The conditional Entropy Power Inequality for bosonic quantum systems
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The conditional Entropy Power Inequality forbosonic quantum systems
Giacomo De Palma , Dario Trevisan QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken5, 2100 Copenhagen, Denmark Universit`a degli Studi di Pisa, I-56126 Pisa, Italy
Abstract:
We prove the conditional Entropy Power Inequality for Gaussianquantum systems. This fundamental inequality determines the minimum quan-tum conditional von Neumann entropy of the output of the beam-splitter or ofthe squeezing among all the input states where the two inputs are conditionallyindependent given the memory and have given quantum conditional entropies.We also prove that, for any couple of values of the quantum conditional en-tropies of the two inputs, the minimum of the quantum conditional entropy ofthe output given by the conditional Entropy Power Inequality is asymptoticallyachieved by a suitable sequence of quantum Gaussian input states. Our proofof the conditional Entropy Power Inequality is based on a new Stam inequalityfor the quantum conditional Fisher information and on the determination of theuniversal asymptotic behaviour of the quantum conditional entropy under theheat semigroup evolution. The beam-splitter and the squeezing are the central el-ements of quantum optics, and can model the attenuation, the amplification andthe noise of electromagnetic signals. This conditional Entropy Power Inequalitywill have a strong impact in quantum information and quantum cryptography.Among its many possible applications there is the proof of a new uncertaintyrelation for the conditional Wehrl entropy.
1. Introduction
The Shannon differential entropy [2] of a random variable X with values in R k and probability density p X ( x )d k x is S ( X ) := − Z R k ln p X ( x ) d p X ( x ) , (1) Giacomo De Palma, Dario Trevisan and quantifies the noise or the information contained in X . Let us consider thelinear combination C := √ η A + p | − η | B , η ≥ A and B with values in R k . The classicalEntropy Power Inequality [13, 21, 22] states that, if A and B have Shannondifferential entropy fixed to the values S ( A ) and S ( B ), respectively, the Shannondifferential entropy of C is minimized when A and B have a Gaussian probabilitydistribution with proportional covariance matrices:exp 2 S ( C ) k ≥ η exp 2 S ( A ) k + | − η | exp 2 S ( B ) k , (3)and is a fundamental element of classical information theory [2].The noncommutative counterpart of probability measures are quantum states,that are linear positive operators on a Hilbert space with unit trace. The counter-part of the probability measures on R k with even k are the quantum states of aGaussian quantum system with n = k modes. Gaussian quantum systems [16,17]model electromagnetic waves in the quantum regime. Electromagnetic wavestraveling through cables or free space provide the most promising platform forquantum communication and quantum key distribution [23]. Gaussian quantumsystems then play a key role in quantum communication and quantum cryp-tography, and provide the model to determine the maximum communicationand key distribution rates achievable in principle by quantum communicationdevices. The noncommutative counterpart of the linear combination (2) is thebeam-splitter of transmissivity 0 ≤ η ≤ η ≥ S (ˆ ρ ) := − Tr [ˆ ρ ln ˆ ρ ] . (4)In this paper we prove the conditional Entropy Power Inequality for Gaussianquantum systems (Theorem 6). Let A and B be the n -mode Gaussian quantumsystems at the input of the beam-splitter of transmissivity 0 ≤ η ≤ η ≥
1, and let C be the n -mode Gaussian quantumsystem at the output. Let us consider a joint quantum input state ˆ ρ ABM suchthat A and B are conditionally independent given the memory system M . Thiscondition is expressed with the vanishing of the quantum conditional mutualinformation: I ( A : B | M ) := S ( A | M ) + S ( B | M ) − S ( AB | M ) = 0 , (5)where S ( X | M ) := S ( XM ) − S ( M ) (6)is the quantum conditional entropy. The conditional Entropy Power Inequalitydetermines the minimum quantum conditional entropy of the output S ( C | M ) he conditional Entropy Power Inequality for bosonic quantum systems 3 among all the quantum input states ˆ ρ ABM as above and with given quantumconditional entropies S ( A | M ) and S ( B | M ):exp S ( C | M ) n ≥ η exp S ( A | M ) n + | − η | exp S ( B | M ) n . (7)We also prove that, for any couple of values of S ( A | M ) and S ( B | M ), the mini-mum (7) for S ( C | M ) is asymptotically achieved by a suitable sequence of quan-tum Gaussian input states (Theorem 7).The conditional Entropy Power Inequality (7) had been conjectured in [18]. Itis the conditional version of the quantum Entropy Power Inequality [5, 6, 19, 20]that provides a lower bound to the von Neumann entropy S ( C ) of the outputof the beam-splitter or of the squeezing for all the product input states ˆ ρ AB =ˆ ρ A ⊗ ˆ ρ B in terms of the entropies of the inputs S ( A ) and S ( B ):exp S ( C ) n ≥ η exp S ( A ) n + | − η | exp S ( B ) n . (8)Contrarily to the classical Entropy Power Inequality (3), the quantum EntropyPower Inequality (8) is not saturated by quantum Gaussian states with propor-tional covariance matrices, unless they have the same entropy [5].In the classical scenario, the conditional Entropy Power Inequality readsexp 2 S ( C | M ) k ≥ η exp 2 S ( A | M ) k + | − η | exp 2 S ( B | M ) k , (9)where A and B are random variables with values in R k and are conditionallyindependent given the random variable M , and C is as in (2). The conditionalEntropy Power Inequality (9) is an easy consequence of its unconditioned ver-sion (3), because the conditional entropy S ( X | M ) coincides with the expecta-tion value with respect to M of the entropy of X given the value of M (seeAppendix A): S ( X | M ) = Z M S ( X | M = m ) d p M ( m ) . (10)The classical conditional Entropy Power Inequality (9) is saturated by any jointprobability measure on ABM such that, conditioning on any value m of M , A and B are independent Gaussian random variables with proportional covariancematrices, and the proportionality constant does not depend on m .In the quantum scenario, conditioning on the value of M is not possible inthe presence of entanglement