AAccessible information of a general quantumGaussian ensemble
A. S. HolevoSteklov Mathematical Institute, RAS, Moscow, Russia
Abstract
Accessible information, which is a basic quantity in quantum infor-mation theory, is computed for a general quantum Gaussian ensembleunder certain “threshold condition”. It is shown that the maximizingmeasurement is Gaussian, constituting a far-reaching generalizationof the optical heterodyning. This substantially extends the previousresult concerning the gauge-invariant case, even for a single bosonicmode.Keywords: quantum Gaussian ensemble, accessible information,Gaussian maximizer, threshold condition, complex structure.
Accessible information of an ensemble of quantum states is a basic quantityin quantum information theory: it is equal to the maximal amount of theShannon information which can be gained from a given quantum ensemble (acollection of “signal” quantum states with fixed probabilities) in a one-stepmeasurement. This quantity is often difficult to compute, the problem liesin finding the global maximum of a convex functional, when the maximizerturns out to be highly non-unique and the standard tools of convex analysisbecome inefficient. The problem becomes still more complicated for continu-ous variable (CV) systems which constitute one of the prospective platformsfor implementation of ideas of quantum information theory (see e.g. [1]).The quantum Shannon theory for CV systems requires mathematical toolsof infinite-dimensional Hilbert spaces and symplectic vector spaces, see [3].1 a r X i v : . [ qu a n t - ph ] F e b he present paper is a continuation and extension of our paper [4] whichgave a solution for the problem going back to 1970-s: it was shown therethat accessible information of a gauge-invariant bosonic Gaussian ensembleis attained by a multimode generalization of heterodyne measurement, andhence can be computed exactly.In the present paper we extend this result to arbitrary Gaussian ensemblessatisfying certain “threshold condition” . This condition is the one thatallows to reduce the classical capacity problem to a simpler minimum outputentropy problem, and it is always fulfilled in the particularly tractable gauge-invariant case. Thus we obtain here a “Gaussian maximizer” result in asituation going beyond gauge invariance (which is often assumed, see e.g.[2], [3] for various aspects of the famous “Gaussian optimizer conjecture” inanalysis and quantum information theory). Main tools will be the infinite-dimensional version of “ensemble-observable duality” developed in [4] andthe multiplication formulas for Gaussian operators from [7]. We refer reader to [3] for definitions of basic notions of quantum statistics.Let H be a separable Hilbert space, X a standard measurable space. An ensemble E = { π ( dx ) , ρ x } consists of a probability measure π ( dx ) on X anda measurable family of density operators (quantum states) x → ρ x on H .The average state of the ensemble is the barycenter of this measure¯ ρ E = (cid:90) X ρ x π ( dx ) , the integral existing in the strong sense in the Banach space of trace-classoperators on H . Let M = { M ( dy ) } be an observable (probability operator-valued measure = POVM) on H with the outcome space Y . There exists a σ − finite measure µ ( dy ) such that for any density operator ρ the probabilitymeasure Tr ρM ( dy ) is absolutely continuous w.r.t. µ ( dy ) , thus having theprobability density p ρ ( y ) (one can take µ ( dy ) = Tr ρ M ( dy ) where ρ is anondegenerate density operator). The joint probability distribution of x, y Loosely speaking, gauge invariance means that the problem has a unique naturalcomplex structure. In quantum optics, this is related to phase-insensitivity of the system. X × Y is uniquely defined by the relation P ( A × B ) = (cid:90) A π ( dx )Tr ρ x M ( B ) = Tr (cid:90) A (cid:90) B p ρ x ( y ) π ( dx ) µ ( dy ) , where A is an arbitrary Borel subset of X and B is that of Y . The classicalShannon information between x, y is equal to I ( E , M ) = (cid:90) (cid:90) π ( dx ) µ ( dy ) p ρ x ( y ) log p ρ x ( y ) p ¯ ρ E ( y )= h ( p ¯ ρ E ) − (cid:90) h ( p ρ x ) π ( dx ) , where h ( p ) = − (cid:90) p ( x ) log p ( x ) µ ( dx )is the differential entropy of a probability density p ( x ) . There is a specialclass