element. Theorems 6.12 and6.13 below provide much simpler and more elegant necessary and sufficient condi-tions on a pair ( A, Λ) as above to determine a mean zero gaussian state. We needa lemma before that. Lemma 6.11.
Let A = A T ∈ M n ( C ) be such that M ( A, > . Then the matrix L ( A ) := 2 M ( A, − − I (6.22) is a n × n positive element of the group Sp (2 n, R ) and L ( A ) − = L ( − A ) .Proof. By Lemma 6.2, both M ( A, and M ( − A, are positive definite and in-vertible matrices. First we show that L ( A ) is a positive definite matrix. Since − / I < A C , we have / I − A C ) < I . Hence M ( A, − > I . To prove that L ( A ) is a symplectic matrix, recall from Lemma 6.2 that M ( A, J = J M ( − A, and hence J M ( A, − = M ( − A, − J . Now L ( A ) T J L ( A ) = L ( A ) L ( − A ) J. (6.23)But L ( A ) L ( − A ) = 4 (cid:26) { M ( A, M ( − A, } − − (cid:8) M ( A, − + M ( − A, − (cid:9) + 14 I (cid:27) . (6.24)Since { M ( A, M ( − A, } − = { I − (2 A C ) } − = 12 { M ( A, − + M ( − A, − } , the right side of (6.24) is the identity matrix and (6.23) completes the proof.In Theorem 6.9 we saw that the E ( H ) -parameters of a mean zero pure gaussianstate is of the form ( c, , A, where A is a symmetric matrix. Our next theoremcharacterizes pure gaussian states using these parameters. Theorem 6.12.
Let c > and A = A T ∈ M n ( C ) . The following statements areequivalent:1. The tuple ( c, , A, are the E ( H ) -parameters of a pure gaussian state ρ ( A, .2. The matrix M ( A, > and the constant c = c ( A, .3. The matrix A is a strict contraction, i.e., k A k < and the constant c = c ( A, .Proof. ⇔
2. The necessity of 2) is a special case of Corollary 6.4 and item 1 inRemarks 6.6. Conversely, assume that M ( A, > . The matrix L ( A ) defined inLemma 6.11 is a positive symplectic matrix. Therefore, (1 / L ( A ) = M ( A, − − I is the covariance matrix of a mean zero pure gaussian state (Proposition 3.10 in[Par10]). Now, Proposition 6.1 and Proposition 6.3 together completes the proof.33 ⇔
3. We look upon the n × n complex matrix A as an operator in threedifferent ways: (i) A as an operator in the complex Hilbert space C n ; (ii) as areal linear operator in C n considered as a n -dimeansional real Hilbert space withscalar product Re ¯ zz ′ , z , z ′ ∈ C n ; (iii) the real matrix A , where A = U AU − , U being the real linear orthogonal transformation U ( x + i y ) = ( x , y ) from C n to R n . Now observe that all these three operators have the same norm and hence k A k = k A k . On the other hand A C is a real symmetric matirx and M ( A, is strictly positive definite. Thus k A C k < . Since C is orthogonal we have k A k = k A k = k A C k < . Theorem 6.13.
Let c > and A, Λ ∈ M n ( C ) with A = A T and Λ ≥ . Thefollowing statements are equivalent:1. The tuple ( c, , A, Λ) are the E ( H ) -parameters of a gaussian state ρ ( A, Λ) .2. The matrix M ( A, Λ) > and the scalar c = c ( A, Λ) .Proof. Necessity of item 2) follows from Corollary 6.4 and item 1 in Remarks 6.6. Toprove the sufficiency, notice by item 3) in Remarks 6.6 that Λ is a strict contraction.Hence M ( A, > in particular and thus by Theroem 6.12, there exists a puregaussian state ρ ( A, with E ( H ) -parameters ( c ( A, , , A, . Hence by Proposition6.7, ≤ M ( − A, − − ( I − iJ ) . Furthermore, < M ( A, Λ) ≤ M ( A, implies M ( A, − ≤ M ( A, Λ) − . Therefore, ≤ M ( − A, − −
12 ( I − iJ ) ≤ M ( − A, Λ) − −
12 ( I − iJ ) . Proposition 6.7 completes this part of the proof.
Corollary 6.14.
Let ( c ( A, Λ) , , A, Λ) be the E ( H ) -parameters of a gaussian state ρ ( A, Λ) . Let Λ ′ be a positive martix such that Λ ′ ≤ Λ . Then there exists a gaussianstate ρ ( A, Λ ′ ) with E ( H ) parameters ( c ( A, Λ ′ ) , , A, Λ ′ ) .Proof. This follows from the relation M ( A, Λ ′ ) ≥ M ( A, Λ) > . E ( H ) In Section 5, we noticed that any positive operator in E ( H ) is determined by aquadruple ( c, µ , A, Λ) in the sense of (5.11). Also, we know that a state is in E ( H ) if an only if it is a gaussian state. This section is devoted to the study of positiveoperators in E ( H ) .A few notations are needed before we proceed. The conventions = 1 = 0! areused in what follows. Notation.
Let ∆ n ( Z + ) = { R | R = [ r ij ] , r ij ∈ Z + ∀ i, j, r ij = 0 , ∀ i > j } denote theset of all n × n upper triangular matrices with nonnegative integer entries. Given R = [ r ij ] ∈ ∆ n ( Z + ) , let ˜ r i := i X j =1 r ji + n X j = i r ij , ˜ r ( R ) := [˜ r , ˜ r , . . . , ˜ r n ] T , (7.1)34 R | := P i,j r ij , R ! := Π i,j r ij ! . Furthermore, for any B = [ b ij ] ∈ M n ( C ) , let B ◦ R := Π i,j b r ij ij (notice that B ◦ R takes account of the uppertriangular entries of B only), ∆( B, t ) := { R = [ r ij ] ∈ ∆ n ( Z + ) | r ij = 0 whenever b ij = 0 , ˜ r ( R ) = t } . (7.2)Since | ˜ r ( R ) | is an even number, ∆( B, t ) = φ , the empty set, whenever | t | is an oddnumber. With the convention that sum over an empty set is zero, we define thefunction ϕ B : Z n + → C by ϕ B ( t ) = √ t ! X R ∈ ∆( B, t ) | R |− Tr R B ◦ R R ! . (7.3)Then ϕ B ( t ) = 0 if | t | is an odd number. Furthermore, we write t ≤ s for two multi-indices t , s ∈ Z n + , t = ( t , t , . . . , t n ) and s = ( s , s , . . . , s n ) to mean t j ≤ s j , ≤ j ≤ n . Lemma 7.1.