between AB and M , and the conditional EntropyPower Inequality is not an easy consequence of the unconditioned Entropy PowerInequality. The saturation conditions are another fundamental difference be-tween the classical and the quantum scenario. The quantum conditional EntropyPower Inequality can be saturated only asymptotically by a suitable sequence ofquantum Gaussian states. Contrarily to the classical scenario, the correlation ofthe inputs A and B with the memory M is necessary for the saturation of theinequality. Indeed, the unconditioned quantum Entropy Power Inequality is notsaturated even asymptotically by quantum Gaussian states.Entropic inequalities are the main tool to prove upper bounds to quantumcommunication rates [16, 24] and to prove the security of quantum key distri-bution schemes [1]. In these scenarios, a prominent role is played by entropic Giacomo De Palma, Dario Trevisan inequalities in the presence of quantum memory, where the entropies are con-ditioned on the knowledge of an external observer holding a memory quantumsystem. The quantum conditional Entropy Power Inequality proven in this pa-per will have a profound impact in quantum information theory and quantumcryptography. The inequality has been fundamental in the proof of a new un-certainty relation for the conditional Wehrl entropy [3]. In section 8, we exploitthe inequality to prove an upper bound to the entanglement-assisted classicalcapacity of a non-Gaussian quantum channel. This implication has been firstconsidered in [18], section III.The proof of the quantum conditional Entropy Power inequality is basedon the evolution with the heat semigroup as in [5, 6, 18, 19]. For simplifyingthe proof, we reformulate the inequality in the equivalent linear version (106)through the Legendre transform. The linear inequality (106) had been provenwith a particular choice of M for Gaussian input states in [18], Theorem 8.1.Our proof consists of two parts: – We prove the quantum conditional Stam inequality (Theorem 3), which pro-vides an upper bound to the quantum conditional Fisher information of theoutput of the beam-splitter or of the squeezing in terms of the quantum con-ditional Fisher information of the two inputs. This inequality implies thatthe difference between the two sides of the linear inequality (106) decreasesalong the evolution with the heat semigroup. The linear version (66) of thequantum conditional Stam inequality in the particular case λ = η had beenproven in [18], eq. (62). The proof of [18], as well as the proofs of the un-conditioned quantum Stam inequality of [5, 6, 19], are affected by regularityissues. Indeed, all these papers define the quantum Fisher information as theHessian of the relative entropy with respect to the displacements, but theydo not prove that this Hessian is well defined. We introduce a new integralversion of the conditional quantum Fisher information, and we define the con-ditional quantum Fisher information as the limit of its integral version. Thisdefinition solves all the previous regularity issues. Our proof of the quantumconditional Stam inequality is based on an integral version of the quantumde Bruijn identity (Theorem 1), that relates the increase of the quantum con-ditional entropy generated by the heat semigroup with the integral quantumconditional Fisher information (Definition 6). – We prove that the quantum conditional entropy has an universal scaling inde-pendent on the initial state (Theorem 5) in the infinite time limit under theevolution with the heat semigroup. This scaling was already known for Gaus-sian states ( [18], Lemma 6.1) and implies that the linear inequality (106)asymptotically becomes an equality. Our proof is based on a more generalresult (Theorem 4), stating that the minimum quantum conditional entropyof the output of any Gaussian quantum channel is asymptotically achievedby the purification of the thermal quantum Gaussian states with infinite tem-perature.The paper is structured as follows. In section 2 we present Gaussian quantumsystems, the beam-splitter and the squeezing. In section 3 we present the quan-tum integral conditional Fisher information, and in section 4 we prove the quan-tum conditional Stam inequality. In section 5 we prove the universal asymptoticscaling of the quantum conditional entropy. In section 6 we prove the quan-tum conditional Entropy Power Inequality, and in section 7 we prove that this he conditional Entropy Power Inequality for bosonic quantum systems 5 inequality is optimal. In section 8 we apply the quantum conditional EntropyPower Inequality to prove an upper bound to the entanglement-assisted clas-sical capacity of a non-Gaussian quantum channel. We conclude in section 9.Appendix A and Appendix B contain the proof of the classical conditional En-tropy Power Inequality and of the auxiliary lemmas, respectively.
2. Gaussian quantum systems
The Hilbert space of a Gaussian quantum system with n modes is the irreduciblerepresentation of the canonical commutation relations h ˆ Q k , ˆ Q l i = 0 , h ˆ P k , ˆ P l i = 0 , h ˆ Q k , ˆ P l i = i δ kl ˆ I , k, l = 1 , . . . , n . (11)We define ˆ R k − := ˆ Q k , ˆ R k := ˆ P k , k = 1 , . . . , n , (12)and (11) becomes h ˆ R i , ˆ R j i = i ∆ ij ˆ I , i, j = 1 , . . . , n , (13)where ∆ := n M k =1 (cid:18) − (cid:19) (14)is the symplectic form. The Hamiltonian of the system isˆ H = 12 n X i =1 (cid:16) ˆ R i (cid:17) − n I . (15) Definition 1 (displacement operators).
For any x ∈ R n we define the dis-placement operator ˆ D ( x ) := exp i n X i =1 x i ∆ − ij ˆ R j ! , (16) the unitary operator satisfying for any i = 1 , . . . , n ˆ D ( x ) † ˆ R i ˆ D ( x ) = ˆ R i + x i ˆ I . (17) Definition 2 (first moments).
The first moments of a quantum state ˆ ρ are r i (ˆ ρ ) := Tr h ˆ R i ˆ ρ i , i = 1 , . . . , n . (18) Definition 3 (covariance matrix).
The covariance matrix of a quantum state ˆ ρ with finite first moments is σ ij (ˆ ρ ) := 12 Tr hn ˆ R i − r i (ˆ ρ ) , ˆ R j − r j (ˆ ρ ) o ˆ ρ i , i, j = 1 , . . . , n , (19) where n ˆ X, ˆ Y o := ˆ X ˆ Y + ˆ Y ˆ X (20) is the anticommutator. Giacomo De Palma, Dario Trevisan
Definition 4 (symplectic eigenvalues).
The symplectic eigenvalues of a realpositive matrix σ are the absolute values of the eigenvalues of ∆ − σ . Definition 5 (heat semigroup).
