Quantum-noise matrix for multimode systems : U ( n ) invariance , squeezing , and normal forms

We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,R) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for n-mode systems, we distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U ( n ) subgroup of Sp(2n,R), the latter to noncompact elements outside U(n) . Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U( n ). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.


I. INTRODUCTION
Squeezed states of the radiation field are distinctly nonclassical in nature [l].Over the past decade their study has developed into a major area of quantum optics [2].
Their experimental realization by several groups  has certainly contributed to the enormous interest and activity in this area.In addition to quadrature squeezing [6lo], attention has also been focused on amplitude [ l l ] as well as higher-order [12] squeezing.While many of these studies have centered around states of singleand two-mode systems [9,10] of immediate relevance to current experimental activity, there nevertheless has been interest in the multimode case as well [13,14].
All the information regarding the quadrature squeezing properties of any state of a multimode (quantum) system is contained in the noise or variance matrix of that state.In the single-mode case this is a 2 x 2 real symmetric positive-definite matrix V: the diagonal entries are the expectation values ( (@ -( @ ) )2 ) and ( (9 -<a ) )* ) while the off-diagonal one is ( +@ ) ) -(Q ) (9 ), 'Also at Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560012, India.
where @ and 9 are the quadrature operator com onents of the mode annihilation operator a=(@ +@ ) / A For a classical probability distribution over a classical two-dimensional phase space with variables q and p , any real symmetric positive-definite matrix is a valid, that is, physically realizable, variance matrix.In the quantum case, however, the variance matrix has to satisfy the additional condition det V 2 $.This is a precise and complete statement of Heisenberg's uncertainty principle for one pair of operator canonical variables.
The group of real linear canonical transformations Sp( 2, R ) = S1( 2, R 1 = SU( 1 , l ) plays a basic role [ 1,2] in the study of squeezing in a single-mode system.(This is so whether or not one wishes to make explicit use of the language and machinery of Lie groups.)This group has as its maximal compact subgroup a one-parameter group U( 1) corresponding to phase-space rotations; its generator is the harmonic-oscillator Hamiltonian (afa-tf2ilt)/2.An important aspect of the definition of squeezing in the single-mode case is that it has a basic built-in invariance under this U(1) subgroup, which is physically reasonable and justifiable.Our interest in this paper is in n-mode systems.We present a comprehensive analysis of the properties of variance matrices for states of such systems.Such matrices are real, symmetric, 2n dimensional, and positive definite.In addition, they obey certain specifically quantum-mechanical inequalities in the Heisenberg sense.We make effective use of elementary concepts and results related to the group Sp( 2n, R ) of real linear canonical transformations in 2n-dimensional phase space, which is naturally available, to analyze both the Heisenberg inequalities and the notion of squeezing for n-mode states.While, as noted above, Sp(2,R) is isomorphic to the pseudounitary group SU ( 1,l) in the single-mode case, Sp( 2n, R ) is not isomorphic to the pseudounitary group SU( n , n ) but rather to a proper subgroup of it when n ? 2.
The material of this paper is organized as follows.In Sec.I1 we address the following basic question: Given a 2n X 2n real symmetric positive-definite matrix, how do we test whether it qualifies to be the variance matrix of some physically realizable state of the (quantum) n-mode system?Clearly this is the same as asking for a complete statement of the Heisenberg uncertainty relations for such systems.We solve this problem by making use of a classic theorem due to Williamson [15] on the normal forms of real symmetric matrices under symmetric symplectic transformations.The nontrivial aspect of this theorem hinges on the facts that only some phase-space rotations are canonical transformations, and that a symmetric symplectic transformation in general is not a similarity transformation.It thus turns out that the normal form is diagonal only for some special subsets of symmetric matrices, and there are several distinct normal forms.What is relevant for our problem is the fact that for symmetric positive-definite matrices the Williamson normal form is a diagonal one, and variance matrices are positive definite.
As can be seen from the work of Caves and Schumaker [6], it is important to be able to describe the variance matrix and squeezing in terms of the real (Hermitian) quadrature components qj,aj as well as in terms of the complex (non-Hermitian) operators aj,aj , and to switch easily between these two descriptions.While a given canonical transformation is specified in terms of a real matrix in the former description, it is specified by a complex matrix in the latter.These two matrices are related through a similarity transformation by a fixed numerical matrix.Similarly the real variance matrix becomes a complex Hermitian one when transcribed to the a, at description.We formulate our complete characterization of the variance matrix in Sec.I1 in the form of necessary and sufficient conditions, and these are expressed in both descriptions.Transformation formulas for passing easily between them are presented.Our notations are somewhat different from those of Caves and Schumaker [6] for the single-mode and two-mode cases, but we believe they are more convenient, typographically and otherwise.
The role of U(1) in the single-mode squeezing group Sp(2,R) is played by the U ( n ) -K ( n ) subgroup of Sp(2n,R) in the n-mode case.This subgroup consists of all those phase-space rotations which are also canonical transformations.It should be appreciated that for n 1 2 most phase-space rotations are not canonical.Motivated by the U( lbinvariant squeezing criterion in the singlemode case and by the familiar division of quantum optical systems into passive and active types, we formulate, in t Sec.111, a U(n )-invariant squeezing criterion for n-mode systems and explain why it is reasonable.Since the K ( n ) subgroup of Sp(2n,R) is too small to diagonalize a general variance matrix, it would appear at first sight that our squeezing criterion is not expressible in terms of the eigenvalue spectrum of the variance matrix.However, the identity of the two coset spaces S0(2n)/S0(2n -1 ) = U ( n ) / U ( n -1)=S2"-', where is the unit sphere in the 2n-dimensional Euclidean S 2 n -1 space R2", enables us to establish that a state is squeezed if and only if the smallest eigenvalue of the corresponding variance matrix is less than +.
We take up in Sec.IV the question of K(n) canonical forms for variance matrices.Though it is true that a generic variance matrix cannot be diagonalized by K ( n ) transformations, there are two special families 8, and S , whose K(n) canonical form is diagonal.The family 8, consists of variance matrices which, apart from a factor of f, are also elements of Sp(2n,R).Members of the family S , are built up from n X n Hermitian matrices H obeying a positivity and spectrum condition.Both families 8, and 8, are contained in the larger family 8, of all those variance matrices whose K( n 1 canonical form is diagonal.We state and prove a simple matrix algebraic necessary and sufficient condition characterizing the elements of 8,.Finally, we also develop canonical (nondiagonal) forms for variance matrices which are not contained in 8, In Sec.V we construct examples of n-mode states and their variance matrices to render transparent the physical meanings of the K ( n ) canonical forms and the families S,, S,, and 8,.It turns out that any acceptable variance matrix can be realized through a suitable squeezed thermal state (zero-mean Gaussian state).Finally, in Sec.VI we present some concluding remarks.
At several places throughout the paper we make use of properties of the symplectic groups Sp(2n,R).We take care to state and explain them clearly at first encounter.(2.12) Consider now a real linear transformation on the variables Qj,aj specified by a 2n X 2n real matrix S ( r ) : The condition that this be canonical means that pry must obey the same commutation relations (2.9a) as do pr'.
This places the following condition on the matrix S ( r ) :

