# A pedagogical note on the computation of relative entropy of two n-mode gaussian states

aa r X i v : . [ qu a n t - ph ] F e b A pedagogical note on the computation of relative entropyof two n -mode gaussian states K. R. Parthasarathy ∗ Indian Statistical Institute, Theoretical Statistics and Mathematics Unit,Delhi Centre, 7 S. J. S. Sansanwal Marg, New Delhi 110 016, India (Dated: February 16, 2021)We present a formula for the relative entropy S ( ρ || σ ) of two gaussian states ρ , σ in the boson Fockspace Γ( C n ). It is shown that the relative entropy has a classical and a quantum part: The classicalpart consists of a weighted linear combination of relative Shannon entropies of n pairs of Bernoulitrials arising from the thermal state composition of the gaussian states ρ and σ . The quantum parthas a sum of n terms, that are functions of the annihilation means and the covariance matricesof 1-mode marginals of the gaussian state ρ ′ , which is equivalent to ρ under disentangling unitarygaussian symmetry operation of the state σ . In memory of Robin L Hudson ∗ [email protected]

1. INTRODUCTION

Every gaussian state in Γ( C n ) is completely determined by its annihilation mean and covariance matrix. A ﬁneranalysis of the covariance matrix reveals that such a guassian state is composed of a product of n thermal states(including the vacuum state) and an entangling unitary gaussian symmetry operator arising from the group generatedby phase translations and symplectic transformations. Every thermal state, when viewed in a product basis, gives riseto a natural Bernoulli (binomial) trial with sucess and failure probabilities corresponding to detection or no-detection of a particle.In this paper we present a formula for the relative entropy S ( ρ || σ ) = Tr ρ (log ρ − log σ ) of two gaussian states ρ, σ in Γ( C n ). It is shown that the relative entropy S ( ρ || σ ) contains two distinguished components: the ﬁrst part is aweighted linear combination of classical Shannon relative entropies H ( p j : p j ) , j = 1 , , . . . , n of n pairs of Bernoulitrials arising from the thermal components of the gaussian states ρ and σ . The second part is purely quantum andit contains a sum of n terms, each of which is arising from the annihilation mean α j ∈ C and the 2 × T j of 1-mode marginal states ρ ( α j , T j ) of a gaussian state ρ ′ = U † ρ U where U = W ( ℓℓℓ ) Γ( L ) corresponds tothe disentangling gaussian symmetry of the state σ .

2. PRELIMINARY CONCEPTS ON GAUSSIAN STATES

We focus our attention on gaussian states in the complex Hilbert space L ( R n ) of a quantum system with n degrees of freedom, which constitute a natural extension of the concept of normal (gaussian) distributions in classicalprobability [Par10, Par13, PS15, CP19].We shall use the Dirac notation h u | v i for the scalar product of any two elements u and v of a complex Hilbert space.The scalar product h u | v i is assumed to be linear in v and conjugate linear in u . For any operator A and elements u , v in the Hilbert space we write h u | A | v i = h A † u | v i whenever v is in the domain of A or u is in the domain of A † ,provided they are well-deﬁned.Consider the Hilbert space L ( R n ), or equivalently, the boson Fock space H = Γ( C n ), over the complex Hilbertspace C n of ﬁnite dimension n . Fix a canonical orthonormal basis { e j , ≤ j ≤ n } in C n with e j = (0 , , . . . , , , , . . . , T where ’1’ appears in the j th position. We shall denote | e j i = | j i .At every element u ∈ C n we associate a pair of operators a ( u ), a † ( u ), called annihilation, creation operators,respectively in the boson Fock space H and W ( u ) = e a † ( u ) − a ( u ) (2.1)denotes the Weyl displacement operator or simply