The heat semigroup is the time evolutiongenerated by the convex combination of displacement operators with Gaussiandistribution and covariance matrix t I n : for any quantum state ˆ ρ N ( t )(ˆ ρ ) := Z R n ˆ D ( x ) ˆ ρ ˆ D ( x ) † e − | x | t d n x (2 π t ) n . (21) For any s, t ≥ N ( s ) ◦ N ( t ) = N ( s + t ) . (22) A quantum Gaussian state is a density operatorproportional to the exponential of a quadratic polynomial in the quadratures:ˆ γ = exp (cid:16) − P ni, j =1 (cid:16) ˆ R i − r i (cid:17) h ij (cid:16) ˆ R j − r j (cid:17)(cid:17) Tr exp (cid:16) − P ni, j =1 (cid:16) ˆ R i − r i (cid:17) h ij (cid:16) ˆ R j − r j (cid:17)(cid:17) , (23)where h is a positive real 2 n × n matrix and r ∈ R n . A thermal Gaussian stateis a Gaussian state with zero first moments ( r = 0) and where the matrix h isproportional to the identity:ˆ ω = e − β ˆ H Tr e − β ˆ H , h = β I n , β > . (24)The von Neumann entropy of a quantum Gaussian state is S = n X k =1 g (cid:18) ν k − (cid:19) , (25)where g ( x ) := ( x + 1) ln ( x + 1) − x ln x , (26)and ν , . . . , ν n are the symplectic eigenvalues of its covariance matrix. Given the n -mode Gaussian quantum systems A , B , C and D , the beam-splitter with inputs A and B , outputs C and D and transmissivity 0 ≤ η ≤ U η : AB → CD acting on the quadratures as [14]ˆ U † η ˆ R iC ˆ U η = √ η ˆ R iA + p − η ˆ R iB , (27a)ˆ U † η ˆ R iD ˆ U η = − p − η ˆ R iA + √ η ˆ R iB , i = 1 , . . . , n . (27b)The beam-splitter is a passive element, and does not require energy for func-tioning. Indeed, the mixing unitary operator preserves the Hamiltonian (15):ˆ U η (cid:16) ˆ H A + ˆ H B (cid:17) ˆ U † η = ˆ H C + ˆ H D . (28) he conditional Entropy Power Inequality for bosonic quantum systems 7 The squeezing unitary operator with parameter η ≥ U † η ˆ Q kC ˆ U η = √ η ˆ Q kA + p η − Q kB , (29a)ˆ U † η ˆ P kC ˆ U η = √ η ˆ P kA − p η − P kB , (29b)ˆ U † η ˆ Q kD ˆ U η = p η − Q kA + √ η ˆ Q kB , (29c)ˆ U † η ˆ P kD ˆ U η = − p η − P kA + √ η ˆ P kB , k = 1 , . . . , n . (29d)The squeezing acts differently on the Q k and on the P k . Indeed, the squeezingis an active operation that requires energy, and the squeezing unitary operatordoes not preserve the Hamiltonian (15).We define for any joint quantum state ˆ ρ AB on AB and any η ≥ B η (ˆ ρ AB ) := Tr D h ˆ U η ˆ ρ AB ˆ U † η i . (30) B η implements the beam-splitter for 0 ≤ η ≤ η ≥ Lemma 1 (compatibility with displacements).
We have for any quantumstate ˆ ρ AB on AB and any x , y ∈ R n B η (cid:16) ˆ D A ( x ) ˆ D B ( y ) ˆ ρ AB ˆ D A ( x ) † ˆ D B ( y ) † (cid:17) = ˆ D C (cid:16) √ η x + p − η y (cid:17) B η (ˆ ρ AB ) ˆ D C (cid:16) √ η x + p − η y (cid:17) † (31) for the beam-splitter with transmissivity ≤ η ≤ , and B η (cid:16) ˆ D A ( x ) ˆ D B ( y ) ˆ ρ AB ˆ D A ( x ) † ˆ D B ( y ) † (cid:17) = ˆ D C (cid:16) √ η x + p η − T y (cid:17) B η (ˆ ρ AB ) ˆ D C (cid:16) √ η x + p η − T y (cid:17) † (32) for the squeezing with parameter η ≥ , where T := n M k =1 (cid:18) − (cid:19) (33) is the time-reversal matrix that leaves the Q k unchanged and reverses the signof the P k . Lemma 2 (compatibility with heat semigroup).
For any s, t ≥ B η ◦ ( N A ( s ) ⊗ N B ( t )) = N C ( ηs + | − η | t ) ◦ B η . (34) Proof.
Follows from Lemma 1.
Giacomo De Palma, Dario Trevisan
3. Quantum integral conditional Fisher information
In this Section, we define the quantum integral conditional Fisher informationthat will permit us to prove the regularity of the quantum Fisher informationof [5, 6, 18, 19].
Definition 6 (quantum integral conditional Fisher information).
Let A be a Gaussian quantum system with n modes, and M a quantum system. Let ˆ ρ AM be a quantum state on AM . For any t ≥ , we define the integral Fisherinformation of A conditioned on M as ∆ A | M (ˆ ρ AM )( t ) := I ( A : X | M ) ˆ σ AMX ( t ) ≥ , t > , (35a) ∆ A | M (ˆ ρ AM )(0) := 0 , (35b) where X is a classical Gaussian random variable with values in R n and proba-bility density function d p X ( t )( x ) = e − | x | t d n x (2 π t ) n , x ∈ R n , (36) and ˆ σ AMX ( t ) is the quantum state on AM X such that its marginal on X is p X ( t ) and for any x ∈ R n ˆ σ AM | X = x ( t ) = ˆ D A ( x ) ˆ ρ AM ˆ D A ( x ) † . (37) Remark 1.
The marginal over AM of ˆ σ AMX isˆ σ AM = ( N ( t ) ⊗ I M )(ˆ ρ AM ) . (38)The fundamental property of the quantum integral conditional Fisher infor-mation is the relation with the increase in the quantum conditional entropygenerated by the heat semigroup. Theorem 1 (quantum integral conditional de Bruijn identity).
The quan-tum integral conditional Fisher information coincides with the increase of thequantum conditional entropy generated by the heat semigroup: for any t ≥ , ∆ A | M (ˆ ρ AM )( t ) = S ( A | M ) ( N A ( t ) ⊗ I M )(ˆ ρ AM ) − S ( A | M ) ˆ ρ AM . (39) Proof. I ( A : X | M ) ˆ σ AMX = S ( A | M ) ˆ σ AMX − S ( A | M X ) ˆ σ AMX = S ( A | M ) ˆ σ AM − Z R n S ( A | M ) ˆ σ AM | X = x d p X ( x )= S ( A | M ) ˆ σ AM − Z R n S ( A | M ) ˆ ρ AM d p X ( x )= S ( A | M ) ˆ σ AM − S ( A | M ) ˆ ρ AM . (40)The goal of the remainder of this Section is proving that the quantum integralconditional Fisher information is a continuous, increasing and concave functionof time (Theorem 2). This result will permit us to prove the regularity of thequantum Fisher information. he conditional Entropy Power Inequality for bosonic quantum systems 9 Lemma 3 (continuity of quantum integral conditional Fisher informa-tion).
For any quantum state ˆ ρ AM such that Tr A h ˆ H A ˆ ρ A i = E < ∞ , S (ˆ ρ M ) < ∞ (41) we have lim t → ∆ A | M (( N ( t ) ⊗ I M )(ˆ ρ AM )) = ∆ A | M (ˆ ρ AM ) . (42) Proof.