Expressions for S ( r ) ( h ( r ) )
in terms of h ( ' ) can be found in Ref. [14].The converse is also true: Given any S"'€Sp(2n,JR), there exists a unitary evolution of the above type, fixed up to a phase which can be narrowed down to a sign ambiguity, which transforms by S ( r ) .The relevance to squeezing problems is now clear, since squeeze operators belon to this class of evolutions.
It is clear that w h e n j y undergoes the linear canonical transformation (2.121, f behaves as follows: s(c)= as(r)aZt , (2.15)This is determined by the relationships (2.6) and (2.8).It is important to realize that S('), though complex, represents the same real linear canonical transformation S'" that appears in Eq. (2.12): it is a complex representation (in the complex aj,aj basis) for the real transformation.

t A. Real form for the noise matrix
Squeezing deals with second-order noise moments.We want to be able to deal collectively with the set of all second-order moments for any state of a multimode system.Earlier studies of such systems have largely concentrated on the squeezed coherent states, or two-photon coherent states (TCS).We shall, however, build up a formalism capable of studying noise and squeezing in an arbitrary (pure or mixed) state specified by a density oRerator p? with the-expectati:n value of any observable 0 being given by ( 0 ) = T r ( P 0 ).
Let us assume without loss of generality (see below) that the state 8 has zero-mean values for the basic variables: ( g r ) ) = ( p c ) ) =O.Consider the operator matrix pr)pr)T.A general element can be written in terms of the anticommutator and commutator of its factors thus: (2.16)We now define the 2n X2n real variance (noise) matrix V'r) for the state @by 1 Y r ) Y r ) i =& 9 6 v 1 +ppv ' ( p r ) p r ) T ) =Tr(ppr)pr)T) (2.17)We can decompose V ( r ) usefully into n Xn blocks in this way: (2.18) ( V , ) j k = ( P j P k ) , j , k = 1 , 2 , . . ., n .
Thus V , gives the noise and correlations among the ljSf variables, V , among the 3 variables, and V 2 comprises the correlations between Tf's and ps.The submatrices V , and V , are individually symmetric, and so is V ( r ) as a whole.
The restriction that the state @ have zero means is easily removed.For, if @ is such that ( p r ' > # O , we simply replace pr) by A F r ) = p r ) -( p " ) in Eq. (2.17) in defining the variance matrix V'r).This corresponds to a rigid translation of the state in the (quantum) phase space by an amount -( Y r ) ), implemented by the displacement operator 8( -( 6) ) familiar from the context of multimode coherent states.Also, such a rigid translation does not affect variances.
As we have mentioned in the Introduction, if we had a classical probability distribution over a classical 2ndimensional phase space, the only restriction on a 2n X 2n variance matrix, for it to be realizable, is that it be positive definite (apart from being real and symmetric).In the present quantum case, however, V'" has to satisfy additional uncertainty inequalities.We wish to now derive them, keeping in evidence always their Sp(2n,R) invariance.
For a single-mode system, the 2 X 2 variance matrix is , & ( 2.19) the means being assumed to vanish.The naive statement of the uncertainty principle, ( Q 2 ) ( P 2 ) .a, (2.20) is, as is well known, a necessary consequence of the commutation relation between Q and a.It is, however, not sufficient to characterize the variance matrix completely.
The correct statement for this purpose is (2.21) Any V c r ) obeying this condition is physically realizable as the variance matrix of some state p.Moreover, the inequality (2.21) is explicitly Sp(2,R) invariant (see below).A special canonical case, which may appear trivial, will turn out to be the most important case in the multimode situation, since any multimode variance matrix will be seen to be reducible essentially to this case.This occurs when the 2 x 2 matrix V c r ) in Eq. (2.19) has the diagonal form Clearly, the statement (2.20) is adequate in this case, and the necessary and sufficient condition for this V ( r ) to be acceptable is  --s ( r ) ( I/( ri + 3 j p ) S ( r ) T (2.25) Finally making use of the defining property (2.13) for symplectic matrices, we obtain the important result (2.26)We term Vcr)' the symmetric symplectic transform of V'" under S"'.This then is the connection between the variance matrices of two states p^ and f l S " ' ) related unitarily through the canonical evolution S'".Incidentally we can now appreciate the Sp( 2, R) invariance of the uncertainty principle (2.21): indeed, for any n, every symplectic matrix is known to be unimodular, so from Eq. (2.26) the determinant of V'" is a symplectic invariant.
We now tackle the question: Given a real symmetric positive-definite 2n X2n matrix V('), what are the necessary and sufficient conditions that ensure that it is the variance matrix of some state of the quantum n-mode system?A similar question was raised by Littlejohn [18] in the limited context of Gaussian Wigner distributions, and solved subsequently by some of the present authors [19].The basic principles behind that solution apply to the present problem as well.
From the arguments leading to Eq. ( 2.26) it is clear that if a given V ( r ) is physically realizable, then so is its symplectic transform V(r)' given by (2.261, for every S("ESp(2n,R).Conversely, the invertibility of S ( r ) implies the opposite statement as well.If V ( r ) does not qualify to be a variance matrix, neither can V(r)' for any s'"'E ~p ( 2 n , R).This suggests the following approach to solve the problem: Given V C r ) , we look for a symplectic transform V(')' which has a particularly simple (canonical or normal) form allowing us to test by inspection whether it is physically realizable or not.Then the given V ( r ) is a variance matrix if and only if V(r)' is a bona fide variance matrix.
The existence of such a canonical form is guaranteed by Williamson's theorem [ 151: For any real symmetric positive-definite 2n X 2n matrix V ( r ) , there exists an  what we have here are simply n copies of the twodimensional example (2.22).It follows that V ( r ) , and hence V('), is a bona fide variance matrix if and only if (2.28) It now remains to write these conditions in a more convenient Sp(2n, R ) invariant form, so that in implementing or checking them we may deal directly with V C r ) .
We must note that the transformation (2.26) is in general not a similarity transformation, so the K~ appearing in Eq. (2.27) are in general not the eigenvalues of V'".However, Eq. ( 2.26) combined with Eq. (2.13) implies that the product matrix V(r)fi does undergo a similarity transformation: Therefore also the square of V"'S transforms by (2.30) Now when V ( r ) takes tht canonical diagonal form e(l) of Eq. (2.?7), the product V'"p takes an off-diagonal form, while (V("fi)2 is diagonal: (e ( r ) p ) 2---diag(Kl,K2,.