of probability densities we will be dealing with for which the differentialentropy is well-defined (see [4] for the detail).The accessible information of the ensemble E is defined as A ( E ) = sup M I ( E , M ) , (1)where the supremum is over all observables M on H .We will systematically use notations and results from the book [3]. Con-sider the finite-dimensional symplectic space ( Z, ∆) with Z = R s and∆ = diag (cid:20) − (cid:21) j =1 ,...,s . (2)In what follows H will be the space of an irreducible representation z → W ( z ); z ∈ Z, of the canonical commutation relations W ( z ) W ( z (cid:48) ) = exp[ − i z t ∆ z (cid:48) ] W ( z + z (cid:48) ) , z, z (cid:48) ∈ Z. (3)Here W ( z ) = exp i Rz are the unitary Weyl operators with the generators Rz = s (cid:88) j =1 ( x j q j + y j p j ) , (4)3 = [ x j y j ] tj =1 ,...,s , and R = [ q j p j ] j =1 ,...,s are the canonical observables of thequantum system in question satisfying q j p k − p k q j = δ jk I . In quantum com-munication theory they describe the relevant modes of the field on receiver’saperture (see, e.g. [1]). The displacement operators D ( z ) = W ( − ∆ − z )satisfy the equation that follows from the canonical commutation relations(3) D ( z ) ∗ W ( w ) D ( z ) = exp (cid:0) iw t z (cid:1) W ( w ) . (5)A centered Gaussian state ρ α is determined by its quantum characteristicfunction Tr ρ α W ( z ) = exp (cid:18) − z t αz (cid:19) , (6)where the covariance matrix α = Re Tr R t ρR is a real symmetric 2 s × s -matrix satisfying α ≥ ± i . (7)Operator J in ( Z, ∆) is called operator of complex structure if J = − I s , (8)where I s is the identity operator in Z , and it is ∆ − positive in the sensethat ∆ J = − J t ∆ , ∆ J ≥ . (9)The Gaussian state ρ α is pure if and only if α = ∆ J where J is an operatorof complex structure. Such state is called J − vacuum and denoted ρ ∆ J . Thenon-centered pure states D ( z ) ρ ∆ J D ( z ) ∗ are called J − coherent states (seesec. 12.3.2 of [3]).Consider the operator A = ∆ − α . The operator A is skew-symmetric inthe Euclidean space ( Z, α ) with the scalar product α ( z, z (cid:48) ) = z t αz (cid:48) . Accord-ing to a theorem from linear algebra, there is an orthogonal basis { e j , h j } in( Z, α ) and positive numbers { α j } (called symplectic eigenvalues of α ) suchthat Ae j = α j h j ; Ah j = − α j e j , j = 1 , . . . , s. Inequality (7) is equivalent to N j ≡ α j − / ≥ j = 1 , . . . , s. Choosing thenormalization α ( e j , e j ) = α ( h j , h j ) = α j gives a symplectic basis in ( Z, ∆). In other words, ∆ is tamed by J . A = ∆ − α, namely, the orthogonal operator J α from the polar decomposition A = | A | J α = J α | A | (10)in the Euclidean space ( Z, α ) . The action of | A | and J α in the symplecticbasis { e j , h j } constructed above is given by the formula | A | e j = α j e j , | A | h j = α j h j ; J α e j = h j , J α h j = − e j . Inequality (7) is equivalent to α ≥
12 ∆ J α (11)because it amounts to α j − / ≥ j = 1 , . . . , s. We will consider the general Gaussian observable (probability operator-valued measure = POVM) on Z = R s (see [4]) (cid:102) M ( d s z ) = D ( Kz ) ρ β D ( Kz ) ∗ | det K | d s z (2 π ) s ; z ∈ R s , (12)where K is a nondegenerate real matrix and ρ β is a centered Gaussian densityoperator with the real symmetric covariance matrix β. In this case µ is justthe normalized Lebesgue measure on Z = R s . Especially important is thecase K = I where M ( d s z ) = D ( z ) ρ β D ( z ) ∗ d s z (2 π ) s . (13)The probability density of the observable (13) in the state ρ α is computedby using the Parceval formula for the quantum Fourier transform (see [6]) p ρ α ( z ) = Tr ρ α D ( z ) ρ β D ( z ) ∗ (14)= (cid:90) exp (cid:18) − w t αw (cid:19) exp (cid:18) − iw t z − w t βw (cid:19) d s w (2 π ) s = 1 (cid:112) (2 π ) s det ( α + β ) exp (cid:18) − z t ( α + β ) − z (cid:19) . An important special case of observable (13) is the (squeezed) heterodynemeasurement M ( d s z ) = D ( z ) ρ ∆ J β D ( z ) ∗ d s z (2 π ) s . (15)5see Appendix of [4] for the gauge-invariant case). Then (13) can be consid-ered as noisy version of the heterodyne measurement, and (12) – as (matrix)rescaling of (13), which describes classical linear post-processing of the mea-surement outcomes. We first prove the lemma:
Lemma 1.