Let µ ∈ C n , B = [ β ij ] ∈ M n ( C ) be a symmetric matrix, and z ∈ C n .Then exp (cid:8) z T B z (cid:9) = X s ∈ Z n + ϕ B ( s ) √ s ! z s , (7.4) exp (cid:8) µ T z + z T B z (cid:9) = X k , s ∈ Z n + k ≤ s µ k k ! ϕ B ( s − k ) p ( s − k )! z s . (7.5) Proof.
First we prove (7.4). Let z = ( x , x , . . . , x n ) , since B is a symmetric matrix,we have ( z T B z ) ℓ ℓ ! = ( P i,j β ij x i x j ) ℓ ℓ ! = ( P i β ii x i + 2 P i Let the linear operators exp (cid:8) µ T a + a T B a (cid:9) and Γ( B ) be definedon the finite particle domain F by (7.6). Then lim N →∞ Γ( B ) ⊕ k ≤ N | z i ⊗ k √ k ! ! = | e ( Bz ) i lim N →∞ exp (cid:8) µ T a + a T B a (cid:9) (cid:18) ⊕ | t |≤ N z t | t i√ t ! (cid:19) = exp (cid:8) µ T z + z T B z (cid:9) | e ( z ) i (7.8) where B in z T B z above denotes the matrix of B . roof. The first equation in (7.8) is immediate from (7.6). To prove the second,notice from (7.6) that exp (cid:8) µ T a + a T B a (cid:9) (cid:18) ⊕ | t |≤ N z t | t i√ t ! (cid:19) = X | t |≤ N z t X k ≤ s ≤ t µ k k ! ϕ B ( s − k ) p ( s − k )! | t − s i p ( t − s )! , write t − s = m on the right side above so that t = s + m and exp (cid:8) µ T a + a T B a (cid:9) (cid:18) ⊕ | t |≤ N z t | t i√ t ! (cid:19) = X | m |≤ N X | s |≤ N −| m | z s X k ≤ s µ k k ! ϕ B ( s − k ) p ( s − k )! z m | m i√ m ! . By Lemma 7.1, for each fixed m the coefficient of m ! − / z m | m i in the equationabove converges to exp (cid:8) µ T z + z T B z (cid:9) as N → ∞ and this completes the proof.In the light of Proposition 7.2 we extend the definition of the operators Γ( B ) and exp (cid:8) µ T a + a T B a (cid:9) to the exponential domain E by Γ( B ) | e ( z ) i = | e ( Bz ) i , exp (cid:8) µ T a + a T B a (cid:9) | e ( z ) i = exp (cid:8) µ T z + z T B z (cid:9) | e ( z ) i . (7.9)Furthermore, for linear operators B and B on H and µ ∈ C , the operator Γ( B ) exp (cid:8) µ T a + a T B a (cid:9) is a well defined linear operator on the exponential do-main with Γ( B ) exp (cid:8) µ T a + a T B a (cid:9) | e ( z ) i = exp (cid:8) µ T z + z T B z (cid:9) | e ( B z ) i . (7.10) Remark 7.3. Is the operator exp (cid:8) µ T a + a T B a (cid:9) defined on the linear span of F ∪ E as above a closable operator in general? We will see in what follows thatthe semigroup E ( H ) provides examples of B ’s where exp (cid:8) µ T a + a T B a (cid:9) closes tobounded operators on Γ( H ) . Theorem 7.4. Let Z ≥ and Z ∈ E ( H ) with parameters ( c, µ , A, Λ) , Define Z = √ c Γ( √ Λ) exp (cid:8) ¯ µ T a + a T ¯ A a (cid:9) (7.11) on span F ∪ E in the sense of (7.6) and (7.10).1. The linear operator Z closes to a bounded operator on Γ( H ) , again denotedby Z with Z ∈ E ( H ) .2. The operator Z admits the factorization Z = Z † Z . (7.12) 3. There exists a partial isometry V such that Z = V √ Z. (7.13) In particular, Z ∈ B (Γ( H )) (i.e., trace class) if and only if Z ∈ B (Γ( H )) (i.e., Hilbert-Schmidt). . Let µ = 0 , then Z | t i = √ c X s ≤ t s(cid:18) ts (cid:19) ϕ ¯ A ( t − s ) √ Λ ⊗ | s | | s i (7.14) where ϕ ¯ A is defined by (7.3). The matrix entry of Z in the n -mode particlebasis, corresponding to t = ( t , t , . . . , t n ) , t ′ = ( t ′ , t ′ , . . . , t ′ n ) ∈ Z n + is given by h t | Z | t ′ i = c X s , s ′ ≤ t ∧ t ′| s | = | s ′ | s(cid:18) ts (cid:19) ϕ A ( t − s ) h s | Λ ⊗ | s | | s ′ i s(cid:18) t ′ s ′ (cid:19) ϕ A ( t ′ − s ′ ) , (7.15) where the j -th coordinate of t ∧ t ′ is min( t j , t ′ j ) , ∀ j = 1 , , . . . , n . In otherwords, the matrix representation of Z in the particle basis is given by, [ Z ] = c [ E A ] [Γ(Λ)] [ E A ] † , (7.16) where the ( t , s ) -th entry of the matrix [ E A ] is given by E A ( t , s ) = (q(cid:0) ts (cid:1) ϕ A ( t − s ) , s ≤ t , , otherwise. (7.17) Proof. 1. By (7.11), Z | e ( v ) i = √ c exp (cid:8) ¯ α T v + v T ¯ A v (cid:9) | e ( √ Λ v ) i (7.18)and thus h Z e ( u ) | Z e ( v ) i = G Z (¯ u, v ) = h e ( u ) | Z | e ( v ) i = D √ Ze ( u ) (cid:12)(cid:12)(cid:12) √ Ze ( v ) E . (7.19)Therefore, for any finite linear combination P kj =1 β j | e ( v j ) i of exponential vectors, k Z k X j =1 β j | e ( v j ) ik = k√ Z k X j =1 β j | e ( v j ) ik ≤ k√ Z k k k X j =1 β j | e ( v j ) i k . Hence the map Z closes to a bounded operator on Γ( H ) . Now by (7.18), Z ∈ E ( H ) with parameters ( √ c, , ¯ α , , √ Λ , ¯ A ) .2. By (7.19) G Z † Z ( u, v ) = G Z ( u, v ) for all u, v ∈ H , hence Z = Z † Z .3. Equation (7.19) asserts that the map √ Z | e ( v ) i 7→ Z | e ( v ) i is scalar productpreserving on Ran √ Z . Hence it extends to a partial isometry V with inital space Ran √ Z satisfying V √ Z | e ( v ) i = Z | e ( v ) i for all v ∈ H .4. Define the operator Z as earlier by Z = √ c Γ( √ Λ) exp (cid:8) a T ¯ A a (cid:9) (7.20)on span F ∪ E . Then by part 1 of the theorem, Z extends to a bounded operator.By (7.7), we have (7.14). Since h t | Z | t ′ i = h Z t | Z t ′ i , equation (7.15) follows from(7.14). 