v ( ~) B v ( ~) ~~.
NOW this last matrix is in general neither symmetric nor manifestly positive definite, and it would clearly be convenient to have such a matrix for which the eigenvalues are the K; for j = 1,2, . . ., n.This is easily achieved.We let ( V(r))1/2 denote the unique real symmetric positivedefinite square root of V ( r ) , and subject Vcr)BVcr'pT to a similarity transformation applying ( on the left and ( Vcr')'" on the right.This will not alter the eigenvalue spectrum at all, so we can say that K:,K:, . . .,K: are the (at least doubly degenerate) eigenvalues of the real symmetric positive-definite matrix ( V(r))1/2BV(r)flT( V(r))1'2.We can now express condition (2.28) in the following invariant manner.Theorem 1.The necessary and sufficient condition for a real symmetric positive-definite 2n X2n matrix V ( r ) to be a bona fide (quantum) variance matrix is that (2.32)In passing we note that the conditions (2.28), namely, K; I + for j = 1,2, . . ., n, can be written in an Sp(2n,R) invariant form in terms of the Sp(2n,R) invariant traces Tr( V(r)fiV(r)fiT)', 1 = 1,2, . . ., n.These traces are evidently the sums of even powers of the K ~, aside from a factor of 2: for 1 = 1, we have the sum of K;; for 1 =2, the sum of K;; and so on.The procedure for expressing the inequalities (2.28) in terms of these traces is the same as that developed earlier for a related problem.The details are given in Ref. [19], so we do not pursue them here.
Let us instead show that our necessary and sufficient conditions (2.28) and ( 2.32) can be cast in yet another instructive form.For this purpose, consider the Hermitian The eigenvalues are easily determined because each Qj is coupled only to its own aj, and not to any Q, or phk for k f j .Thus this matrix has the spectrum of eigenvalues ~~& f for j = 1 , 2 , . . .,n.It follows that our condition (2.28) amounts to demanding that e ( " + ( i / 2 ) p be positive semidefinite.Now the transformation (2.34) connecting the two Hermitian matrices is not a similarity transformation but a real symmetric one by a nonsingular matrix.Thus,, while it may not preserve the spectrum of eigenvalues, it certainly preserves positive semidefiniteness (in fact, it preserves the signature of the eigenvalues and the rank of the concerned matrix).We have thus established the following result as fully equivalent to (2.28) and (2.32): Theorem 2. A real symmetric positive-definite 2n X 2n matrix V ( r ) is a bona fide (quantum) variance matrix if and only if the Hermitian matrix V c r ) + ( i /2 )B is positive semidefinite: We may remark that the necessity of this condition (2.35) is actually quite obvious when we look at the structure of Eq. (2.17) defining V ( r ) , and also remember that each entry in pr) is Hermitian.What is quite nontrivial is therefore the sufficiency of this condition.

B. Complex form for the noise matrix
To conclude this section we derive some results concerning the complex representation V c c ) of the real variance matrix V"'.Analogous to Eq. ( 2.17), and in view of (2.91, we define V c c ' by (2.36) -[ Written out in detail in terms of n X n blocks this reads: (2.37) B,, =Bk, = ( a , a k ) .
t (Remember again that the means of a, and a, are assumed, without loss of generality, to be zero.)Thus A is Hermitian, so A * = A T; and B is symmetric, so B * = B '.
These imply the hermiticity of V c c ) .Conversely, ( V"')'= V'" implies A = A , and B T= B .
We relate V") to V"' easily by using Eq.(2.6): V'C'=RV"'R' , V"'=flntV'C'~ . (2.38) Here we have also used the connections (2.39) Passing between the real and complex forms of the variance matrix via Eq.( 2.38) is consistent with the fact that reality of V c r ' implies the special form (2.37) for V").
Further, the real symmetric positive-definite nature of V'" implies that Vcc' is complex Hermitian positive definite.Written out in terms of the blocks of Vcr' and V"', we have
2 Thus, going back to Eqs. (2.32) and (2.351, we arrive at a complete characterization of V"): Theorem 3. The necessary and sufficient conditions for a 2n X2n complex matrix V c c ) to be a bona fide (quantum) variance matrix (in the complex representation) is that it be Hermitian positive definite with the special form (2.37) and in addition satisfy (one ofl the following equivalent conditions: (2.47)These are a complete statement of the uncertainty principles in complex representation; and as before with V ( r ) , the necessity of the second inequality above is obvious from Eq. (2.36).