Let (cid:102) M be the Gaussian observable (12) where ρ β is acentered Gaussian density operator with the real symmetric covariance matrix β. Assume that α is covariance matrix of a Gaussian state satisfying thecondition α ≥
12 ∆ J β . (16) Then max E :¯ ρ E = ρ α I ( E , ˜ M ) = 12 log det ( α + β ) −
12 log det (cid:18) β + 12 ∆ J β (cid:19) (17)= 12 log det ( α + β ) (cid:18) β + 12 ∆ J β (cid:19) − , which is attained on the ensemble E ∗ of J β − coherent states D ( z ) ρ ∆ J β D ( z ) ∗ ,where z has the centered Gaussian probability distribution π γ with the covari-ance matrix γ = α −
12 ∆ J β . (18)We would like to stress that in this paper we do not assume the gauge sym-metry: α and β need not share the common complex structure, J α need notcoincide with J β . In the gauge-invariant case, where the complex structureis unique, we have the correspondence J α = J β = ∆ − → i , α → Σ + I s / β → N + I s / , ∆ − β → i ( N + I s /
2) [3], and (17) turns into the formula oftheorem 1 in [4]: C χ ( (cid:102) M ; Σ) = log det (cid:0) I s + ( N + I s ) − Σ (cid:1) . (19) Proof (sketch).
We will need the formula for the differential entropy of amultidimensional Gaussian probability density p γ with the covariance matrix γ : h ( p γ ) = 12 log det γ + C, (20)6here the constant C depends on the normalization of the Lebesgue measureinvolved in the definition of the differential entropy (cf. [8]).In [4] it is shown that the result does not depend on K so that we cantake K = I s and consider the POVM (13). Then the proof is similar toproof of theorem 1 in [6]. We havemax E :¯ ρ E = ρ α I ( E , ˜ M ) = h ( p ρ α ) − min ρ h ( p ρ ) . (21)Let us show that the maximum is attained on the ensemble E ∗ = (cid:110) π γ ( dz ) , D ( z ) ρ ∆ J β D ( z ) ∗ (cid:111) . The condition (16) ensures existence of the centered Gaussian distribution π γ ( dz ) on Z with the covariance matrix γ = α − ∆ J β . The average state is¯ ρ E = (cid:90) R s D ( z ) ρ ∆ J β D ( z ) ∗ π γ ( dz ) = ρ α , One can check this equality by computing the quantum characteristic func-tions. The probability density of (13) is given by (14). Thus according to(20) h ( p ρ α ) = 12 log det ( α + β ) + C. (22)The result of the paper [5] (Proposition 4; see also [4]) concerning theminimal output entropy of the Gaussian measurement channel implies thatthe minimizer can be taken as the vacuum state ρ ∆ J β related to the complexstructure J β . Substituting α = ∆ J β into (22), we getmin ρ h ( p ρ ) = h (cid:16) p ρ
12 ∆ Jβ (cid:17) = 12 log det (cid:18) β + 12 ∆ J β (cid:19) + C. (23)Substituting (22) and (23) into (21), we get (17). (cid:3) Now we can prove the main result of the paper.