From Theorem 1, the claim is equivalent tolim t → S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) = S ( A | M ) ˆ ρ AM . (43)We will then proceed along the same lines of the proof of the continuity of theentropy in the set of the quantum states with bounded average energy ( [16],Lemma 11.8). We have (see e.g. [4], Lemma 2)lim t → k ( N ( t ) ⊗ I M )(ˆ ρ AM ) − ˆ ρ AM k = 0 . (44)Since the quantum entropy is lower semicontinuous ( [16], Theorem 11.6) and S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) = S (( N ( t ) ⊗ I M )(ˆ ρ AM )) − S (ˆ ρ M ) , (45)we have lim inf t → S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) ≥ S ( A | M ) ˆ ρ AM . (46)On the other hand, we have for any β > ≤ t < ǫS ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) = β Tr A h ˆ H A N ( t )(ˆ ρ A ) i + ln Tr A e − β ˆ H A − S ( N ( t ) ⊗ I M )(ˆ ρ AM ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e − β ˆ H A Tr A e − β ˆ H A ⊗ ˆ ρ M ! = β ( E + n t ) + ln Tr A e − β ˆ H A − S ( N ( t ) ⊗ I M )(ˆ ρ AM ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e − β ˆ H A Tr A e − β ˆ H A ⊗ ˆ ρ M ! , (47)where S (ˆ ρ k ˆ σ ) = Tr [ˆ ρ (ln ˆ ρ − ln ˆ σ )] (48)is the quantum relative entropy [16]. Since the quantum relative entropy is lowersemicontinuous ( [16], Theorem 11.6),lim sup t → S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) ≤ β ( E + n t ) + ln Tr A e − β ˆ H A − S ˆ ρ AM (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e − β ˆ H A Tr A e − β ˆ H A ⊗ ˆ ρ M ! = S ( A | M ) ˆ ρ AM + β (cid:16) E + n t − Tr A h ˆ H A ˆ ρ A i(cid:17) , (49)and the claim follows taking the limit β → Lemma 4.
For any s, t ≥ , ∆ A | M (( N A ( s ) ⊗ I M )(ˆ ρ AM ))( t ) = I ( A : X | M ) ( N A ( s ) ⊗ I M )(ˆ σ AMX ( t )) , (50) where ˆ σ AMX ( t ) is as in Definition 6.Proof. We have ∆ A | M (( N A ( s ) ⊗ I M )(ˆ ρ AM ))( t ) = I ( A : X | M ) ˆ τ AMX ( s,t ) , (51)where X is as in Definition 6, and ˆ τ AMX ( s, t ) is the quantum state on AM X such that its marginal on X is p X ( t ), and for any x ∈ R n ˆ τ AM | X = x ( s, t ) = ˆ D A ( x ) ( N A ( s ) ⊗ I M )(ˆ ρ AM ) ˆ D A ( x ) † = ( N A ( s ) ⊗ I M ) (cid:16) ˆ D A ( x ) ˆ ρ AM ˆ D A ( x ) † (cid:17) = ( N A ( s ) ⊗ I M )(ˆ σ AM | X = x ( t )) . (52)Hence for any t ≥ τ AMX ( s, t ) = ( N A ( s ) ⊗ I M )(ˆ σ AMX ( t )) , (53)and the claim follows. Lemma 5.
For any s, t ≥ ∆ A | M (( N A ( s ) ⊗ I M )(ˆ ρ AM ))( t ) ≤ ∆ A | M (ˆ ρ AM )( t ) . (54) Proof.
From Lemma 4, the claim is equivalent to I ( A : X | M ) ( N A ( s ) ⊗ I M )(ˆ σ AMX ( t )) ≤ I ( A : X | M ) ˆ σ AMX ( t ) , (55)that follows from the data-processing inequality for the quantum mutual infor-mation. Lemma 6.
For any s, t ≥ ∆ A | M (ˆ ρ AM )( s + t ) = ∆ A | M (ˆ ρ AM )( s ) + ∆ A | M (( N A ( s ) ⊗ I M )(ˆ ρ AM ))( t ) ≥ ∆ A | M (ˆ ρ AM )( s ) . (56) Proof.
Follows from Theorem 1.
Theorem 2 (regularity of quantum integral conditional Fisher infor-mation).
For any quantum state ˆ ρ AM on AM such that Tr A h ˆ H A ˆ ρ A i < ∞ , S (ˆ ρ M ) < ∞ , (57) the quantum integral conditional Fisher information S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) is acontinuous, increasing and concave function of time. he conditional Entropy Power Inequality for bosonic quantum systems 11 Proof.
The continuity follows from Lemma 3 and Lemma 6. From Lemma 6, S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) is increasing. We then have to prove that for any s, t ≥ ∆ A | M (ˆ ρ AM ) (cid:18) s + t (cid:19) ? ≥ ∆ A | M (ˆ ρ AM )( s ) + ∆ A | M (ˆ ρ AM )( t )2 . (58)Without lost of generality we can assume s ≤ t . We can rephrase (58) as ∆ A | M (ˆ ρ AM ) (cid:18) s + t (cid:19) − ∆ A | M (ˆ ρ AM )( s ) ? ≥ ∆ A | M (ˆ ρ AM )( t ) − ∆ A | M (ˆ ρ AM ) (cid:18) s + t (cid:19) , (59)that thanks to Lemma 6 is equivalent to ∆ A | M (ˆ ρ AM ( s )) (cid:18) t − s (cid:19) ? ≥ ∆ A | M (cid:18)(cid:18) N A (cid:18) t − s (cid:19) ⊗ I M (cid:19) (ˆ ρ AM ( s )) (cid:19) (cid:18) t − s (cid:19) , (60)where ˆ ρ AM ( s ) := ( N A ( s ) ⊗ I M )(ˆ ρ AM ) . (61)Finally, (60) holds from Lemma 5.
4. Quantum conditional Fisher information and quantum Staminequality
In this Section, we derive the quantum conditional Fisher information and thequantum conditional de Bruijn identity from their integral versions presented insection 3, and we prove that the quantum conditional Fisher information satisfiesthe quantum Stam inequality.
Definition 7 (quantum conditional Fisher information).
Let ˆ ρ AM be aquantum state on AM satisfying the hypotheses of Theorem 2. The Fisher in-formation of A conditioned on M is J ( A | M ) ˆ ρ AM := lim t → ∆ A | M (ˆ ρ AM )( t ) t . (62) Remark 2.
Since from Theorem 2 the function t ∆ A | M (ˆ ρ AM )( t ) is continuousand concave, the limit in (62) always exists (finite or infinite). Remark 3.
Definition 7 is equivalent to the definition of [18], eq. (53).
Proposition 1 (quantum conditional de Bruijn identity).