U( II ) INVARIANCE AND THE n-MODE SQUEEZING CRITERION
Having completely characterized the variance matrices of n-mode systems from the point of view of the quantum-mechanical uncertainty principles, it is now natural to pose the following question.Given an acceptable variance matrix, how shall we decide whether it describes a squeezed state or not?The aim of this section is to motivate and develop an answer to this question.
It is clear that we need a squeezing criterion possessing certain desirable properties.Given V ( r ) , suppose one of its diagonal elements is already less than +.For guidance let us turn again to a single-mode system.To be specific, consider the variance matrix for some real 7 > 0. Both diagonal elements exceed +, so there is no manifest evidence of squeezing.Yet we know that squeezing is buried in this variance matrix, since the elements here are the variances of the squeezed coherent state where la ) is any coherent state.
To understand the situation in a form that will help us to generalize to the n-mode case, we note that Hermitian quadratic Hamiltonians in the single-mode case are of two types.The first type, labeled by one real parameter 8, is &e)=-ca e t a+m+).Now in this single-mode case squeezing is defined modulo passive transformations of the type (3.4).That is, one considers not only sf and $, but also the-rotated versions@, and$, for all 8, where (@e+@e)/V'2=ae-ie.A state is deemed to be squeezed if, for some 8, either @, or a, is squeezed.Actually it is sufficient to consider just Be since a,=@,+,/,.We regard @e and a, for every 8 in the full range -r I 8 5 7~ of SO(2) to be just as good quadrature components as @ and $ since they are related by passive transformations.
In terms of V ( r ) this means that we look at all the matrices &z) = +(za+2+z *a2) .8) =s(')( 8) v(Wr)( 8 F , (3.6) obtained from V ( r ) by Eq. ( 2.261, with S'" (8) going over the (maximal) compact subgroup SO(2) = U(1) of Sp(2,R).We see whether for some 8=8,, any one of the diagonal elements of V"'(8) falls below 4 and so shows manifest squeezing.(In fact it is enough to fix one's attention on a specific diagonal element as 8 is varied.)If the answer is in the affirmative, then the state giving rise to the original V") is deemed to be squeezed, the squeezed component being @eo oras, as the case may be.
To re-emphasize the point: the definition of squeezing in the single-mode case is set up so as to be U(1) invariant.The variance matrices in (3.1) and (3.7) are U(1) related, and one views the state with variances (3.1) to be as much squeezed as the state leading to (3.7), notwithstanding the fact that the latter alone shows manifest squeezing.
These considerations generalize to the n-mode case governed by the n (2n -t 1 )-dimensional group Sp( 2n, W ).  Conversely, every element of K( n ), the maximal compact subgroup of Sp(2n,R), can be written in the form S"'(X, Y) with X and Y obeying conditions (3.12).
The conditions (3.12) transcribed in terms of U read u u + = 1 . (3.16) Thus we have established the isomorphism between K( n ) and U(n): (3.17 . ., o ) ~, x = ( x ] , x z , . . ., X Z n I T , x T x = l . (3.18) Here e , is the unit vector in the pth direction ( 9 , or p F P n as the case may be), its column having unity at the pth position and zero elsewhere.The set of all unit vectors x constitutes the unit sphere S Z n -' .Now it is well known that SO(2n) acts transitively on S Z n -' .For later application we now ask whether the much smaller group K ( n ) also acts transitively on S Z n -' .To answer this question we shall compute the orbit of some unit vector, say, e,, under K ( n ) , and see whether it exhausts all of .
[The orbit of e , under K( n 1 is the set of all vectors x ES2"-' which are obtained from e , by action with elements of K( n ); the group automatically acts transitively thereon.]From Eq. ( 3.11) it is clear that if a canonical rotation S"'(X, Y ) leaves a particular qj unchanged, it necessarily also leaves the conjugate Pj unchanged, because the former property means that in the jth row of S'"(X,Y) we have unity at the jth column and zero elsewhere.The specific structure of S"'(X,Y) then means that in the ( j + n )th row also we have unity at the ( j f n )th column and zero elsewhere.Thus the stability group of e , is just K ( n -1 ), the group of canonical rotations in the (remaining) 2n -2 dimensions involving q k , j j k for k f j .This means that the orbit of e , under K ( n ) is the coset space U(n )/U(n -11, but this is known to be S 2 n -1 : We have thus proved that not only SO(2n) but also (the much smaller subgroup) K( n 1 acts transitively on the unit sphere S2n-'.The reasons in the two cases are different: in the first case it is because the coset space S0(2n)/S0(2n -1 )=S2n-1, in the second because the coset space ~( n ) / ~( n -1 )=s2"-'.
We now have all the necessary tools to handle the nmode squeezing criterion.Generalizing from the singlemode case, we have seen that the subgroup of Sp(2n,R) corresponding to passive systems is K( n 1.Therefore, as far as squeezin is concerned, we must treat every comq y e n t of cqi ) 4 r ) and also every component of 6 z S ( ~) ( X , Y)( for equally good quadrature components.Therefore a state is to be deemed as squeezed if the fluctuation of some component of pr)', for some S'"(X, Y ) , is less than + in the concerned state.Now the fluctuation in &?)' is simply the pth diagonal element in the transformed variance matrix V(r)'=S(r)(X, Y) V(r)S(r)(X, Y)? Thus our explicitly K( n binvariant n-mode squeezing criterion reads: V ( r ) is squeezedmin [S'"(X, Y)V'"S"'(X, Y)T],, < f , ,2n)

S(')(X,Y)EK(n)
is just the same as the statement

S'"CX, Y ) E K ( n ) min [ X ~V " ) X ] < + .
x € S * n -' (3.20) (3.21) It is here in the last step that we made use of the result embodied in Eq. (3.191, namely, that K ( n ) acts transitively on S Z n -' .Finally, we see immediately that this last form of our squeezing criterion can be expressed in terms of the eigenvalue spectrum of V ( r ) , namely, we have the following theorem.We stress that our squeezing criterion has been set up based on the reasonable premise that all quadrature components related by elements of the U(n) subgroup of Sp(2n,R) should be treated on equal footing, since these compact elements conserve total photon number, and hence correspond to passive optical systems, while the rest of Sp(Zn,R) corresponds to active systems.As a result we have a U( n )-invariant squeezing criterion.Further, we have been aware all along of the fact that we do not have at our disposal the entire rotation group S0(2n), with whose help any V c r ) could have been diagonalized, but only the subgroup K( n ) of canonical rotations.Hence, as one would have rightly suspected, diagonalization of V ( r ) just using elements of K ( n ) is in general not possible.It is therefore our result on the transitive action of K( n ) on S2" -that has nevertheless allowed us to express our squeezing criterion in terms of the smallest eigenvalue of Pr).(3.22)

IV. CANONICAL FORMS FOR VARIANCE MATRICES
Now we examine the question of canonical forms for variance matrices in the context of our squeezing criterion.To begin, we note two obvious simple forms which, however, for reasons to be made clear, are not suitable for the present purpose.