Theorem 1.
Let γ be a real positive definite matrix and let E be theGaussian ensemble { π γ ( d s z ) , ρ β,z } , where π γ ( d s z ) = exp (cid:18) − z ∗ γ − z (cid:19) d s z (2 π ) s √ det γ , (24) ρ β,z = D ( z ) ρ β D ( z ) ∗ (25) For the clarity of proofs we assume that the covariance matrix γ of the Gaussiandistribution π γ is nondegenerate, although this restriction can be relaxed by using moreformal computations with characteristic functions. hen the accessible information (1) of this ensemble is equal to A ( E ) = 12 log det (cid:16) ˜ α + ˜ β (cid:17) (cid:18) ˜ β + 12 ∆ J ˜ β (cid:19) − , (26) where ˜ α = γ + β, (27)˜ β = ˜ α (cid:113) I s + (2∆ − ˜ α ) − γ − (cid:113) I s + (2 ˜ α ∆ − ) − ˜ α − ˜ α (28) provided the threshold condition ˜ α −
12 ∆ J ˜ β ≥ holds.The supremum in (1) is attained on the squeezed heterodyne observable M ∗ ( d s z ) = D ( Kz ) ρ β ∗ D ( Kz ) ∗ | det K | d s z (2 π ) s , (30) where K is a nondegenerate matrix and β ∗ = ˜ α (cid:113) I s + (2∆ − ˜ α ) − (cid:18) ˜ α −
12 ∆ J ˜ β (cid:19) − (cid:113) I s + (2 ˜ α ∆ − ) − ˜ α − ˜ α. (31)Notice that the condition (29) is automatically fulfilled in the gauge-invariant case where the complex structure is unique: J ˜ β = J α = J β =∆ − → i , and the statement reduces to theorem 2 in [4]. Otherwise, apartfrom the single-mode case considered in the following section, the condition(29) might be difficult to check, therefore the following simple sufficient con-dition could be useful. Proposition 1. If γ ≥ β then (29) holds.Proof. Consider the inequality˜ α ≥ (cid:113) I s + (2∆ − ˜ α ) − ˜ α (cid:113) I s + (2 ˜ α ∆ − ) − , which is easily established in the symplectic basis of eigenvectors of the oper-ator ∆ − ˜ α, where it amounts to λ j ≥ (cid:2) − (2 λ j ) − (cid:3) λ j , ( λ j are the symplectic8igenvalues of the matrix ˜ α ) . Then the inequality γ ≥ β and (27) imply con-secutively 2 γ ≥ (cid:113) I s + (2∆ − ˜ α ) − ˜ α (cid:113) I s + (2 ˜ α ∆ − ) − , (cid:113) I s + (2∆ − ˜ α ) − γ − (cid:113) I s + (2 ˜ α ∆ − ) − ≤ α − , ˜ α ≥ ˜ α (cid:113) I s + (2∆ − ˜ α ) − γ − (cid:113) I s + (2 ˜ α ∆ − ) − ˜ α − ˜ α = ˜ β. But ˜ β ≥ ∆ J ˜ β , which implies (29). (cid:3) Proof of theorem 1 . By using the characteristic function and (5), we findthe average state of the ensemble E ¯ ρ E ≡ (cid:90) ρ β,z π γ ( d s z ) = ρ γ + β = ρ ˜ α . (32)Proof of (26) uses ensemble-observable duality from [4], which is sketchedbelow (see [4] for detail of mathematically rigorous description).Let E = { π ( dx ) , ρ x } be an ensemble, µ ( dy ) a σ − finite measure and M = { M ( dy ) } an observable having operator density m ( y ) = M ( dy ) /µ ( dy ) withvalues in the algebra of bounded operators in H . The dual pair ensemble-observable ( E (cid:48) , M (cid:48) ) is defined by the relations E (cid:48) : π (cid:48) ( dy ) = Tr ¯ ρ E M ( dy ) , ρ (cid:48) y = ¯ ρ / E m ( y ) ¯ ρ / E Tr ¯ ρ E m ( y ) ; (33) M (cid:48) : M (cid:48) ( dx ) = ¯ ρ − / E ρ x ¯ ρ − / E π ( dx ) , (34)Then the average states of both ensembles coincide¯ ρ E = ¯ ρ E (cid:48) (35)and the joint distribution of x, y is the same for both pairs ( E , M ) and ( E (cid:48) , M (cid:48) )so that I ( E , M ) = I ( E (cid:48) , M (cid:48) ) . (36)Moreover, sup M I ( E , M ) = sup E (cid:48) :¯ ρ E(cid:48) =¯ ρ E I ( E (cid:48) , M (cid:48) ) , (37)where the supremum in the right-hand side is taken over all ensembles E (cid:48) satisfying the condition ¯ ρ E (cid:48) = ¯ ρ E . 9ow define the POVM dual to ensemble (24), (25): M (cid:48) ( d s z ) = ¯ ρ − / E ρ β,z ¯ ρ − / E π γ ( d s z )= D ( ˜ Kz ) ρ ˜ β D ( ˜ Kz ) ∗ (cid:12)(cid:12)(cid:12) det ˜ K (cid:12)(cid:12)(cid:12) d s z (2 π ) s , (38)where ˜ K is a nondegenerate matrix (given explicitly by (52)). The secondequality follows with the help of results in [7], Sec. 3.2 (see also Appendix).In particular, for z = 0 it amounts to ρ ˜ β ∼ ρ − / α ρ β ρ − / α , or ρ / α ρ ˜ β ρ / α ∼ ρ β ( ∼ means “proportional”). The correlation matrix of the operator ρ / ρ ρ / where ρ , ρ are Gaussian is given in [7], eq. (3.27). In our case ( ρ = ¯ ρ E = ρ ˜ α , ρ = ρ ˜ β ) it reads β = ˜ α − (cid:113) I s + (2 ˜ α ∆ − ) − ˜ α (cid:16) ˜ β + ˜ α (cid:17) − ˜ α (cid:113) I s + (2∆ − ˜ α ) − . (39)Reversing (39) and using ˜ α − β = γ, we get˜ β = (cid:113) I s + (2 ˜ α ∆ − ) − ˜ αγ − ˜ α (cid:113) I s + (2∆ − ˜ α ) − − ˜ α. (40)By noticing that (cid:113) I s + (2 ˜ α ∆ − ) − ˜ α = ˜ α (cid:113) I s + (2∆ − ˜ α ) − , see [7], we ar-rive at (28). Then by (37) and by lemma 1 above A ( E ) = sup M I ( E , M ) = max E (cid:48) :¯ ρ E(cid:48) = ρ ˜ α I ( E (cid:48) , M (cid:48) )= 12 log det (cid:16) ˜ α + ˜ β (cid:17) (cid:18) ˜ β + 12 ∆ J ˜ β (cid:19) − , (41)provided the condition (29) is fulfilled.The statement concerning the optimal observable is obtained from thecorresponding statement of lemma 1 replacing α, β by ˜ α, ˜ β. Here the optimalensemble consists of J ˜ β − coherent states D ( z ) ρ ∆ J ˜ β D ( z ) ∗ , and it is dual tothe observable of the form (30) with some K and ρ β ∗ ∼ ρ − / α ρ ∆ J ˜ β ρ − / α . Byusing (40) with γ replaced by ˜ α − ∆ J ˜ β we obtain (31). (cid:3) It is interesting to compare the quantity (41) with the lower bound ob-tained by taking the heterodyne observable (15). According to (14), the prob-ability density of outcomes of this observable for the Gaussian input state ρ ˜ α
10s centered Gaussian with the covariance matrix ˜ α + ∆ J β = γ + β + ∆ J β = α + β, where at the last step we used (18).Computation using (22) and (23) gives the Shannon information I ( E , M ) = h ( p ρ ˜ α ) − h ( p ρ
12 ∆ Jβ ) (42)= 12 log det ( α + β ) −