Weyl operator in H . For any state ρ in H the complex-valuedfunction on C n given by ˆ ρ ( u ) = Tr W ( u ) ρ, u ∈ C n (2.2)is the quantum characteristic function of ρ at u . The state ρ is said to have ﬁnite second moments if h a † ( u ) a ( u ) i ρ = Tr a † ( u ) a ( u ) ρ < ∞ , ∀ u ∈ C n . (2.3)The annihilation mean , or simply the mean m ρ ∈ C n in the state ρ ∈ H is deﬁned through h a ( u ) i ρ = Tr a ( u ) ρ = h u | m ρ i , u ∈ C n . (2.4)Then h a † ( u ) i ρ = h m ρ | u i . (2.5)Deﬁne the observables q ( u ) = a ( u ) + a † ( u ) √ p ( u ) = a ( u ) − a † ( u ) i √ . (2.7)Then the quantum characteristic function of ρ at u can be expressed asˆ ρ ( u ) = Tr e − i √ p ( u ) ρ. (2.8)Let u = x + i y , x , y ∈ R n . Then x → q ( x ), y → p ( y ) are position and momentum ﬁelds obeying the commutationrelations [ q ( x ) , p ( y )] = i x T y . (2.9)Then q j = q ( e j ) , p j = p ( e j ) , j = 1 , , . . . , n yield the canonical commutation relations (CCR) [Par10][ q j , q k ] = 0 , [ p j , p k ] = 0 , [ q j , p k ] = i δ jk . (2.10)Variance of p ( u ) in the state ρ yields a quadratic form in ( x , x , . . . , x n ; y , y , . . . , y n ) T ∈ R n so thatVar ρ p ( x + i y ) = (cid:0) x , y (cid:1) T C ρ (cid:18) xy (cid:19) . (2.11)Here C ρ is a real 2 n × n positive deﬁnite matrix, satisfying the matrix inequality C ρ + i J ≥ J = (cid:18) n I n − I n n (cid:19) (2.13)being the canonical symplectic matrix in the real symplectic matrix group of order 2 n :Sp(2 n, R ) = (cid:8) L | L T J L = J (cid:9) . In (2.13) the right hand side is expressed in the block notation, with 0 n , I n denoting n × n null and identity matricesrespectively. The correspondence t W ( t u ) = e − i t √ p ( u ) (2.14)is a strongly continuous one-parameter group of unitary operators in t ∈ R , and ˆ ρ ( t, u ) = Tr W ( t u ) ρ is the charac-teristic function of the observable p ( u ), which has the normal or gaussian distribution with mean value h m ρ | u i − h u | m ρ i i √ (cid:0) x T , y T (cid:1) C ρ (cid:18) xy (cid:19) , x + i y = u . (2.16)Then ˆ ρ ( t, ( x + i y )) = exp (cid:20) t ( h m ρ | u i − h u | m ρ i ) − t (cid:0) x T , y T (cid:1) C ρ (cid:18) xy (cid:19)(cid:21) , (2.17)for all t ∈ R , u ∈ C n with u = x + i y . Thus ρ is a quantum gaussian state with mean m ρ and covariance matrix C ρ if and only if ˆ ρ ( x + i y ) = exp (cid:20) i Im( x − i y ) T m ρ − (cid:0) x T , y T (cid:1) C ρ (cid:18) xy (cid:19)(cid:21) , = exp (cid:20) i (cid:0) x T Im m ρ − y T Re m ρ (cid:1) − (cid:0) x T , y T (cid:1) C ρ (cid:18) xy (cid:19)(cid:21) , (2.18)for all x , y ∈ R n . We write ρ ( m , C ) = n − mode gaussian state with mean m and covariance matrix C. The covariance matrix may be expressed in the n × n block matrix notation as C = (cid:18) C C C C (cid:19) (2.19)where C = Cov ( p , p , . . . p n ) C = Cov ( q , q , . . . q n ) (2.20) C = ( − Cov ( p i , q j )) = C T . Note that W ( z ) ρ ( m , C ) W ( z ) † = ρ ( m + z , C ) , z ∈ C n . (2.21)

3. WILLIAMSON’S THEOREM APPLIED TO n -MODE GAUSSIAN COVARIANCE MATRIX Let C be a covariance matrix of an n -mode gaussian state ρ as described above. Then there exist t ≥ t ≥ . . . ≥ t n ≥ / L ∈ Sp(2 n, R ) such that C = L T diag ( t , t , . . . , t n ) 0 n n diag ( t , t , . . . , t n ) L (3.1) Theorem 1.

To every L ∈ Sp(2 n, R ) , there exists a unitary operator Γ( L ) in the boson Fock space H = Γ( C n ) satisfying Γ( L ) W ( u ) Γ( L ) − = W ( L ◦ u ) , u ∈ C n (3.2) where L ◦ u = (cid:0) I n , i I n (cid:1) L (cid:18) xy (cid:19) = ( A x + B y ) + i ( C x + D y ) , (3.3) with A, B, C, D denoting n × n real matrices constituting L as L = (cid:18) A BC D (cid:19) , (3.4) and u = x + i y , x , y ∈ R n . The unitary operator Γ( L ) is unique upto a scalar multiple of modulus unity.Proof. Follows as a consequence of Stone-von Neumann theorem on Weyl operator [Par10, Par13].