The quantumconditional Fisher information coincides with the time derivative of the quantumconditional entropy under the heat semigroup evolution: J ( A | M ) ˆ ρ AM = dd t S ( A | M ) ( N A ( t ) ⊗ I M )(ˆ ρ AM ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 . (63) Proof.
Follows from Theorem 1.
Remark 4.
Proposition 1 had been proven in [18], Theorem 7.3 in the particularcase where ˆ ρ AM is a quantum Gaussian state. Theorem 3 (quantum conditional Stam inequality).
Let A , B and C beGaussian quantum systems with n modes, M a quantum system, and B η : AB → C the beam-splitter with transmissivity ≤ η ≤ or the squeezing with parameter η ≥ . Let ˆ ρ ABM be a quantum state on
ABM such that Tr AB h(cid:16) ˆ H A + ˆ H B (cid:17) ˆ ρ AB i < ∞ , S (ˆ ρ M ) < ∞ , (64) and let us suppose that A and B are conditionally independent given M : I ( A : B | M ) ˆ ρ ABM = 0 , (65) Then, for any ≤ λ ≤ the quantum linear conditional Stam inequality holds: J ( C | M ) ˆ ρ CM ≤ λ η J ( A | M ) ˆ ρ AM + (1 − λ ) | − η | J ( B | M ) ˆ ρ BM , (66) where ˆ ρ CM := ( B η ⊗ I M )(ˆ ρ ABM ) . (67) The quantum conditional Stam inequality follows minimizing over λ the right-hand side of (66) : J ( C | M ) ˆ ρ CM ≥ ηJ ( A | M ) ˆ ρ AM + | − η | J ( B | M ) ˆ ρ BM . (68) Remark 5.
The linear Stam inequality (66) had been proven in the particularcase λ = η in [18], eq. (62). Proof.
We will prove the following inequality for the quantum integral condi-tional Fisher information: ∆ C | M (ˆ ρ CM )( t ) ≤ ∆ A | M (ˆ ρ AM ) (cid:18) λ tη (cid:19) + ∆ B | M (ˆ ρ BM ) (cid:18) (1 − λ ) t | − η | (cid:19) . (69)The quantum linear conditional Stam inequality (66) follows taking the deriva-tive of (69) in t = 0. The quantum conditional Stam inequality (68) followschoosing λ = η J ( B | M ) ˆ ρ BM η J ( B | M ) ˆ ρ BM + | − η | J ( A | M ) ˆ ρ AM , (70)that minimizes the right-hand side of (66).For any t ≥ ∆ C | M (ˆ ρ CM )( t ) = I ( C : Z | M ) ˆ σ CMZ ( t ) , (71)where Z is a Gaussian random variable with values in R n and probability densityfunction d p Z ( t )( z ) = e − | z | t d n z (2 π t ) n , z ∈ R n , (72) he conditional Entropy Power Inequality for bosonic quantum systems 13 and ˆ σ CMZ ( t ) is the quantum state on CM Z such that its marginal on Z is p Z ( t )and for any z ∈ R n ˆ σ CM | Z = z ( t ) = ˆ D C ( z ) ˆ ρ CM ˆ D C ( z ) † . (73)We define the quantum state ˆ σ ABMZ ( t ) on ABM Z such that its marginal on Z is p Z ( t ) and for any z ∈ R n ˆ σ ABM | Z = z = ˆ D A (cid:18) λ z √ η (cid:19) ˆ D B (cid:18) (1 − λ ) z √ − η (cid:19) ˆ ρ ABM ˆ D A (cid:18) λ z √ η (cid:19) † ˆ D B (cid:18) (1 − λ ) z √ − η (cid:19) † (74)if 0 ≤ η ≤
1, andˆ σ ABM | Z = z = ˆ D A (cid:18) λ z √ η (cid:19) ˆ D B (cid:18) (1 − λ ) T z √ η − (cid:19) ˆ ρ ABM ˆ D A (cid:18) λ z √ η (cid:19) † ˆ D B (cid:18) (1 − λ ) T z √ η − (cid:19) † (75)if η ≥
1, where T is the time-reversal matrix defined in (33). We then have forany t ≥ σ CMZ ( t ) = ( B η ⊗ I MZ )(ˆ σ ABMZ ( t )) . (76)We have I ( A : B | M Z ) ˆ σ ABMZ = Z R n I ( A : B | M ) ˆ σ ABM | Z = z d p Z ( z )= Z R n I ( A : B | M ) ˆ ρ ABM d p Z ( z ) = 0 . (77)We then have from the data-processing inequality for the quantum mutual in-formation I ( C : Z | M ) ˆ σ CMZ ≤ I ( AB : Z | M ) ˆ σ ABMZ = I ( A : Z | M ) ˆ σ AMZ + I ( B : Z | M ) ˆ σ BMZ + I ( A : B | M Z ) ˆ σ ABMZ − I ( A : B | M ) ˆ σ ABM ≤ I ( A : Z | M ) ˆ σ AMZ + I ( B : Z | M ) ˆ σ BMZ . (78)Proceeding as in the proof of Theorem 1 we get I ( A : Z | M ) ˆ σ AMZ ( t ) = S ( A | M ) ˆ σ AM ( t ) − S ( A | M ) ˆ ρ AM , (79)where ˆ σ AM ( t ) = Z R n ˆ D A (cid:18) λ z √ η (cid:19) ˆ ρ AM ˆ D A (cid:18) λ z √ η (cid:19) † e − | z | t d n z (2 π t ) n = (cid:18) N A (cid:18) λ tη (cid:19) ⊗ I M (cid:19) (ˆ ρ AM ) , (80)hence I ( A : Z | M ) ˆ σ AMZ ( t ) = ∆ A | M (ˆ ρ AM ) (cid:18) λ tη (cid:19) . (81)We can analogously show that I ( B : Z | M ) ˆ σ BMZ ( t ) = ∆ B | M (ˆ ρ BM ) (cid:18) (1 − λ ) t | − η | (cid:19) , (82)and (78) becomes the claim (69).
5. Universal asymptotic scaling of quantum conditional entropy
In this Section, we prove that the quantum conditional entropy has a universalasymptotic scaling in the infinite-time limit under the heat semigroup evolution(Theorem 5). The proof is based on the following more general result, that pro-vides a new universal lower bound for the conditional entropy of the output ofany quantum Gaussian channel.
Theorem 4 (universal lower bound for quantum conditional entropy).