12 log det (cid:18) β + 12 ∆ J β (cid:19) = 12 log det ( α + β ) (cid:18) β + 12 ∆ J β (cid:19) − . for the ensemble E and observable M defined by (15) thus giving a lowerbound for the accessible information A ( E ) . We thus have the inequality between (26) and the lower bound (42)12 log det ( α + β ) (cid:18) β + 12 ∆ J β (cid:19) − ≤
12 log det (cid:16) ˜ α + ˜ β (cid:17) (cid:18) ˜ β + 12 ∆ J ˜ β (cid:19) − , (43)which becomes equality in the gauge-invariant case. We start with the case of lemma 1. Let the measurement noise covariancematrix be β = (cid:20) β β (cid:21) ; β β ≥ . The corresponding complex structure is J β = (cid:20) − (cid:112) β /β (cid:112) β /β (cid:21) , Notice, that when β = β , we are in the gauge-invariant case with thestandard complex structure J = (cid:20) −
11 0 (cid:21) . The covariance matrix of the squeezed vacuum is12 ∆ J β = 12 (cid:20) (cid:112) β /β (cid:112) β /β (cid:21) . .0 0.5 1.0 1.5 2.00.00.51.01.52.0 γ γ β = Figure 1: (color online) The “threshold condition” domain for β = 1 / β + 12 ∆ J β = (cid:20) β + (cid:112) β /β β + (cid:112) β /β (cid:21) , so that det (cid:0) β + ∆ J β (cid:1) = (cid:0) √ β β + 1 / (cid:1) , hence the second term in theinformation quantity (17) is − log (cid:0) √ β β + 1 / (cid:1) . Let us restrict to the diagonal input covariance matrices α = (cid:20) α α (cid:21) , (cid:18) α α ≥ (cid:19) . Then the condition (16) amounts to α ≥ (cid:112) β /β , α ≥ (cid:112) β /β , The matrix α + β = (cid:20) β + α β + α (cid:21) has the determinant ( β + α ) ( β + α ) , so that the information quantity(17) (and the lower bound in (42)) is12 log ( β + α ) ( β + α ) (cid:16)(cid:112) β β + 1 / (cid:17) − . (44)12 .0 0.5 1.0 1.5 2.00.00.51.01.52.0 γ γ β = Figure 2: (color online) The “threshold condition” domain for β = 1.Let us now turn to the theorem 1 for a Gaussian ensemble E with thecovariance matrix γ = (cid:20) γ γ (cid:21) ≥ . By (27)˜ α = β + γ , ˜ α = β + γ . (45)Let us find the matrix ˜ β = (cid:20) ˜ β
00 ˜ β (cid:21) . According to (28) we have˜ β = ( β + γ ) γ (cid:20) β −
14 ( β + γ ) (cid:21) , ˜ β = ( β + γ ) γ (cid:20) β −
14 ( β + γ ) (cid:21) . (46)Note that β β ≥ / β ˜ β = 1 γ γ (cid:20) β ( β + γ ) − (cid:21) (cid:20) β ( β + γ ) − (cid:21) ≥ α ≥ (cid:113) ˜ β / ˜ β , ˜ α ≥ (cid:113) ˜ β / ˜ β , which after substituting (45), (46) gives (cid:26) ( β + γ ) ( γ − β ) + ≥ , ( β + γ ) ( γ − β ) + ≥ . . (47)13he accessible information (26) is A ( E ) = 12 log (cid:16) ˜ α + ˜ β (cid:17) (cid:16) ˜ α + ˜ β (cid:17) (cid:18)(cid:113) ˜ β ˜ β + 1 / (cid:19) − . Computation of the parameters (31) of the optimal Gaussian observable(30) gives β ∗ = 12 (cid:115) ˜ β ˜ β ˜ α ˜ α ˜ α − (cid:113) ˜ β / ˜ β ˜ α − (cid:113) ˜ β / ˜ β , β ∗ = 12 (cid:115) ˜ β ˜ β ˜ α ˜ α ˜ α − (cid:113) ˜ β / ˜ β ˜ α − (cid:113) ˜ β / ˜ β . Notice that β ∗ β ∗ = 1 / β = β = β ≥ /