38 emark 7.5. It maybe noticed from (7.17) that the matrix [ E A ] appearing in (7.16)is a multiindex lower triangular matrix. Furthermore, since Γ(Λ) | H s (cid:13) k = Λ ⊗ k | H s (cid:13) k , Γ(Λ) leaves H s (cid:13) k invariant for k = 0 , , , . . . , and the matrix of Γ(Λ) is a block diagonalmatrix in the particle basis. Theorem 7.6. Let c ∈ C , α ∈ C n , A, Λ ∈ M n ( C ) , where A is complex symmetricand Λ is positive semidefinite. Then there exists a positive operator Z ∈ E ( H ) withparameters ( c, α , A, Λ) if and only if the operator Z defined by (7.11) on E extendsto a bounded operator on Γ( H ) .Proof. If there exists a positive operator Z ∈ E ( H ) with parameters ( c, α , A, Λ) then Theorem 7.4 provides the required result. Conversely, if Z defined by (7.11)on E extends to a bounded operator on Γ( H ) which is again denoted by Z , then Z † Z is the required operator. Remark 7.7. The factorization of Z in (7.12) of Theorem 7.4 tempts us to expressa gaussian state ρ with E ( H ) -parameters ( c, , A, Λ) , where A = [ α rs ] as ρ = c exp (X r,s α r,s a † r a † s ) Γ(Λ) exp (X r,s ¯ α r,s a r a s ) on the exponential domain E but the operator exp nP r,s α r,s a † r a † s o makes sense onlywhen A is in a special region in the space of complex symmetric matrices of order n . We had seen in Proposition 6.5 that a positive operator Z ∈ E ( H ) with param-eters ( c, µ , B, Λ) is trace class if and only if the matrix M ( B, Λ) defined by (6.3) ispositive definite and in this case Λ must be a strict contraction (item 3) of Remarks6.6). The following lemma analyses the situation for a general positive element in E ( H ) . Lemma 7.8. Let Z ≥ and Z ∈ E ( H ) with parameters ( c, µ , B, Λ) . Let M ( B, Λ) be the n × n real symmetric matrix defined by (6.3). Then,1. M ( B, > or equivalently k B k < ;2. M ( B, Λ) ≥ ;3. Λ is a contraction;4. if Λ = D , where D = Diag ( λ , λ , . . . , λ n , , , . . . , n − times ) , λ j < , ≤ j ≤ n , n = n − n , then in the direct sum decomposition C n = C n ⊕ C n , the matrix B has the block diagonal form B = (cid:20) (cid:21) B 00 0 . (7.21)39 roof. 1. Write Z = Z † Z as in Theorem 7.4, then Z = Γ(0) Z is trace classand ( Z ) † Z is a positive trace class element in E ( H ) with parameters ( c, µ , B, .Hence by Proposition 6.5, M ( B, > .2. Let < θ < then Γ( θ · I ) is trace class and thus Γ( θ · I ) Z Γ( θ · I ) † is a positivetrace class element of E ( H ) with parameters ( c, µ, θ B, θ Λ) . Hence by Proposition6.5, M ( θ B, θ Λ) > . The result follows because M ( θ B, θ Λ) → M ( B, Λ) as θ → .3. Follows from 2) as in item 3) of Remarks 6.6.4. Let B = (cid:20) B B B B (cid:21) in the decomposition C n = C n ⊕ C n and D λ =Diag ( λ , λ , . . . , λ n ) , then by definition, M ( B, Λ) = R n R n R n R n I n − D λ − B − B − B − B − B − B − B − B − B − B I n − D λ + 2 Re B B − B − B B B . By 2) we know that M ( B, Λ) ≥ . This implies in particular that the real symmetricmatrices − B and B both are positive matrices. So B = 0 and bythe positivity of M ( B, Λ) , we see that B , B and B are all zero matrices. Theorem 7.9. Let Z ≥ and Z ∈ E ( H ) with parameters ( c, µ, A, Λ) . Then Z liesin the weak-closure of the set of all positive scalar multiples of gaussian states.Proof. Let < θ < , define ρ θ = Γ( θ · I ) Z Γ( θ · I ) † as in the proof of 2) in Lemma7.8, then ρ θ is a positive scalar multiple of a gaussian state for all θ ∈ (0 , . Observethat ρ θ → Z weakly as θ → . Remark 7.10. Let Z ∈ E ( H ) be a positive operator with parameters ( c, , A, Λ) .By Lemma 7.8 Λ is a positive contraction. Then there exists a unitary U suchthat Γ( U ) Z Γ( U ) † has E ( H ) -parameters given by ( c, , B, D ) , where D is a diagonalcontraction and B = U AU T . If D = Diag ( D λ , I n ) as in item 4) of Lemma 7.8,then write H = H ⊕ H . Then keeping the same notations of item 4) of Lemma7.8, Γ( U ) Z Γ( U ) † = Z ⊗ Γ( I n ) , where Z ∈ E ( H ) is a positive operator with E ( H ) -parameters ( c, , B , D λ ) , Γ( I n ) is the identity operator on Γ( H ) . The operator Z is a positive scalar multipleof a gaussian state if and only if M ( B , D λ ) > . For the purpose of illustration,it may be noted that the -mode state operator ρ ( α, tends to an unboundedoperator with domain including exponential vectors when | α | → from below and M ( , 0) = 0 . In Section 6, we analysed the E ( H ) -parameters of a gaussian state and found thata mean zero gaussian state is completely determined by a pair of n × n complexmatrices ( A, Λ) , A being symmetric and Λ being positive definite. A necessary and40ufficient condition on the pair ( A, Λ) to determine a gaussian state is the property M ( A, Λ) > , (Theorem 6.13) where M ( A, Λ) is defined by 6.3. Furthermore, it wasalso observed that the