Since a variance matrix y C r )
is always real symmetric, it can definitely be transformed to a diagonal matrix by conjugation (similarity transformation) with a suitable SO(2n) rotation.The resulting diagonal elements will be the eigenvalues of V").However, as already noted, general elements of SO( 2n) are not canonical transformations, so the diagonal form achieved in this way is not relevant to the squeezing problem.Indeed, an SO(2n) transform of a variance matrix may well fail to be a bona fide variance matrix.We therefore do not consider this SO( 2n ) normal form any further.
There is yet another normal form which is provided by the Williamson theorem referred to in Sec.11.This form obtains because in addition to being real symmetric, V"' is also positive definite.Therefore by a suitable element of Sp(2n,R) the symmetric symplectic transform of V ( r ) can be made diagonal.Recognizing that after achieving diagonal form there is the freedom to make reciprocal scale changes in qj and independently for each canonical pair, we can achieve Eq. ( 2.27) and call this the Williamson normal form.In contrast to the diagonal form considered in the preceding paragraph, the present one is achieved through canonical transformations.Even so this is not suitable for our present purpose, for two reasons.First, the diagonal entries in the Williamson normal form are generally not the eigenvalues of the original V'I).Second, passage to this form makes use in principle of the full group Sp(2n,R) inclusive of squeezing transformations, whereas our squeezing criterion is only Against this background we may ask: What is the most natural and simplest canonical form into which a general V'" can be cast, if we use only transformations by elements of the K(n) group of canonical rotations?It is clear at the outset that this form cannot generically be diagonal, because a diagonal matrix has 2n free parameters, while K( n ) is an n '-parameter group.Thus matrices V t r ) that can be brought to diagonal form using K( n ) action can at the most be an n ( n +2)-parameter family.But the totality of all V c r ) constitutes an n ( 2 n + 1 )-parameter family, being restricted only by symmetry; and for n 1 2 this is a much larger family.(The positive definiteness conditions and the uncertainty principles are all inequalities and so do not cut down the number of parameters).
In this way we are led also to ask the supplementary question: Is it possible to characterize in a concise manner the family of variance matrices which can be brought to diagonal form using K( n transformations alone?It turns out that this can be done, and in a rather elegant way.We shall denote this family of variance matrices by s, , reserving for the set of all variance matrices the symbol 8.But before giving a characterization of S , , we first discuss two subfamilies of S,, to be denoted That 8, indeed lies within 8, is a consequence of the property that every matrix of the type S ( u , v ) can be diagonalized through conjugation by a suitable element of K ( n ) :
[For ease in writing we have denoted by R rather than by

S'"(X, Y ) the necessary element of K(n)
here.]Clearly, the diagonal entries in A are the eigenvalues of S ( u , u ) , and we see that they occur in reciprocal pairs.It now follows that the K( n ) canonical form of V"' E SG is diagonal; for any such V"', there is a suitable R E K ( n ) such It is now again evident that the elements of 8, do satisfy, in fact they saturate, the uncertainty principles (2.32), and thus are indeed bona fide variance matrices.We also see that except for the isolated case V " ' = ~l ESG, every other V"'E 8, is squeezed by our criterion.
In passing we may note that this n ( n + 1 )-parameter family can be trivially extended to an ( n ( n + 1 1 -t-1 1parameter family of variance matrices of the form (T V ' r ) , with V"'ES, and CT real positive.The uncertainty principle (2.32) imposes the condition 1 I CT < 0 0 , and then (T V"' is physically realizable.

B. The family JJH
In this second special family each element is fully determined by an n X n Hermitian positive-definite matrix H subject to further conditions to be derived below: Ic -Dl (4.5)The real matrices C and D are, respectively, symmetric and antisymmetric.The complex form of this variance matrix is given by Eq.

V ( c ' ( H ) -+ S ( c ' ( U)V"'(H)S(')( UIt
It is this K ( n ) transformation law, together with the fact that any Hermitian matrix can be diagonalized by a unitary transformation, that has motivated the definition of this family 8,.Thus the K ( n ) canonical forms of V"'(H) and y"'(H) are obtained by choosing U E U ( n ) so that UHU is diagonal, and so are themselves diagonal:

U = X -i Y E U ( n ) , (4.8)
We see that the eigenvalues K ~, j = 1,2, . . ., n, of H are also the doubly degenerate eigenvalues of V"'(H) [and of

V " ) ( H ) ] ,
and the uncertainty principle (2.32) imposes the condition trj Z +, j = 1, . . ., n; that is, it demands H 5 ( + ) 1.The definition of 8, can now be given in full detail as (4.9) It is evident that by our criterion no element of 8, is squeezed.
The normal form (4.8) of an element in 8, has n independent parameters.The collection 8, is the union of the orbits under K ( n ) of all such variance matrices in normal form.However, 8, is an n2-parameter family, rather than an n ( n + 1 )-parameter family, of matrices.This happens because a variance matrix in the diagonal form (4.8) generically possesses invariance under the subgroup SO( 2)XSO(2)X * * * X S 0 ( 2 ) C K ( n ) , the factors being independent phase-space rotations in the qj and pi planes for j = 1,2, . . ., n.This is precisely how n parameters are lost.
It is easy to examine to what extent the two special families 8, and 8, intersect.We have seen above that every V " ' E 8 , is not squeezed, while every V ' " E 8 , oth- We can see from Eqs. (2.18), (2.371, and (2.40) that a variance matrix V'r) being diagonal corresponds exactly to the submatrices A and B entering the complex form V") being simultaneously real diagonal.It turns out that a complete characterization of elements V'"E 8, is most concisely stated in terms of A and B entering V").But we first state a property of complex symmetric matrices which is needed for this purpose.Lemma I .If M is a complex symmetric matrix of dimension n, M ' = M , it possesses an Euler decomposition of the form M = U T M , U , (4.12) where M , is real diagonal positive-semidefinite and The proof of this lemma is a straightforward analysis of the process of diagonalizing the Hermitian matrices M M t = M M * and M t M = M * M .Note that while M , in (4.12) is unique up to the sequence of its diagonal ele- First we prove the necessity of this condition, and next the sufficiency (which is where the lemma comes in).Under the action by U E K ( n ) = U ( n ) , we know from Eq. ( 2.44) and ( 3

.15) that A -U A U ' , B + U B U T , Thus the rule for AB is A B + U A B U T .
Further, when V ( r ) is diagonal, we have noted above that A and B are simultaneously real diagonal, so A B is real diagonal, hence symmetric.Hence by the transformation rule just given for AB, the necessity is proved.
The sufficiency involves a moderate amount of effort.We begin with Hermitian A and symmetric B obeying   where the sign at each entry may be chosen independently.Then we see that  Under the action of U , , A ' remains unaltered and diagonal, while B' has been carried to B" which is diagonal with real non-negative entries.Hence S"'(X, Y ) corre-