Let A , B be quantum Gaussian systems with m and n modes, respectively, and Φ : A → B a quantum Gaussian channel. Let ˆ ρ AM be a quantum state on AM such that Tr A h ˆ H A ˆ ρ A i < ∞ , S (ˆ ρ M ) < ∞ . (83) Then, for any quantum system M and any joint quantum state ˆ ρ AM S ( B | M ) ( Φ ⊗ I M )(ˆ ρ AM ) ≥ lim ν →∞ S ( B | A ′ ) ( Φ ⊗ I A ′ )(ˆ ω AA ′ ( ν )) , (84) where A ′ is a Gaussian quantum system with m modes, and for any ν ≥ , ˆ ω AA ′ ( ν ) is a purification of the thermal Gaussian state ˆ ω A ( ν ) on A with covari-ance matrix ν I m .Proof. Since the quantum conditional entropy is concave, we can restrict to ˆ ρ AM pure. Let K : R m → R n be the matrix such that for any x ∈ R n ( Φ ⊗ I M ) (cid:16) ˆ D A ( x ) ˆ ρ AM ˆ D A ( x ) † (cid:17) = ˆ D B ( K x ) ( Φ ⊗ I M )(ˆ ρ AM ) ˆ D B ( K x ) † . (85)Let ˆ ρ A be the marginal of ˆ ρ AM on A . Since the quantum conditional mutualinformation is invariant under local unitaries, we can assume that ˆ ρ A has zerofirst moments. We have S ( B | M ) ( Φ ⊗ I M )(ˆ ρ AM ) = S (( Φ ⊗ I M )(ˆ ρ AM )) − S (ˆ ρ M ) = S (cid:16) ˜ Φ (ˆ ρ A ) (cid:17) − S (ˆ ρ A ) , (86)where ˆ ρ M is the marginal state of ˆ ρ AM , and ˜ Φ is the complementary channelof Φ . Let ˆ γ A be the quantum Gaussian state with the same first and secondmoments as ˆ ρ A . Since ˜ Φ is a Gaussian channel, ˜ Φ (ˆ γ A ) is the quantum Gaussianstate with the same first and second moments as ˜ Φ (ˆ ρ A ). We then have fromLemma 8 S (cid:16) ˜ Φ (ˆ ρ A ) (cid:17) − S (ˆ ρ A )= S (cid:16) ˜ Φ (ˆ γ A ) (cid:17) − S (ˆ γ A ) + S (ˆ ρ A k ˆ γ A ) − S (cid:16) ˜ Φ (ˆ ρ A ) (cid:13)(cid:13)(cid:13) ˜ Φ (ˆ γ A ) (cid:17) ≥ S (cid:16) ˜ Φ (ˆ γ A ) (cid:17) − S (ˆ γ A ) , (87)where we have used the data-processing inequality for the quantum relativeentropy. Let α be the covariance matrix of ˆ γ A . We then haveˆ ω A ( k α k ∞ ) = Z R m ˆ D A ( x ) ˆ γ A ˆ D A ( x ) † d p X ( x ) , (88) he conditional Entropy Power Inequality for bosonic quantum systems 15 where p X is the probability distribution of the classical Gaussian random variable X with values in R m , zero mean and covariance matrix k α k ∞ I m − α . We thenhave from Lemma 10 S (cid:16) ˜ Φ (ˆ γ A ) (cid:17) − S (ˆ γ A ) ≥ S (cid:16) ˜ Φ (ˆ ω A ( k α k ∞ )) (cid:17) − S (ˆ ω A ( k α k ∞ )) . (89)Lemma 10 also implies that the function ν S (cid:16) ˜ Φ (ˆ ω A ( ν )) (cid:17) − S (ˆ ω A ( ν )) , ν ≥
12 (90)is decreasing, hence S (cid:16) ˜ Φ (ˆ ω A ( k α k ∞ )) (cid:17) − S (ˆ ω A ( k α k ∞ )) ≥ lim ν →∞ (cid:16) S (cid:16) ˜ Φ (ˆ ω A ( ν )) (cid:17) − S (ˆ ω A ( ν )) (cid:17) = lim ν →∞ ( S (( Φ ⊗ I A ′ )(ˆ ω AA ′ ( ν ))) − S (ˆ ω A ′ ( ν ))) = lim ν →∞ S ( B | A ′ ) ( Φ ⊗ I A ′ )(ˆ ω AA ′ ( ν )) . (91) Lemma 7.
For any t > , lim ν →∞ S ( A | A ′ ) ( N ( t ) ⊗ I A ′ )(ˆ ω AA ′ ( ν )) = n ln t + n . (92) Proof.
For any ν ≥ , the quantum Gaussian state ˆ ω AA ′ ( ν ) is the tensor productof n identical two-mode squeezed quantum Gaussian states, each with covariancematrix α ( ν ) := ν q ν − ν − q ν − q ν − ν − q ν − ν , (93)where the block decomposition refers to the AA ′ bipartition. For any t ≥ N ( t ) ⊗ I A ′ )(ˆ ω AA ′ ( ν )) is the tensor product of n identical two-mode quantum Gaussian states, each with covariance matrix α ( ν, t ) := ν + t q ν − ν + t − q ν − q ν − ν − q ν − ν . (94)The symplectic eigenvalues of α ( ν, t ) are ν ± ( ν, t ) = 12 q ν t ± t p ν t + t + 1 + 2 t + 1 = √ ν t + O (1) (95) for ν → ∞ , hencelim ν →∞ S ( A | A ′ ) ( N ( t ) ⊗ I A ′ )(ˆ ω AA ′ ( ν )) = lim ν →∞ n (cid:18) g (cid:18) ν + ( ν, t ) − (cid:19) + g (cid:18) ν − ( ν, t ) − (cid:19) − g (cid:18) ν − (cid:19)(cid:19) = n ln t + n , (96)where we used that for ν → ∞ g (cid:18) ν − (cid:19) = ln ν + 1 + O (cid:18) ν (cid:19) . (97) Theorem 5 (universal asymptotic scaling of quantum conditional en-tropy).
Let A be a Gaussian quantum system with n modes, and M a quantumsystem. Let ˆ ρ AM be a quantum state on AM such that its marginal on A hasfinite first and second moments, and its marginal on M has finite entropy. Then, lim t →∞ (cid:0) S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) − n ln t − n (cid:1) = 0 . (98) Remark 6.
The scaling (98) had been proven in [18], Lemma 6.1 in the particularcase where ˆ ρ AM is a Gaussian state. Proof.