constant parameter c in the E ( H ) parameterization of a meanzero gaussian state has to be c ( A, Λ) := p det M ( A, Λ) . In Section 7, we describedan important factorization property of positive operators in E ( H ) . In this section,we see the consequences of Theorem 7.4 in the case of gaussian states. We draw theattention of the reader to equations (8.2) and (8.5) which describe respectively theparticle basis expansion for a mean zero pure gaussian state and the density matrixformula (DMF) for an arbitrary mean zero gaussian state in the same basis. Theseresults lead to Theorem 8.6, which describes the architecture of a gaussian state. Proposition 8.1. If c > , and A = [ α ij ] is a complex symmetric matrix, then ( c, , A, are the E ( H ) -parameters of a positive operator Z ∈ B (Γ( H )) if and onlyif X t ∈ Z n + | ϕ A ( t ) | < ∞ , (8.1) where ϕ A is given by (7.3). In this case, Z = | ψ ih ψ | , where | ψ i = √ c P t ∈ Z n + ϕ A ( t ) | t i and | ψ A i = p c ( A, X t ∈ Z n + ϕ A ( t ) | t i (8.2) is a pure gaussian state.Proof. Assume first that ( c, , A, are the E ( H ) -parameters of a positive operator Z . Then by Theorem 7.4, Z = √ c Γ(0) exp (cid:8) a T ¯ A a (cid:9) defined on span F ∪ E in thesense of (7.6) and (7.10) extends to a bounded operator on Γ( H ) . Furthermore, by(7.7), Z | t i = √ cϕ ¯ A ( t ) | Ω i , ∀ t ∈ Z n + . In other words, Z is the rank one operator | Ω ih ψ | , where h ψ | t i = √ cϕ ¯ A ( t ) , ∀ t ∈ Z n + . So (8.1) is satisfied and Z = | ψ ih ψ | . Conversely, if (8.1) is satisfied, define | ψ i = √ c P t ∈ Z n + ϕ A ( t ) | t i . Then Z = | ψ ih ψ | ∈ E ( H ) with the required properties.The state defined by (8.2) is a gaussian state because of item 1 in Remarks 6.6. Corollary 8.2. If ( c ( A, , , A, are the E ( H ) -parameters of a mean zero puregaussian state | ψ A ih ψ A | , then ϕ A ( t ) = h t | ψ A i p c ( A, . (8.3) Proof. Equation (8.3) is immediate from (8.2). Proposition 8.3. Let c > and A, Λ ∈ M n ( C ) with A = A T and Λ ≥ . The tuple ( c, , A, Λ) are the E ( H ) -parameters of a gaussian state ρ ( A, Λ) if and only if theoperator Z defined by Z = √ c Γ( √ Λ) exp (cid:8) a T ¯ A a (cid:9) (8.4) on span F ∪ E in the sense of (7.6) and (7.10) extends to a Hilbert-Schmidt operator Z A, Λ1 on Γ( H ) and Tr( Z A, Λ1 ) † ( Z A, Λ1 ) = 1 . In this case, ρ ( A, Λ) = ( Z A, Λ1 ) † ( Z A, Λ1 ) . roof. This follows from Theorem 7.4 and Theorem 7.6. Theorem 8.4. Let ρ be a mean zero gaussian state. Then there exists a pair of n × n complex matrices ( A, Λ) , A being symmetric and Λ positive, such that thematrix representation of ρ = ρ ( A, Λ) in the particle basis is given by the densitymatrix formula (DMF), ρ mat ( A, Λ) = c ( A, Λ) [ E A ] [Γ(Λ)] [ E A ] † , (8.5) where c ( A, Λ) is defined by (6.4), and the ( t , s ) -th entry of the matrix [ E A ] is givenby E A ( t , s ) = (q(cid:0) ts (cid:1) ϕ A ( t − s ) , if s ≤ t , , otherwise, (8.6) with ϕ A defined by (7.3).Proof. Equation (8.5) follows from item 4) in Theorem 7.4. Definition 8.5. Given s ∈ Z n + , consider the mode-shift isometry S s at s ∈ Z n + defined by S s | t i = | t + s i and the corresponding homomorphism τ s on B (Γ( H )) defined by τ s ( X ) = S s XS † s , X ∈ B (Γ( H )) . It may be noticed that { τ s | s ∈ Z n + } is an n -parameter semigroup of homomorphisms on B (Γ( H )) . Given a vector λ =( λ , λ , . . . , λ n ) ∈ R n + , define the mixing kernel M λ on Z n + by M λ ( t , t ′ ) = X s ≤ t ∧ t ′ s(cid:18) ts (cid:19)(cid:18) t ′ s (cid:19) λ s τ s , (8.7)where s ≤ t ∧ t ′ is meant entrywise. Theorem 8.6. Let D λ be the positive diagonal matrix Diag ( λ , λ , . . . , λ n ) and ( c ( A, D λ ) , , A, D λ ) be the E ( H ) -parameters of a gaussian state ρ ( A, D λ ) . Let ρ ( A, 0) = | ψ A ih ψ A | with | ψ A i as in (8.2). Then ρ ( A, is a pure gaussian statewith E ( H ) -parameters ( c ( A, , , A, and the matrix entries of ρ ( A, D λ ) are givenby h t | ρ ( A, D λ ) | t ′ i = c ( A, D λ ) c ( A, h t | M λ ( t , t ′ )( ρ ( A, | t ′ i , ∀ t , t ′ ∈ Z n + . (8.8) Proof. By (7.15), and Proposition 6.5 we have h t | ρ ( A, D λ ) | t ′ i = c ( A, D λ ) √ t ! t ′ ! X s ≤ t ∧ t ′ ϕ A ( t − s ) p ( t − s )! ϕ ¯ A ( t ′ − s ) p ( t ′ − s )! λ s s ! , (8.9)Now by (8.3), h t | ρ ( A, D λ ) | t ′ i = c ( A, D λ ) c ( A, √ t ! t ′ ! X s ≤ t ∧ t ′ h t − s | ψ A i p ( t − s )! h ψ A | t ′ − s i p ( t ′ − s )! λ s s != c ( A, D λ ) c ( A, X s ≤ t ∧ t ′ s(cid:18) ts (cid:19)(cid:18) t ′ s (cid:19) λ s h t | S s | ψ A ih ψ A | S † s | t ′ i . This is same as (8.8). 42 emarks 8.7. 