A ~+ A ~~= U , A ~U
and the proof is complete.
One can analyze the extent to which each U p ) used in Eq. (4.19) is nonunique by counting the degeneracies in the diagonal elements b (la), b Y), . . ., b;:' when B '"I is carried to diagonal form.This will then disclose the stability group of V('), but we forego the details.It is evident that the generic situation in SK is the nondegenerate one described by Eq. ( 4.171, and then the stability group of the K ( n ) canonical diagonal form of V ( r ) is discrete (we may assume for definiteness that the nondegenerate eigenvalues of A have been ordered, say, as an increasing sequence a , < a , < . . .<a, ).Since, again generically, we have 2n independent eigenvalues for V c r ) appearing in Eq. ( 4.171, and n 2 parameters in K(n), we see that 8, is an n ( n + 2 )-parameter family.
For V"'ES, and V'"ES,, respectively, we have expressed the diagonal forms of V(r) as in Eqs.(4.4) and

D. The family 8
Finally, we turn our attention to 8, the full set of all physically realizable n-mode variance matrices.We want to find K ( n ) canonical forms for general V ( r ) € 8 .We have shown already through consideration of dimensionality that such a form cannot generically be diagonal, for K(n) is too small a group and S too large a family.
We now develop two interesting canonical forms.The number of independent parameters left here can be counted as n in V',' + V y ; n ( n +1)/2 in V y -V y ; and n ( n -l ) / 2 i n V;l.Thisaddsupton(n+l),exactlythe difference between n (2n -t 1 1, the dimension of 8, and n *, the dimension of K(n).Therefore we indeed have in Eq. (4.23) a K( ) canonical form for general V ( r ) € 8.However, none of the diagonal entries in this form is expected to be an eigenvalue of V"), so we turn our attention to another possibility.

Second canonical form
This is constructed by repeated appeal to the fact, proved in Sec. 111,that  Transitive K(n) action on S 2 n -1 means that there exists an element R , E K ( n ) [in fact, an entire K(n -1 ) coset of elements] such that R l x ( l ) = e l , the first unit basis vector defined in Eq. (3.18).So, transformation by R , reduces V(') to this form: . . . . . .Here the leading matrix element is the chosen eigenvalue h,, and all other entries in the first row and first column vanish.Now we iterate,and perform a second canonical rotation R , leaving x ( l ) , the direction of the new q l , unaffected.As noted in Sec.111, it is a property of canonical rotations that R , will necessarily leave the new p1 also unaffected.That is, the elements in Eq. ( 4 That is, in this form the submatrix V , is diagonal, while V , is lower diagonal with zeros along the diagonal.That this is a canonical form can be checked by counting the number of free parameters left: n in the diagonal V , , n ( n -1 ) / 2 in the lower diagonal V,, and n ( n + 1 ) / 2 in the symmetric Vj, adding up to n ( n + 1 ).
While the diagonal elements A;, A;, . . . in Eq. (4.27) are not expected to be eigenvalues of V'", 31, is one of its eigenvalues.For squeezing problems it is natural to choose 31, to be the least eigenvalue of V ' r ) , so that from the canonical form (4.27) one can decide by inspection whether the given V ' r i corresponds to a squeezed state or not.( p ) = ( P " ) =o (5.3)Conversely, given a variance matrix of the canonical form V"'=diag(R + f, R + + ) .
V"'=dlag(K,K) , (5.4) we can associate it with a thermal state of mean photon number H = K - and thermal parameter The uncertainty principle limit K 1 f ensures that p is well defined and non-negative.The fact that (5.4) is the variance matrix of a thermal state means that V""=diag(s2~,s -' K ) (5.6) is the variance matrix of a squeezed thermal state with squeeze factor s and the same thermal factor (5.5) as before.Indeed, since V ( 1)' =sir) y(r)s( 1)' , ] . (5.8) In the single-mode case governed by Sp( 2, R ) = SU( 1,1), the maximal compact subgroup is SO( 2 ) -U( 1 ).Thus all phase-space rotations are canonical, SO( 2 ) n Sp( 2, R ) = 0 ( 2 ) , a situation that does not generalize for n > 1.
The K(1) canonical form for a single-mode variance matrix is always diagonal.The variance matrix (5.6) arose from (5.4) by squeezing along @ or 8.If the squeezing was done along a general phase-space direction, we would get a rotated version Vcr)" of V")' in (5.6): it would be symmetric positive definite but nondiagonal.Even for such a general squeezed thermal state it is true that (detV""')1'2 plays the role of K in fixing the thermal parameter, while the fourth root of the ratio of the eigenvalues of V(r)" would determine the squeeze factor.In this rotated case 52'+af2 in the exponent in Eq. (5.8) is replaced by e21ea2+e -21.at2 , so the operator in the exponent multiplying -p would be the most general Hermitian quadratic expression in 52 and 52 consistent with positive definiteness.We can summarize these considerations for the singlemode case by saying that the allowed variance matrices and squeezed thermal states are in one-to-one correspondence.This connection helps us in the multimode case to which we now turn.We consider first the family 8,.To see our results at a glance, we have collected them in Table I.To conclude this section, we note that the examples we have given are zero-mean Gaussian states, and we have provided illustrative density operators for every conceivable variance matrix.It is the zero-mean and Gaussian conditions that make our examples unique.But the variance matrix V ' r ) by itself cannot specify the state.For instance, the multimode displacement operator changes the moment) of F" and hence the state, without V ' r ) at all.Beyond this, two states with the same means and variances can still differ in their higher moments.This is yet another respect in which the family 8, enjoys a special status.Specification of a V " ' E S G determines the state up to a displacement (first moments).This is so because every V"'ESG saturates the uncertainty principle and hence has to necessarily correspond to some Gaussian (squeezed coherent or squeezed vacuum) state.Gaussian states are fully determined by first and second moments.Thus there are no non-Gaussian states with V"'ES,.