2. Then the set of solutions ( γ , γ ) of this system, as well as the setmin { γ , γ } ≥ β corresponding to the condition of proposition 1, are shownon Figs. 1, 2. One can see that the difference between the two sets becomesmarginal already for β = 1.The inequality (43) becomes12 log ( β + γ + 1 /
2) ( β + γ + 1 /
2) ( β + 1 / − ≤
12 log (cid:16) β + γ + ˜ β (cid:17) (cid:16) β + γ + ˜ β (cid:17) (cid:18)(cid:113) ˜ β ˜ β + 1 / (cid:19) − , which turns into equality iff γ = γ (the gauge-invariant case). In our notations the statement of Lemma 5 of the paper [7] readsTr W ( z ) √ ρ α W ( − z ) √ ρ α = exp (cid:18) − z t αz − z t αz + z t κz (cid:19) , (48)where κ = (cid:113) I s + (2 α ∆ − ) − α = α (cid:113) I s + (2∆ − α ) − . (49)14 ketch of proof . The quantum Fourier transform of √ ρ α computed in [9]is f ( w ) = Tr √ ρ α W ( w ) = (cid:112) det (2 ˆ α ) exp (cid:18) − w t ˆ αw (cid:19) , (50)where ˆ α = α + κ = α (cid:18) I s + (cid:113) I s + (2∆ − α ) − (cid:19) . HenceTr ( W ( z ) √ ρ α ) W ( w ) = Tr √ ρ α W ( w ) W ( z ) = exp (cid:18) − i w t ∆ z (cid:19) f ( w + z ) . By using Parceval relation for the quantum Fourier transform [10], we haveTr W ( z ) √ ρ α W ( − z ) √ ρ α = Tr ( W ( z ) √ ρ α ) ( √ ρ α W ( z )) ∗ = 1(2 π ) s (cid:90) exp (cid:18) − i w t ∆ z (cid:19) f ( w + z )exp (cid:18) i w t ∆ z (cid:19) f ( w + z ) d s w = 1(2 π ) s (cid:90) exp (cid:20) − i w t ∆ ( z + z ) (cid:21) f ( w + z ) f ( w + z ) d s w. Substituting (50), computing a Gaussian integral and using the relationˆ α −
14 ∆ ˆ α − ∆ = 2 α from the paper [11] gives (48). (cid:3) Corollary 1.
Tr ( √ ρ α ρ β,z √ ρ α ) W ( z ) = c exp (cid:18) iz t Kz − z t α z (cid:19) , (51)where c = (det ( α + β )) − / exp (cid:18) − z t ( α + β ) − z (cid:19) ,α = α − κ ( α + β ) − κ, K = κ ( α + β ) − . Proof.
By the inversion formula for the quantum Fourier transform [10] ρ β,z = 1(2 π ) s (cid:90) exp (cid:18) iz t z − z t βz (cid:19) W ( − z ) d s z . √ ρ α ρ β,z √ ρ α W ( z ) = 1(2 π ) s (cid:90) exp (cid:18) iz t z − z t βz (cid:19) × exp (cid:18) − z t αz − z t αz + z t κz (cid:19) d s z . Computation of the Gaussian integral results in (51). (cid:3)
Replacing in (51), (49) α, β by ˜ α, ˜ β , we rederive (39). Replacing addi-tionally z by ˜ Kz, where˜ K = (cid:16) ˜ α + ˜ β (cid:17) κ − = (cid:16) ˜ α + ˜ β (cid:17) ˜ α − (cid:104) I s + (cid:0) α ∆ − (cid:1) − (cid:105) − / , (52)after some routine calculations using (40) we obtainTr ( √ ρ ˜ α ρ ˜ β, ˜ Kz √ ρ ˜ α ) W ( z ) = 1 (cid:12)(cid:12)(cid:12) det ˜ K (cid:12)(cid:12)(cid:12) √ det γ exp (cid:18) − z t γ − z (cid:19) Tr ρ β,z W ( z ) , implying (38). References [1] A. Serafini,
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