1. The total particle number of any basis vector of the form | t i isdecreased by the strictly upper triangular matrix E † A − I , preserved by the blockdiagonal matrix Γ(Λ) and increased by the strictly lower triangular matrix E A − I in their respective actions. Thus our DMF in (8.5) as a factorizationpreserves the spirit of Wick ordering in quantum stochastic calculus [Par92].2. Theorem 8.6 throws light on the architecture of a general gaussian state. Let ρ be a gaussian state with E ( H ) -parameters ( c ( α , A ′ , Λ ′ ) , α , A ′ , Λ ′ ) . Conjugat-ing ρ with an appropriate Weyl operator followed by the second quantizationof a unitary matrix operator in H , as suggested by Remarks 6.6 and Lemma5.11 respectively, transforms ρ to a gaussian state with E ( H ) parameters ( c ( A, D λ ) , , A, D λ ) , where D λ is a positive diagonal matrix. By Theorem 8.6,this transformed state is completely determined by the action of a positivityand trace class preserving kernel M λ ( t , t ′ ) on the pure gaussian state | ψ A ih ψ A | which is constructed from the -particle annihilation amplitude matrix A (cf.Section 2 and equation (5.5)). In this section, we discuss some interesting examples of gaussian states using the E ( H ) -parameters. An easy condition to check the entanglement of a pure gaussianstate is obtained in Corollary 9.11. Furthermore, a whole class of completely entan-gled pure gaussian states is obtained. This yields examples of such entangled stateswhich are also invariant under the action of the permutation group S n on the set ofall the n modes. Example 9.1. [ -mode mean zero pure gaussian states] Let n = 1 , α ∈ C . Then byTheorem 6.12, ( c, , α, are the E ( H ) -parameters of a pure gaussian state ρ ( α, if and only if | α | < / , c = c ( α, 0) = (1 − | α | ) / . The index set ∆( α, t ) definedby (7.2) is given by ∆( α, t ) = ( φ, if t is odd { t/ } , if t is even , where φ denote the empty set. Now the function φ α ( t ) defined by (7.3) is given by ϕ α ( t ) = ( , if t is odd (cid:0) tt/ (cid:1) / α t/ , if t is even . Therefore the gaussian state ρ ( α, 0) = | ψ α ih ψ α | , where | ψ α i = (1 − | α | ) / X t ∈ Z + (cid:18) tt (cid:19) / α t | t i . (9.1)Equation (9.1) has the following interpretation: if the observable a † a (which mea-sures the number of particles) is measured in the state | ψ α i then the possible43utcomes are , , , . . . , t, . . . and the probability for the outcome t is equal to (1 − | α | ) / (cid:0) tt (cid:1) | α | t . Simple algebra shows that this is equal to Pr( { t } ) = q (1 − | α | ) ( + 1) · · · ( + ( t − t ! (4 | α | ) t , t = 0 , , , . . . , where the right hand side as a function of t on Z + is the well known negative binomialdistribution [Fel68] on Z + with index − / and parameter p = 4 | α | , < p < . Example 9.2. [ -mode mean zero mixed gaussian states] Let n = 1 , α ∈ C , λ > .Then by Theorem 8.4, ( c, , α, λ ) are the E ( H ) -parameters of a gaussian state ρ ( α, λ ) if and only if | α | < (1 − λ ) / , c = c ( α, λ ) = { (1 − λ ) − | α | } / . Then by the DMF(8.5) ρ mat ( α, λ )( t, t ′ ) = { (1 − λ ) − | α | } / X s ≤ t ∧ t ′ t − s,t ′− s even √ t ! t ′ ! s !( t − s )!( t ′ − s )! ¯ α t − s α t ′− s λ s . (9.2)Thus, in the particle basis measurement, the probability for a t -particle count isequal to ρ mat ( α, λ )( t, t ) = { (1 − λ ) − | α | } / X s ≤ t t − s even t ! s !( t − s !) | α | t − s λ s . (9.3) Example 9.3. [2-mode mean zero pure gaussian states] Let ρ ( A, 0) = | ψ A ih ψ A | bea general -mode mean zero pure gaussian state parametrized by a matrix A = (cid:20) α ββ γ (cid:21) ∈ M ( C ) as in Theorem 6.12. Using (8.2) we will describe the particle basisexpansion of | ψ A i . First step in this direction is the description of the matrix indexset ∆( A, t ) in (7.2). We have ∆( A, t ) = (cid:26) R = (cid:20) r r r (cid:21) | r , r , r ∈ Z + , r + r = t , r + r = t (cid:27) . (9.4)Recall from the definition of ∆( A, t ) that ∆( A, t ) = 0 if | t | is odd. Let t = ( t , t ) ∈ Z be such that t + t is even. Then t and t are both even or both odd There aretwo cases, namely, (i) t = 2 k, t = 2 ℓ , (ii) t = 2 k + 1 , t = 2 ℓ + 1 , where k, ℓ ∈ Z + . Case (i) : The nonnegative integer equations r + r = 2 k and r + r = 2 ℓ in(9.4) imply that r must be even and ≤ r ≤ k ∧ ℓ . If r = 2 r, ≤ r ≤ k ∧ ℓ ,then r = k − r and r = ℓ − r . Case (ii) : A similar argument as in Case (i) shows that, r = 2 r + 1 and ≤ r ≤ k ∧ ℓ .Thus the set ∆( A, t ) has ( k ∧ ℓ ) + 1 matrices in both cases and44 ( A, t ) = (" k − r r ℓ − r | r ∈ Z + , ≤ r ≤ k ∧ ℓ ) , if t = [2 k, ℓ ] T , (" k − r r + 10 ℓ − r | r ∈ Z + , ≤ r ≤ k ∧ ℓ ) , if t = [2 k + 1 , ℓ + 1] T φ, otherwise. (9.5)So by (7.3), ϕ A ( t ) = p (2 k )!(2 ℓ )! k ∧ ℓ P r =0 2 r α k − r β r γ ℓ − r ( k − r )!(2 r )!( ℓ − r )! if t = [2 k, ℓ ] T , p (2 k + 1)!(2 ℓ + 1)! k ∧ ℓ P r =0 2 r +1 α k − r β r +1 γ ℓ − r ( k − r )!(2 r +1)!( ℓ − r )! if t = [2 k + 1 , ℓ + 1] T . , otherwise. (9.6)Now by (8.2) | ψ A i = p c ( A, X k,l ∈ Z + p (2 k )!(2 ℓ )! k ∧ ℓ X r =0 α k − r (2 β ) r γ ℓ − r ( k − r )!(2 r )!