4. STRUCTURE THEOREM FOR n -MODE GAUSSIAN STATES For 0 < s ≤ ∞ , we may associate a single mode gaussian thermal state ρ ( s ) in Γ( C ) by ρ ( s ) = (1 − e − s ) e − s a † a (4.1)where s denotes inverse temperature and ρ ( ∞ ) = | Ω ih Ω | , the vacuum state or the thermal state at zero temperature.Every thermal state ρ ( s ) yields a natural binomial distribution with probability for success equal to e − s , where successstands for the event that the number of particles detected is greater than or equal to 1. Then, failure is equivalent to no particle count with probability 1 − e − s .The von Neumann entropy of the thermal state ρ ( s ) is given by S ( ρ ( s )) = − Tr ρ ( s ) log ρ ( s )= H ( e − s )1 − e − s . (4.2)Here H ( p ) = − p log p − (1 − p ) log (1 − p ) , ≤ p ≤ ρ ( s ) → ρ ( ∞ ) = | Ω ih Ω | as s → ∞ . The state ρ ( ∞ ) being pure, S ( ρ ( ∞ )) = 0.Consider any n -mode gaussian state ρ ( m , C ) with mean m ∈ C n and real symmetric positive deﬁnite 2 n × n covariance matrix C . Then, there exist 0 < s ≤ s ≤ . . . ≤ s n ≤ ∞ , and a symplectic matrix L ∈ Sp(2 n, R ) suchthat ρ ( m , C ) = W ( m ) Γ( L ) ρ ( s ) ⊗ ρ ( s ) ⊗ . . . ⊗ ρ ( s n ) Γ( L ) − W ( m ) − . (4.3)The sequence s , s , . . . , s n is unique. The covariance matrix C is given by C = (cid:0) L − (cid:1) T diag (cid:0) coth s j , j = 1 , , . . . , n (cid:1) n n diag (cid:0) coth s j , j = 1 , , . . . , n (cid:1) L − . (4.4)Thus the von Neumann entropy S ( ρ ) is given by S ( ρ ) = X j : s j < ∞ H ( e − s j )1 − e − s j . (4.5)

5. RELATIVE ENTROPY S ( ρ | σ ) OF TWO GAUSSIAN STATES

To set the stage ready for the computation of relative entropy S ( ρ | σ ) = Tr ρ (log ρ − log σ ) of two gaussian states ρ , σ we present the following lemmas. Lemma 1.

Let | ψ i ∈ H , || ψ || = 1 , and ρ be a state in H such that h ψ | ρ | ψ i < . Then S ( ρ || | ψ ih ψ | ) := ∞ . (5.1) Proof.

Write | ψ i = | ψ i , | ψ i , | ψ i , . . . such that {| ψ n i , n = 0 , , . . . } forms an orthonormal basis (ONB). Then ∞ X j =0 h ψ j | ρ | ψ j i = 1 . (5.2)For all j : j ≥ h ψ j | ρ | ψ j i > , | ψ j i is an eigenvector of − log | ψ ih ψ | with eigenvalue ∞ . Thus − Tr ρ log | ψ ih ψ | ≥ h ψ j | ρ | ψ j i × ∞ = ∞ . Then S ( ρ || | ψ ih ψ | ) = Tr ρ log ρ − Tr ρ log | ψ ih ψ | = ∞ . (5.3) Lemma 2.

Consider Hilbert spaces H j , j = 1 , , . . . , n and states σ j ∈ H j . Relative entropy of any arbitrary state ρ with respect to the state σ = σ ⊗ σ ⊗ . . . ⊗ σ n , in H ⊗ H ⊗ . . . ⊗ H n is given by S ( ρ || σ ) = − S ( ρ ) − n X j =1 Tr ρ j log σ j (5.4) where ρ j is the j -th marginal of the state ρ in H j . Proof.

Write ˜ σ j = I ⊗ I ⊗ . . . ⊗ I ⊗ σ j ⊗ I ⊗ . . . ⊗ I, where σ j appears at the j -th position. Thenlog σ = X j log ˜ σ j (5.5)and Tr ρ log σ = X j Tr ρ log ˜ σ j = X j Tr ρ j log σ j (5.6)So S ( ρ || σ ) = Tr ρ log ρ − Tr ρ log σ = − S ( ρ ) − X j Tr ρ j log σ j . (5.7) Lemma 3.

Let ρ ( α, T ) be a 1-mode gaussian state with annihilation mean α ∈ C and × covariance matrix T .Consider a thermal state ρ ( t ) = (1 − e − t ) e − t a † a with a, a † denoting 1-mode annihilation and creation operatorsrespectively. Then Tr ρ ( α, T ) log ρ ( t ) = log (1 − e − t ) − (cid:0) Tr T + 2 | α | − (cid:1) , if 0 < t < ∞ . (5.8) Proof.