VI. CONCLUDING REMARKS
We have presented in this paper a comprehensive analysis of variance matrices and squeezing properties of arbitrary states, pure or mixed, of n-mode quantum systems.The symplectic group Sp(2n,R) underlying the kinematics and dynamics of such systems has been exploited, and the special role played by the maximal compact subgroup K( n ) in squeezing problems brought out.
The necessary and sufficient conditions on a given 2n X 2 n matrix for it to be physically realizable as the variance matrix of some state have been derived and expressed as simple matrix inequalities.A K( n )-invariant squeezing criterion has been motivated, and its surprising connection with the eigenvalues of the variance matrix made clear.The K ( n ) canonical forms for general variance matrices have been worked out, and the subfamily of those matrices diagonalizable within K( n ) has been characterized in a concise manner.Squeezed thermal states have been given as examples to illustrate multimode variance matrices and their K( n ) canonical forms.
The entire treatment has been given in such a manner as to apply to the real representation in terms of the quadrature components qi,jjj as well as to the complex representation in terms of ai,aj.Formulas for passing between these representations, in a simple form, have been given.It may be of interest to note that this passage is strictly analogous to that between the linear and the circular polarization bases in polarization optics.
We have shown that a multimode state is squeezed if and only if the least eigenvalue 1 ( V"') of its variance matrix V") is less than the coherent state or vacuum fluctuation limit of f .When this happens we can define the squeeze factor as We have shown that for any given variance matrix we can construct a unique zero-mean Gaussian (squeezed thermal) state which reproduces it.Since such states have Gaussian Wigner distributions, it follows that with every variance matrix there is associated a unique (modulo rigid phase-space displacements) Gaussian Wigner distribution.Squeezed thermal states correspond to distributions centered at the origin, and displaced states to displaced distributions.It should be appreciated that these are one-to-one onto correspondences.
In this paper we have only considered quadrature squeezing.It is of interest to extend this analysis to higher orders.The next leading one involves the fourth moments.It may at first appear that analysis of these (and higher) moments would be considerably more complex than that of the variance matrix.We believe, however, that judicious exploitation of the Sp(2n, R ) structure underlying the problem in the spirit of the present work, and due appreciation of the special role played by the maximal compact subgroup K( n ), may render the problem tractable.We plan to return to this elsewhere.
well-known defining property for the elements of the real symplectic group Sp(2n,R): the set of all matrices s'" obeying Eq.(2.13)  gives us in fact the defining representation of this group[ 16,171.Thus real canonical linear transformations in 2n dimensions and Sp( 2n,R matrices are in one-to-one correspondence.The importance of such transformations for squeezing problems arises from the well-known fact that unitary evolutions generated by Hermitian Hamiltonians which are quadratic in Qj a. (equivalently, in aj,aj ) produce these very transformations on the canonical variables: gi c ) ~ g c)' = s c c , p c ) S"'ESp(2n,IW) .
any such K , we can exhibit a thermal state with variance matrix (2.22) (see Sec. VI).Now we turn to n-mode systems.Given a state 8 with variance matrix V"), consider the state f l S ( ' ) ) = o ( S ' r ) ) @ ( S ' r ' ) t , where o ( S ' r ' ) is a unitary operator implementing the symplectic transfor,mation S'"ESp(2n,W) in the sense of Eq. (2.14).Let V ( r ) be the variance matrix of this latter state: T r [ f l S ( r l ) p r ) p r ) the cyclic invariance of traces, and Eqs.
the matrices on the left-hand sides of Eqs.(2.32) and (2.35) can be expressed thus in terms of Vcc': 4( y(r) ) 1 / 2 p l / ( r ) p T ( v(r) )1/2 = 4 0 + ( ~( c ) ) I /2= ~C C ) 3 space rotations: S'"( 8) E SO( 2) C Sp( 2, R 1, which is the same as changing a by the U( 1) phase e -' ,.The second type of Hamiltonians consists of a pair of generators labeled by a complex number z:(3.5)Such Hamiltonians give rise to scaling (squeezing) transformations in phase space.The three independent Hermitian generators contained in Eqs.(3.3)  and (3.5) together build up, on exponentiation, the well-known squeezing group Sp( 2, R ) = SL( 2, R ) = SU( 1 , l ) for the single-mode case.[More precisely, they lead to the metaplectic group Mp(2)I.In technical terms the two types of generators described above are, respectively, compact and noncompact ones.In physical terms, and more importantly for us, the first (compact) type(3.3)conserves photon number and hence correspond to passive systems.Examples of such systems are free evolution, and action by lossless beam splitters in homodyne detection.The second (noncompact) type of Hamiltonians(3.5),does not conserve photon number, and hence correspond to active systems.
Hamiltonians which are compact generators are combinations of the following n 2 independent ones: t t $(ajaj+ajaj), j = 1 , 2 ) . . ., n ; (3.8)All of these n ' operators conserve the total photon number since they commute with the total number operator 8: (3.9) so they correspond to passive systems.The subgroup of Sp( 2n, W) generated by these compact-type Hamiltonians is the n 2-dimensional unitary group U( n 1, as can be seen from the action on pc).This is the maximal compact subgroup of Sp(2n,R).Stated in another way, these generators produce the (maximal) rotation subgroup K ( n ) = S 0 ( 2 n ) n S p ( 2 n , R ) of Sp(2n,R), as can be seen from the action on Fr'.The remaining n ( n + 1 ) linearly independent Hermitian quadratic generators are of the noncompact, nonphoton number conserving type; we can take them to be (3.10)i i t -(a, ak -akaj ) , j i k = 1,2, . . ., n .4 These generators as Hamiltonians correspond to active systems.Taken together, the expressions in (3.8) 1 ) generators of Sp(2n,lR).We now display the manner in which the maximal compact subgroup of Sp(2n,R) can be explicitly seen in the defining representation of Sp(2n,W).Consider a real 2n X 2n matrix of the form S'"(X, Y ) = p Y :I ' (3.11)where X and Yare real n X n matrices obeying X X T + Y Y T = l , such a matrix is both orthogonal (in 2n dimensions) and symplectic.It is therefore contained in the intersection K(n 1 =SO( 2 n ) n Sp(2n,R 1 .

)
It is necessary to emphasize again that not all rotations in the 2n-dimensional phase space [elements of SO(2n)I are canonical transformations.Only those which have the special form (3.11) are canonical.Therefore we may refer to K( n C SO( 2n as the subgroup of canonical rotations.Let us for a moment view the 2n-dimensional phase space as RZn and define basis and general unit vectors e , = ( O , . . ., o , ~, o , .