( ℓ − r )! × | k, ℓ i + p (2 k + 1)(2 ℓ + 1)2 β r + 1 | k + 1 , ℓ + 1 i ! (9.7)Thus, in the particle basis measurement, the probability of counting k -particles inthe first mode and ℓ -particles in the second mode is Pr(2 k, ℓ ) = c ( A, k )!(2 ℓ )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∧ ℓ X r =0 α k − r (2 β ) r γ ℓ − r ( k − r )!(2 r )!( ℓ − r )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (9.8)and that of counting k + 1 -particles in the first mode and ℓ + 1 -particles in thesecond mode is Pr(2 k + 1 , ℓ + 1) = c ( A, k + 1)!(2 ℓ + 1)! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∧ ℓ X r =0 α k − r (2 β ) r +1 γ ℓ − r ( k − r )!(2 r + 1)!( ℓ − r )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (9.9)Next three examples are special cases of the previous example.45 xample 9.4. Let A = (cid:20) α ββ (cid:21) ∈ M ( C ) with k A k < / . By analysing the set ∆( A, t ) and demanding γ = 0 in (9.6) gives ϕ A ( t ) = p t !( t − k )! α k (2 β ) t − k k !( t − k )! if t = ( t, t − k ) , ≤ k ≤ t , otherwise. (9.10)Thus the number of particles in the second mode cannot exceed that in the first mode .Furthermore, Pr( t, t − k ) = c ( A, (cid:18) tk (cid:19)(cid:18) t − kt − k (cid:19) | α | k (4 | β | ) t − k , ≤ k ≤ t (9.11) Example 9.5. Let B = (cid:20) ββ α (cid:21) ∈ M ( C ) with k B k < / . This case is similar tothe previous example and it may be noticed that the number of particles in the firstmode cannot exceed that in the second mode and ϕ B ( t , t ) = ϕ A ( t , t ) , where A issame as that in Example 9.4 and Pr( t − k, t ) is same as the value on the right sideof (9.11). Example 9.6. Let A = (cid:20) ββ (cid:21) ∈ M ( C ) with k A k < / , i.e., | β | < / . In thiscase, ϕ A ( t ) = ( k β k , if t = t = k, , if t = t . Hence | ψ A i = q − | β | ∞ X k =0 (2 β ) k | k, k i . (9.12)Thus the number of particles in both the modes must be the same . Furthermore, Pr( k, k ) = (1 − | β | ) | β | k , (9.13)which is a geometric distribution with parameter p = | β | . Equation (9.12) givesan example of a -mode, mean zero, pure gaussian state which is invariant underthe permutation of modes and has the mixed one-mode marginal gaussian state (1 − | β | ) ∞ X k =0 (4 | β | ) k | k ih k | . (9.14)This is a well known example of an entangled gaussian state which is called a photonnumber entangled state (PNES) by some authors. Refer [DGCZ00, EP03, SIS11]for more details. We shall meet a heirarchy of such arbitrary mode gaussian statesin the following discussions. Example 9.7. Let n = 3 , and A = α α α α α α k A k < / so that there exists a pure gaussian state ρ ( A, with parameters ( c ( A, , , A, by Theorem 6.12. If | t | = 2 k the elements of ∆( A, t ) have theproperty P i ≤ j r ij = k , we have t j ≤ k, ∀ j and ∆( A, t ) = k − t k − t k − t . For t , such that t + t + t = 2 k , ϕ A ( t ) = p t ! t ! t ! 2 k α k − t α k − t α k − t ( k − t )!( k − t )!( k − t )! Hence | ψ A i = c ∞ X k =0 k X t + t + t =2 k max i t i ≤ k √ t ! t ! t !( k − t )!( k − t )!( k − t )! α k − t α k − t α k − t | t , t , t i , (9.15)where c = p c ( A, is such that k ψ A k = 1 . Example 9.8. [ n -mode mean zero gaussian state with A and Λ as diagonal matri-ces] Let D α = Diag { α , α , . . . , α n } and D λ = Diag { λ , λ , . . . , λ n } . Let A = D α and Λ = D λ , be the parameters of a gaussian state. Furthermore, let Z α j ,λ j bethe bounded extension of the -mode operator c ( α j , λ j )Γ( p λ j ) e α j a j a j defined on span F ∪ E in the sense of (7.6) and (7.10). If t = ( t , t , . . . , t n ) ∈ Z n + , then theoperator Z A, Λ1 in (8.4) satisfies Z A, Λ1 | t i = Z D α ,D λ | t i = ⊗ nj =1 Z α j ,λ j | t j i . Hence ρ A, Λ = ρ D α ,D λ = ⊗ nj =1 ρ α j ,λ j , where ρ α j ,λ j is the -mode gaussian state with parameters ( c ( α j , λ j ) , , α j , λ j ) , as inExample 9.2, j = 1 , , . . . , n .Now we obtain a hierarchy of completely entangled pure gaussian states invariantunder the action of the permutation group S n on the set of all the n modes. Definition 9.9. Let H = H ⊕ H . A state ρ on Γ( H ) = Γ( H ) ⊗ Γ( H ) is saidto be separable if it can be written in the form ρ = X j p j ρ j ⊗ ρ j where p j ≥ , P j p j = 1 , ρ jk is a state on Γ( H k ) , k = 0 , , ∀ j . The state ρ is saidto be entangled if it is not separable. It follows that if ρ is a pure state, then itis separable if and only ρ = ρ ⊗ ρ , where ρ k is a pure state on Γ( H k ) , k = 0 , .Fix an orthonormal basis { e , e , . . . , e n } of H , a state ρ on Γ( H ) is said to be completely entangled , if ρ remains entangled for any decomposition H = H ⊕ H where H = span { e , . . . , e k } , H = H ⊥ , ≤ k < n .47 roposition 9.10. Let H = H ⊕ H . Let ρ ( A, Λ) be the gaussian state with E ( H ) -parameters ( c ( A, Λ) , , A, Λ) . If A = (cid:20) A A A A (cid:21) and Λ = (cid:20) Λ Λ Λ Λ (cid:21) in the directsum decomposition