From the subadditivity of the quantum entropy we have for any t ≥ S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) ≤ S ( N ( t )(ˆ ρ A )) . (99)Let E := 1 n Tr A h ˆ H A ˆ ρ A i (100)be the average energy per mode of ˆ ρ A , and let ˆ ω A be the thermal quantum Gaus-sian state with average energy per mode E and covariance matrix (cid:0) E + (cid:1) I n .For any t ≥ N ( t )(ˆ ω A ) is the thermal quantum Gaussian state with the sameaverage energy as N ( t )(ˆ ρ A ). We then have from Lemma 9 S ( N ( t )(ˆ ρ A )) ≤ S ( N ( t )(ˆ ω A )) = n g ( E + t ) , (101)hencelim sup t →∞ (cid:0) S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) − n ln t − n (cid:1) ≤ lim sup t →∞ n ( g ( E + t ) − ln t − . (102)On the other hand, from Theorem 4 and Lemma 7 we have for any t ≥ S ( A | M ) ( N ( t ) ⊗ I M )(ˆ ρ AM ) ≥ lim ν →∞ S ( A | A ′ ) ( N ( t ) ⊗ I A ′ )(ˆ ω AA ′ ( ν )) = n ln t + n . (103) he conditional Entropy Power Inequality for bosonic quantum systems 17
6. Quantum conditional Entropy Power Inequality
In this Section we prove the conditional Entropy Power Inequality, the mainresult of this paper.
Theorem 6 (quantum conditional Entropy Power Inequality).
Let A , B and C be Gaussian quantum systems with n modes, M a quantum system, and B η : AB → C the beam-splitter with transmissivity ≤ η ≤ or the squeezer ofparameter η ≥ . Let ˆ ρ ABM be a quantum state on
ABM such that Tr AB h(cid:16) ˆ H A + ˆ H B (cid:17) ˆ ρ AB i < ∞ , S (ˆ ρ M ) < ∞ , (104) and let us suppose that A and B are conditionally independent given M : I ( A : B | M ) ˆ ρ ABM = 0 . (105) Then, for any ≤ λ ≤ the quantum linear conditional Entropy Power Inequal-ity holds: S ( C | M ) ˆ ρ CM n ≥ λ S ( A | M ) ˆ ρ AM n + (1 − λ ) S ( B | M ) ˆ ρ BM n + λ ln ηλ + (1 − λ ) ln | − η | − λ . (106) The quantum conditional Entropy Power Inequality follows maximizing over λ the right-hand side of (106) : exp S ( C | M ) ˆ ρ CM n ≥ η exp S ( A | M ) ˆ ρ AM n + | − η | exp S ( B | M ) ˆ ρ BM n . (107) Remark 7.
The quantum linear inequality (106) had been proven for λ = η inthe special case where ˆ ρ ABM is a quantum Gaussian state in [18], Theorem 8.1.
Proof.
Let us define for any t ≥ ρ ABM ( t ) := (cid:18) N A (cid:18) λ tη (cid:19) ⊗ N B (cid:18) (1 − λ ) t | − η | (cid:19) ⊗ I M (cid:19) (ˆ ρ ABM ) , ˆ ρ CM ( t ) := ( B η ⊗ I M )(ˆ ρ ABM ( t )) . (108)We have for any t ≥ ρ AM ( t ) = (cid:18) N A (cid:18) λ tη (cid:19) ⊗ I M (cid:19) (ˆ ρ AM ) , (109a)ˆ ρ BM ( t ) = (cid:18) N B (cid:18) (1 − λ ) t | − η | (cid:19) ⊗ I M (cid:19) (ˆ ρ BM ) , (109b)ˆ ρ CM ( t ) = ( N C ( t ) ⊗ I M ) (ˆ ρ CM ) , (109c)where we have set ˆ ρ CM := ˆ ρ CM (0). The time evolution preserves the condition I ( A : B | M ) = 0. Indeed, we have from the data-processing inequality for thequantum mutual information0 ≤ I ( A : B | M ) ˆ ρ ABM ( t ) ≤ I ( A : B | M ) ˆ ρ ABM = 0 . (110) We define the function φ ( t ) := S ( C | M ) ˆ ρ CM ( t ) − λ S ( A | M ) ˆ ρ AM ( t ) − (1 − λ ) S ( B | M ) ˆ ρ BM ( t ) . (111)We have from Proposition 1 and Theorem 3 φ ′ ( t ) = J ( C | M ) ˆ ρ CM ( t ) − λ η J ( A | M ) ˆ ρ AM ( t ) − (1 − λ ) | − η | J ( B | M ) ˆ ρ BM ( t ) ≤ . (112)From Theorem 2, φ is a linear combination of continuous concave functions,hence it is almost everywhere differentiable and for any t ≥ φ ( t ) − φ (0) = Z t φ ′ ( s ) d s ≤ . (113)We then have from Theorem 5 φ (0) ≥ lim t →∞ φ ( t )= lim t →∞ (cid:0) S ( C | M ) ˆ ρ CM ( t ) − λ S ( A | M ) ˆ ρ AM ( t ) − (1 − λ ) S ( B | M ) ˆ ρ BM ( t ) (cid:1) = n (cid:18) λ ln ηλ + (1 − λ ) ln | − η | − λ (cid:19) , (114)and the quantum linear conditional Entropy Power Inequality (106) follows. Thequantum conditional Entropy Power Inequality (107) follows choosing λ = η exp S ( A | M ) ˆ ρAM ( t ) n η exp S ( A | M ) ˆ ρAM ( t ) n + | − η | exp S ( B | M ) ˆ ρBM ( t ) n , (115)that maximizes the right-hand side of (106).
7. Optimality of the quantum conditional Entropy Power Inequality
In this Section, we prove that the quantum conditional Entropy Power Inequalityis asymptotically saturated by a suitable sequence of quantum Gaussian inputstates.
Theorem 7 (optimality of the quantum conditional Entropy Power In-equality).