Theorem 4 .
A state with variance matrix V ( r ) is squeezed according to the criterion (3.20) if and only if the least eigenvalue I ( V")) of V'r) obeys 1( V ( ' ) ) < + , and conversely.

by 8 ,
and 8,.Each of these, then, consists of certain kinds of V C r ) , diagonalizable within K ( n 1.After dealing with 8, and S , , we take up S,, and thereafter explore possible canonical forms for general Vcr' in 8. A. The family 53G This family consists of those V(r' which apart from a factor of + are also elements of Sp( 2n, R 1: S,=[I/"'=tS/SESp(2n,R),ST=S,S>0) , (4.1)It is known that symmetric positive-definite symplectic matrices correspond one-to-one to points of the coset space O=Sp(2n,R)/K(n).(This is seen, for instance, from the polar decomposition.)It is a fact that in the definition (4.1) do obey the uncertainty principles; this is most easily seen by applying Williamson's theorem to S. Finally, it is also known that this family of Sp( 2n, R ) matrices can be parametrized globally and smoothly by two real symmetric n X n matrices u and u, with u being positive definite: S € S p ( 2 n , R ) , S T = S , S>O [ 1111 u + u u -All this is consistent with the dimensionality of the coset space O= Sp( 2n, R ) / K ( n 1, and so of the family SG, being n ( n + 1 ).

8 ,
er than (+ 11 is squeezed.Now the matrix ( 4 11 is present in S , , too, so we conclude This common element is the variance matrix of any nmode coherent state.C 8,.The definition is 8,=( V"'ES(RTV'"R =(diagonal) , suitable R E K ( n ) ] .(4.11) ments, U is in general not unique.Its arbitrariness is to the extent of O(n, ) X O ( n 2 ) X * * * X O ( n k ) transformations, where n 1 , n 2 , . . ., nk are the multiplicities of the distinct eigenvalues of the diagonal M,, With the help of this lemma, we can characterize the elements of 8, very simply and elegantly.We have the result Theorem 5.The K ( n ) canonical form of a variance matrix V ' " E 8 is diagonal, V'"E8,, if and only if its complex form V'') has blocks A and B obeying A B = B A * .Note that the properties A t = A , B T = B allow us to express the condition AB = B A * in the equivalent form ( AB)T= AB.Choose U , E U ( n ) to diagonalize A .,na=n.A B =( ABF.=diag(a 1 , a 2 , . . .,a, , aj > 0 . of A are nondegenerate, then the condition ( A'B')T= A'B', where B'= U,BUT, implies that B' is also diagonal.However this could continue to be complex: B' = U , BUT =diag(b,eim',b2ei4', . . .,b,e i4 " ) , b, 1 0 .
+B''= U,B'U?=diag(b,, . . ., b, 1 .This completes the proof of sufficiency in this nondegenerate case: U = U 2 U , E U ( n ) carries A and B to A" and B", which are both real diagonal; hence the corresponding S'"(X,Y) (where U = X -i Y ) takes V ( r ) to diagonal form:y(r) = 1 v -4 V 0 l K = S ' " ( X , Y)V"S'"(X, YIT = 4 diag(a + b,,a, + b,, I . .,a, + b,,a I -b l , a , -b 2 , .. .,a, -b, 1 , a j > bj 1 0 , j = 1,2, . . ., that in this nondegenerate case there is a Z, X Z , X . . .X Z , freedom in the choice of U , (apart from the freedom in U , corresponding to the ordering of the diagonal elements of A ' ) , so this is the nature of the stability group of the canonical form.On the other hand, if has degenerate eigenvalues, we modify the argument as follows.Let the distinct eigenvalues ( a . . ., ak ) have respective multiplicities n 1 , n 2 , . . ., n k such that X$=,n,=n.When we pass from A to its diagonal form A' through Eq. (4.13), the condition ( ABIT= A B forces B' into a block-diagonal form: A ' = U , A U ~ =diag(a, * .* a l a z * * * a , -* ak * *ak 1 -B ' = U , B U : =block-diag(B")B'2) -* -B ' k ' ) , d i d ( " ) --n, , a = 1 , 2 , . . ., k .
Take a general complex variance matrix V(') with block matrices A and B .Choose U , E U ( n ) to put A ' = U , A UT into diagonal form.Since we no longer have the symmetry ( A B ) T = AB, the resulting B ' = U , B U T does not have any specific form, though it 0 \ O will be complex symmetric.Let the diagonal elements of B' (when nonzero) have phases el",el", . . . .e l p n .Now choose a diagonal U2 E U ( n ) , as in Eq. (4.151, to make the diagonal elements of B " = U,B ' Ur real non-negative.In the process A " = U, A'UL = A ' is unchanged.Then the combined transformation U = U, U, E U( n ) takes the original V(') to a new V(')" for which A" is real diagonal positive definite and B" is complex symmetric with real non-negative diagonal elements.Throufh Eqs.(2.40) this implies for the real variance matrix V ( r ) : V',' + V y = (diagonal) , V p V ; , , j = 1 , 2 , . . ., n .
K( rn) acts transitively on S 2 m -l , for every m.Given any V ( " E 8, choose any one of its eigenvalues , A,, say, and a corresponding normalized real eigenvector x .25) marked 0 are left unchanged by R,.Thus we isolate an (2n -2 )-dimensional symmetrical submatrix in R , V'"R T by dropping the first and ( n + 1 )th rows, and the first and( n + 1 )th columns.We can choose R , now to attain 0 It will be realized that h; is not an eigenvalue of the original V C r ) (in general), but of the (2n -2 )-dimensional submatrix of R , V("R referred to above.This process can be repeated n times and then one has V to construct explicit physical examples to illustrate the various canonical forms for variance matrices derived in Sec.IV.All of them thermal states.Let us consider first the single-mode case.The thermal state density operator for an oscillator is p = ( l -e -~) e x p ( -~' a ) .

5
f i w / k T the usual thermal parameter.We can rewrite p^ in terms of Ti, the mean occupation number: diagonal in the number representation, both C ( r ) and c ( c ) have zero means, and further V"' has a simple form: ' is the complex representation (2.15) of the scaling transformation S i r ' connecting V ( r i and V'"', we see that V'r)' is produced by the density operator p" = o ( S ( r ) ) p o ( s ( r i ) t =( 1 -e -P)exp[ -PI (cosh2q )ail?+ ( sinh2q)mt +(coshq ) ( ~i n h q ) ( a ~+ i ?' ~) ]