H = H ⊕ H , then the marginal state ρ = Tr ρ ( A, Λ) in thetensor product decomposition Γ( H ) = Γ( H ) ⊗ Γ( H ) has the E ( H ) -parameters ( c , , A , Λ ) where c = c ( A, Λ) c ( A , Λ ) ,A = A + 14 C M ( A , Λ ) − C T , Λ = Λ + 14 C M ( A , Λ ) − C † , (9.16)with C = (cid:2) (Λ + 2 A ) i (Λ − A ) (cid:3) . (9.17) Proof. Notice first that M ( A , Λ ) > as it is a principal submatrix of the positivematrix M ( A, Λ) . The proof follows from a routine computation using (3.23) andthe gaussian integral formula.In the context of Proposition 9.10, it is evident from equations (9.16) and (9.17)that Λ = Λ if and only if C = 0 , and in such a case A = A . But bydefinition, C = 0 if and only if A = 0 and Λ = 0 , i.e., both A and Λ are blockdiagonal in the decomposition H = H ⊕ H . Since M ( A jj , Λ jj ) > , ρ ( A jj , Λ jj ) is agaussian state on Γ( H j ) , j = 0 , . If A and Λ were block diagonal in the first place,then by item 4) in Examples 5.2, ρ ( A, Λ) = ρ ( A , Λ ) ⊗ ρ ( A , Λ ) is a separablestate. Furthermore, the positive value Tr C M ( A , Λ ) − C † may be consideredas a measure of the ‘influence’ of the second party on the first. The discussion abovehas turned out to be a useful test for the entanglement of a bipartite pure gaussianstate. Corollary 9.11. Let H = H ⊕ H . Let ρ be a pure gaussian state with E ( H ) -parameters ( c ( A, , , A, . Then ρ is separable if and only if A is a block diagonalmatrix in the direct sum decomposition H ⊕ H . Moreover we have the following. Proposition 9.12. Let ρ ( A, be a n -mode pure gaussian state. If the singularvalues of A are { α , α , · · · , α n } , then there exists a unitary U on C n such that ρ ( A, 0) = Γ( U ) ⊗ nj =1 ρ ( α j , U ) † , (9.18) where ρ ( α j , 0) = (cid:12)(cid:12) ψ α j (cid:11)(cid:10) ψ α j (cid:12)(cid:12) is a -mode gaussian state with parameters ( α j , as inExample 9.1, j = 1 , , . . . , n .Proof. By Autonne’s theorem in linear algebra (Corollary 2.6.6 in [HJ12]), the com-plex symmetric matrix A has a singular value decomposition of the form A = U D α U T . Lemma 5.11 and Example 9.8 together completes the proof.48 emark 9.13. In the light of Lemma 5.11, Example 9.8 and Proposition 9.12, apair of matrices ( A, Λ) satisfying M ( A, Λ) > produces a product state upto aconjugation by a second quantization unitary if and only if there exists a singleunitary U such that A = U D α U T and Λ = U D λ U † , where D α and D λ are diagonalmatrices. Theorem 9.14. Let A = [ α ij ] be a complex n × n symmetric matrix satisfying thefollowing conditions:1. α ij = 0 for all i = j ;2. k A k < .Then the associated pure gaussian state | ψ A i (or equivalently ρ ( A, 0) = | ψ A ih ψ A | )with ψ A as in (8.2) is completely entangled.Proof. The matrix of A is not block diagonal in any decomposition H = H ⊕ H asin the definition of completely entangled states. Now Corollary 9.11 completes theproof.The following corollary provides a genrealization of photon number entangledstates (PNES) in Example 9.6. Corollary 9.15. Let θ ∈ C be such that | θ | < n − and A be the matrix with alldiagonal entries equal to and all non diagonal entries equal to θ , i.e., A = θ · · · 11 0 1 · · · ... · · · · · · · · · ... · · · . Then | ψ A i is a completely entangled zero mean pure gaussian state which is invariantunder the action of the permutation group S n on the set of all the modes.Proof. Observe that k A k < . So there exists a gaussian state | ψ A i given by equa-tion (8.2). Furthermore, P AP T = A for any permutation matrix P , and hence Γ( P ) | ψ A ih ψ A | Γ( P ) † = | ψ A ih ψ A | by Lemma 5.11. In other words, | ψ A ih ψ A | is invari-ant under the action of the permutation group S n on the set of all the modes. Remark 9.16. As a special case of Corollary 9.15, we have a completely entangled -mode pure gaussian state which is invariant under the action of S on the modeswhen A = θ , | θ | < . By Example 9.7, we have in this case, | ψ A i = p c ( A, ∞ X k =0 k θ k X t + t + t =2 k max i t i ≤ k √ t ! t ! t !( k − t )!( k − t )!( k − t )! | t , t , t i . In this section, we make some remarks on the tomography of an unknown gaussianstate in Γ( C n ) through the estimation of its E ( H ) -parameters ( c, α , A, Λ) by usingfinite set valued-measurements. For tomography based on the estimation of the meanannihilation and position-momentum covariance matrix parameters with countableset-valued measurements, we refer to [PS15].Let α = ( α , α , . . . , α n ) , A = [ a jk ] , Λ = [ λ jk ] . Recall the notations defined in(2.13), we have the following relations from (5.5) h Ω | ρ | Ω i = c, h χ jj | ρ | Ω i = √ c ( α j a jj ) , h χ j | ρ | Ω i = cα j , h χ jk | ρ | Ω i = 2 c ( α j α k a jk ) , h χ j | ρ | χ k i = c ( α j ¯ α k + λ jk ) , (10.1)where ≤ j, k ≤ n and the equations involving χ jk are valid only for