It is readily seen that Tr ρ ( α, T ) log ρ ( t ) = Tr ρ ( α, T ) (cid:8) log (1 − e − t ) − t a † a (cid:9) = log (1 − e − t ) − t (cid:10) a † a (cid:11) ρ ( α,T ) . (5.9)Expressing (cid:10) a † a (cid:11) ρ ( α,T ) = 12 (cid:10) (cid:0) p + q − (cid:1) (cid:11) ρ ( α,T ) = 12 (cid:16) Var ( p ) ρ ( α,T ) + Var ( q ) ρ ( α,T ) + h p i ρ ( α,T ) + h q i ρ ( α,T ) − (cid:17) = 12 Tr T + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) q + ip √ (cid:29) ρ ( α,T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − = 12 (cid:0) Tr T + 2 | α | − (cid:1) (5.10)we obtain (5.8). Lemma 4.

Let ρ = ρ ( m , C ) , σ be gaussian states in Γ( C n ) , where m is the annihilation mean i.e., h a ( u ) i ρ = h u | m i and C is the covariance matrix of ρ . Let the standard structure of the gaussian state σ be σ = W ( ℓℓℓ ) Γ( L ) ρ ( t ) ⊗ ρ ( t ) ⊗ . . . ⊗ ρ ( t n ) Γ( L ) − W ( ℓℓℓ ) − (5.11) so that ℓℓℓ is the annihilation mean of σ . Here L ∈ Sp( n, R ) with Γ( L ) W ( u ) Γ( L ) − = W ( L ◦ u ) , ∀ u ∈ C n and < t ≤ t ≤ . . . ≤ t n ≤ ∞ as before in the structure theorem. Then S ( ρ || σ ) = S ( ρ ′ || ρ ( t ) ⊗ ρ ( t ) ⊗ . . . ⊗ ρ ( t n )) (5.12) where ρ ′ ≡ ρ ′ ( m ′ , C ′ ) is the gaussian state characterized by the annihilation mean m ′ = L − ◦ ( m − ℓℓℓ ) (5.13) and the covariance matrix C ′ = L T C L. (5.14)

Proof.

For any unitary U in Γ( C n ) and for ρ ′ = U † ρ U, σ ′ = U † σ U we have S ( ρ ′ || σ ′ ) = S ( ρ || σ ) , (5.15)Choose U = W ( ℓℓℓ ) Γ( L ) to be the gaussian symmetry leading to the standard form (5.11) of σ i.e., σ ′ = ( W ( ℓℓℓ ) Γ( L )) − σ W ( ℓℓℓ ) Γ( L ) = ρ ( t ) ⊗ ρ ( t ) ⊗ . . . ⊗ ρ ( t n ) . Noting that W ( z ) − ρ ( m , C ) W ( z ) = ρ ( m − z , C ) , z ∈ C n Γ( M ) − ρ ( m , C ) Γ( M ) = ρ (cid:0) M − ◦ m , M T C M (cid:1) , M ∈ Sp(2 n, R )we recover (5.13), (5.14) respectively for the annihilation mean and the covariance matrix of the transformed state ρ ′ ( m ′ , C ′ ) = ( W ( ℓℓℓ ) Γ( L )) − ρ ( W ( ℓℓℓ ) Γ( L )). Hence (5.12) follows.Let us denote ααα = L − ◦ ( m − ℓℓℓ ) = ( α , α , . . . , α n ) T T j = the j − th 1 − mode covariance matrix of order 2 in L T C L.

The 1-mode covariance matrix T j is made up of the ( jj )-th, ( j, j + n )-th, ( j + n, j )-th, ( j + n, j + n )-th elements ofthe transformed 2 n × n covariance matrix C ′ = L T C L .Following observations are useful for computing relative entropy S ( ρ || σ ) of ρ with respect to σ . Observation 1 : The j -th mode marginal of the transformed state ρ ′ ( m ′ , C ′ ) = ( W ( ℓℓℓ ) Γ( L )) − ρ W ( ℓℓℓ ) Γ( L ) is the1-mode gaussian state denoted by ρ ( α j , T j ). Observation 2 : Using Lemma 4 we have S ( ρ || σ ) = S ( ρ ′ || ρ ( t ) ⊗ ρ ( t ) ⊗ . . . ⊗ ρ ( t n )). Since ρ ′ and ρ are unitarilyequivalent it follows that S ( ρ ′ ) = S ( ρ ). Observation 3 : By Lemma 2 it follows that S ( ρ || σ ) = − S ( ρ ) − X j Tr ρ ( α j , T j ) log ρ ( t j ) . If t j = ∞ , then ρ ( t j ) = | Ω ih Ω | (vacuum state). If, in addition, ρ ( α j , T j ) = | Ω ih Ω | , then by Lemma 1 we have − Tr ρ ( α j , T j ) log ρ ( t j ) = ∞ . Since − Tr ρ ( α j , T j ) log ρ ( t j ) > ∀ j, it follows that S ( ρ || σ ) = ∞ . Observation 4 : If ρ ( t j ) = | Ω ih Ω | , whenever t j = ∞ , and by Lemma 3Tr ρ ( α j , T j ) log ρ ( t j ) = log (1 − e − t j ) − t j (cid:0) Tr T j + 2 | α j | − (cid:1) , if t j < ∞ . we have S ( ρ || σ ) = − S ( ρ ) + X j : t j < ∞ (cid:26) − log (1 − e − t j ) + t j (cid:2) Tr T j + 2 | α j | − (cid:3)(cid:27) . Thus we obtain S ( ρ || σ ) = ∞ , if ∃ j such that t j = ∞ and ρ ( α j , T j ) = | Ω ih Ω | , − S ( ρ ) + X j : t j < ∞ (cid:18) − log (1 − e − t j ) + t j (cid:0) Tr T j + 2 | α j | − (cid:1)(cid:19) , otherwise . (5.16)Let 0 < s ≤ s ≤ . . . ≤ s n ≤ ∞ , t ≤ t ≤ . . . ≤ t n ≤ ∞ and S ( ρ || σ ) < ∞ . Then S ( ρ || σ ) = − S ( ρ ) + n X j =1 (cid:26) − log (1 − e − t j ) + t j (cid:2) Tr T j + 2 | α j | − (cid:3)(cid:27) = n X j =1 (cid:26) − H ( e − s j )(1 − e − s j ) − log (1 − e − t j ) + t j (cid:2) Tr T j + 2 | α j | − (cid:3)(cid:27) . (5.17)Drop the suﬃx j for the j -th term of the summation in (5.17) for convinience of computation and express it as − H ( e − s )(1 − e − s ) − log (1 − e − t ) + t (cid:0) Tr T + 2 | α | − (cid:1) . (5.18)The ﬁrst two terms of (5.18) can be expressed as − H ( e − s )(1 − e − s ) − log (1 − e − t ) = H ( p : p )(1 − e − s ) − t e − s (1 − e − s ) (5.19)where H ( p : p ) = p log p + (1 − p ) log (1 − p ) − p log p − (1 − p ) log (1 − p ) , p = e − s , p = e − t (5.20)is the classical Shannon relative entropy of a binomial trial with probability of success p with respect to another withprobability of sucess p . Thus (5.18) takes the form H ( e − s : e − t )(1 − e − s ) + t (cid:20) Tr T − (cid:18) e − s − e − s (cid:19) + 2 | α | (cid:21) = H ( e − s : e − t )(1 − e − s ) + t h Tr T − coth (cid:16) s (cid:17) + 2 | α | i . (5.21)Substituting (5.21) in (5.17) leads to S ( ρ || σ ) = n X j =1 (cid:26) H ( e − s j : e − t j )(1 − e − s j ) + t j h Tr T j − coth (cid:16) s j (cid:17) + 2 | α j | i(cid:27) (5.22)whenever S ( ρ || σ ) < ∞ . Remark 1.

Relative entropy S ( ρ || σ ) of two gaussian states ρ and σ consists of a classical part P nj =1 H ( e − sj : e − tj )(1 − e − sj ) and a quantum part P nj =1 t j (cid:2) Tr T j − coth (cid:0) s j (cid:1) + 2 | α j | (cid:3) . Remark 2.

Note that coth (cid:0) s j (cid:1) is equal to the trace of the × covariance matrix of the single mode gaussian state ρ ( s j ) . Remark 3.

Note that e − s j = Success probability= P r ( ≥ ρ ( s j )) ,e − t j = Success probability= P r ( ≥ ρ ( t j )) whereas − e − s j and − e − t j are respectively the probabilities of the ”no particle count” in the thermal states ρ ( s j ) and ρ ( t j ) . For alternate approaches on the computation of relative entropy between two gaussian states see Refs. [Sch01,Chen05, SP17].

ACKNOWLEDGEMENT

I thank A R Usha Devi for making a readable manuscript out of my scribbled algebra.

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