j = k . Ouraim is to estimate the E ( H ) -parameters by making measurements in the state ρ .To estimate c , consider the projection P = | Ω ih Ω | and the yes-no measurement M = {P , I − P } . Measurement of M in the state ρ yields a classical random variable X on the twopoint set M with values in { , } and Pr( X = 1) = Tr ρ P = c. Hence by the law of large numbers, estimates of c can be obtained by makingmeasurements M ⊗ k in ρ ⊗ k , k ∈ N . The parameters α , A and Λ are functions of thescalars h u | ρ | v i , where u, v vary over the set B = {| Ω i , | χ j i , | χ jk i | ≤ j ≤ k ≤ n } . Towards estimating these parameters, recall the polarisation formula h u | ρ | v i = h u + v √ | ρ | u + v √ i − i h u + iv √ | ρ | u + iv √ i − − i h u | ρ | u i + h v | ρ | v i ) . (10.2)For each j, ≤ j ≤ n , let P j = | χ j ih χ j | , P j = | ψ j ih ψ j | , P ′ j = (cid:12)(cid:12) ψ ′ j (cid:11)(cid:10) ψ ′ j (cid:12)(cid:12) , where | ψ j i = | χ j i + | Ω i√ and (cid:12)(cid:12) ψ ′ j (cid:11) = | χ j i + i | Ω i√ . Consider the yes-no measurements M j = {P j , I − P j } , M j = {P j , I − P j } , M ′ j = {P ′ j , I − P ′ j } , (10.3)To estimate α j , take u = χ j and v = Ω in (10.2). Each term on the right hand sideof (10 . can be estimated using the procedure described to estimate c but using themeasurements M j , M ′ j , M j and M respectively. Thus α can be estimated usingthe measurements in (10.3).For each j, k, ≤ j ≤ k ≤ n , let P jk = | ψ jk ih ψ jk | , P ′ jk = (cid:12)(cid:12) ψ ′ jk (cid:11)(cid:10) ψ ′ jk (cid:12)(cid:12) , where | ψ jk i = | χ jk i + | Ω i√ and (cid:12)(cid:12) ψ ′ jk (cid:11) = | χ jk i + i | Ω i√ . Now consider the measurements M jk = {P jk , I − P jk } , M ′ jk = {P ′ jk , I − P ′ jk } . (10.4)50ssume that we have already estimated c and α . Now by (10.1), to estimate A and Λ , it is enough to estimate the scalars h χ jk | ρ | Ω i and h χ j | ρ | χ k i , ≤ j ≤ k ≤ n . Tothis end, let P denote the projection onto the subspace spanned by B and M bethe von-Neumann measurement M = {| ζ ih ζ | | ζ ∈ B } ∪ { I − P} , which contains N := ( n +1)( n +2)2 + 1 mutually orthogonal projections. Now we labelthe elements of M using numbers from to N − , as follows | Ω ih Ω | 7→ , | χ j ih χ j | 7→ j, ≤ r ≤ n, | χ jk ih χ jk | 7→ n + (2 n − j )( j − k, ≤ j ≤ k ≤ n,I − P 7→ N − . (10.5)Thus we get a classical random variable X on the N point sample space M , takingvalues , , , . . . , N − with respective probabilities Pr( X = 0) = h Ω | ρ | Ω i , Pr( X = r ) = h χ r | ρ | χ r i , ≤ r ≤ n, Pr( X = n + (2 n − j )( j − k ) = h χ jk | ρ | χ jk i , ≤ j ≤ k ≤ n, Pr( X = N − 1) = 1 − N − X t =0 Pr( X = t ) . (10.6)Hence the scalars h χ j | ρ | Ω i and h χ j | ρ | χ k i can be approximated by using the polari-sation formula (10.2) and making the measurements M ⊗ k in ρ ⊗ k , k ∈ N .Denoting by Q the projection P + P + P where P j is the projection on the j -th particle subspace and tomographing the finite dimensional density operator (Tr ρQ ) − QρQ , it is possible to reduce the number of measurements. We leave thisproblem open for the present. Conclusions 1. A Klauder-Bargmann integral representation of all gaussian symmetries in an n -mode boson Fock space is obtained.2. The notion of generating function of a bounded operator in the boson Fockspace Γ( C n ) over the n -dimensional Hilbert space C n is introduced and a † -closed multiplicative semigroup E ( H ) with H = C n is constructed. Thesemigroup E ( H ) is closed under the weak operator topology and contains allthe gaussian states and their symmetries in Γ( C n ) .3. Using the properties of the semigroup E ( H ) , the set of all n -mode gaussianstates is parametrized by a set of scalars derived from the matrix entries ofthe gaussian state at , , and -particle vectors of a particle basis in Γ( C n ) .The exact relations between these new parameters and the conventional set ofmeans and covariances of position and momentum observables are obtained.51. A positive element Z in the semigroup E ( H ) is factorised as Z † Z , where Z = √ c Γ( √ Λ) exp nP nr =1 λ r a r + P nr,s =1 α rs a r a s o on the dense linear mani-fold generated by all exponential vectors.5. An explicit particle basis expansion of an arbitrary mean zero pure gaussianstate vector along with a density matrix formula for a general mean zerogaussian state is obtained in terms of the E ( H ) -parameters.6. A whole class of completely entangled n -mode pure gaussian states is con-structed. This yields examples of such entangled states which are also in-variant under the action of the permutation group S n on the set of all the n modes.7. The new parametrization enables the tomography of an unknown n -mode gaus-sian state by O ( n ) measurements with a finite number of outcomes. 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