The quantum conditional Entropy Power Inequality (107) is optimal.In other words, for any a, b ∈ R there exists a sequence of quantum Gaussianinput states n ˆ γ ( n ) ABA ′ B ′ o n ∈ N of the form ˆ γ ( n ) ABA ′ B ′ = ˆ γ ( n ) AA ′ ⊗ ˆ γ ( n ) BB ′ , n ∈ N , (116) such that lim n →∞ exp (cid:16) S ( A | A ′ B ′ ) ˆ γ ( n ) AA ′ B ′ − (cid:17) = a , (117a)lim n →∞ exp (cid:16) S ( B | A ′ B ′ ) ˆ γ ( n ) BA ′ B ′ − (cid:17) = b (117b) he conditional Entropy Power Inequality for bosonic quantum systems 19 and lim n →∞ exp (cid:16) S ( C | A ′ B ′ ) ˆ γ ( n ) CA ′ B ′ − (cid:17) = η a + | − η | b , (118) where ˆ γ ( n ) CA ′ B ′ := ( B η ⊗ I A ′ B ′ ) (cid:16) ˆ γ ( n ) AA ′ ⊗ ˆ γ ( n ) BB ′ (cid:17) , (119) and A , A ′ , B , B ′ and C are one-mode Gaussian quantum systems.Proof. Let ˆ γ ( n ) AA ′ and ˆ γ ( n ) BB ′ be the quantum Gaussian states with covariance ma-trices σ ( n ) AA ′ = n na q n a − na − q n a − q n a − na − q n a − na , (120a) σ ( n ) BB ′ = n nb q n b − nb − q n b − q n b − nb − q n b − nb , (120b)where the block decompositions refer to the bipartitions AA ′ and BB ′ , respec-tively. The symplectic eigenvalues of σ ( n ) AA ′ are ( n, n ), hence S ( AA ′ ) ˆ γ ( n ) AA ′ = 2 g (cid:18) n − (cid:19) = ln n + 2 + O (cid:18) n (cid:19) , (121a) S ( A ′ ) ˆ γ ( n ) AA ′ = g (cid:18) n a − (cid:19) = ln n a + 1 + O (cid:18) n (cid:19) , (121b)and lim n →∞ S ( A | A ′ B ′ ) ˆ γ ( n ) AA ′ B ′ = lim n →∞ S ( A | A ′ ) ˆ γ ( n ) AA ′ = 1 + ln a . (122)Analogously, lim n →∞ S ( B | A ′ B ′ ) ˆ γ ( n ) BA ′ B ′ = 1 + ln b . (123)For 0 ≤ η ≤
1, the covariance matrix of ˆ γ ( n ) CA ′ B ′ is σ ( n ) CA ′ B ′ = n n (cid:0) ηa + − ηb (cid:1) I q η (cid:0) n a − (cid:1) T q (1 − η ) (cid:0) n b − (cid:1) T q η (cid:0) n a − (cid:1) T na I q (1 − η ) (cid:0) n b − (cid:1) T nb I , (124)where I = (cid:18) (cid:19) , T = (cid:18) − (cid:19) , (125) and the block decomposition refers to the tripartition CA ′ B ′ . We have on onehand S ( A ′ B ′ ) ˆ γ ( n ) CA ′ B ′ = g (cid:18) n a − (cid:19) + g (cid:18) n b − (cid:19) = ln n a b + 2 + O (cid:18) n (cid:19) . (126)Since σ ( n ) CA ′ B ′ n is still the covariance matrix of a positive quantum Gaussian state,its symplectic eigenvalues are all larger than , and the symplectic eigenvalues (cid:16) ν ( n )1 , ν ( n )2 , ν ( n )3 (cid:17) of σ ( n ) CA ′ B ′ are all larger than n . We then have S ( CA ′ B ′ ) ˆ γ ( n ) CA ′ B ′ = g (cid:18) ν ( n )1 − (cid:19) + g (cid:18) ν ( n )2 − (cid:19) + g (cid:18) ν ( n )3 − (cid:19) = ln (cid:16) ν ( n )1 ν ( n )2 ν ( n )3 (cid:17) + 3 + O (cid:18) n (cid:19) = 12 ln det σ ( n ) CA ′ B ′ + 3 + O (cid:18) n (cid:19) = ln n ( η a + (1 − η ) b ) a b + 3 + O (cid:18) n (cid:19) , (127)and lim n →∞ S ( C | A ′ B ′ ) ˆ γ ( n ) CA ′ B ′ = ln ( η a + (1 − η ) b ) + 1 . (128)Similarly, for η ≥ γ ( n ) CA ′ B ′ is σ ( n ) CA ′ B ′ = n n (cid:0) ηa + − ηb (cid:1) I q η (cid:0) n a − (cid:1) T q ( η − (cid:0) n b − (cid:1) I q η (cid:0) n a − (cid:1) T na I q ( η − (cid:0) n b − (cid:1) I nb I . (129)If (cid:16) ν ( n )1 , ν ( n )2 , ν ( n )3 (cid:17) are its symplectic eigenvalues, S ( CA ′ B ′ ) ˆ γ ( n ) CA ′ B ′ = g (cid:18) ν ( n )1 − (cid:19) + g (cid:18) ν ( n )2 − (cid:19) + g (cid:18) ν ( n )3 − (cid:19) = ln (cid:16) ν ( n )1 ν ( n )2 ν ( n )3 (cid:17) + 3 + O (cid:18) n (cid:19) = 12 ln det σ ( n ) CA ′ B ′ + 3 + O (cid:18) n (cid:19) = ln n ( η a + ( η − b ) a b + 3 + O (cid:18) n (cid:19) , (130)and lim n →∞ S ( C | A ′ B ′ ) ˆ γ ( n ) CA ′ B ′ = ln ( η a + ( η − b ) + 1 . (131) he conditional Entropy Power Inequality for bosonic quantum systems 21
8. Entanglement-assisted classical capacity
In this section, we exploit the quantum conditional Entropy Power Inequalityto prove an upper bound to the entanglement-assisted classical capacity of thefollowing non-Gaussian quantum channel. This implication has been first con-sidered in [18], section III.Let us fix a quantum state ˆ σ B on the n -mode Gaussian quantum system B .We consider the channel Φ : A → C that mixes with ˆ σ B the input state ˆ ρ A onthe n -mode Gaussian quantum system A through a beam-splitter or a squeezingoperation: Φ (ˆ ρ A ) = B η (ˆ ρ A ⊗ ˆ σ B ) , η ≥ . (132)If the sender can use an unlimited amount of energy, the entanglement-assisted classical capacity is infinite. Since this scenario is not physical, weassume that the sender can use at most an energy E per each mode. Theentanglement-assisted classical capacity [16, 24] of Φ is then equal to the supre-mum of the quantum mutual information: C ea ( Φ ) = sup n I ( C : M ) ( Φ ⊗ I M )(ˆ ρ AM ) : ˆ ρ AM pure , Tr A h ˆ H A ˆ ρ A i ≤ n E o . (133)Let E := 1 n Tr B h ˆ H B ˆ σ B i , S := S (ˆ σ B ) n (134)be the average energy and the entropy per mode of ˆ σ B , respectively. The averageenergy per mode of Φ (ˆ ρ A ) is1 n Tr C h ˆ H C Φ (ˆ ρ A ) i = ηn Tr A h ˆ H A ˆ ρ A i + | − η | E + η + | − η | − ≤ η E + | − η | E + η + | − η | − . (135)From Lemma 9, S ( Φ (ˆ ρ A )) ≤ n g (cid:18) η E + | − η | E + η + | − η | − (cid:19) . (136)From the quantum conditional Entropy Power Inequality we have (we recall that M is correlated only with A and that ˆ ρ AM is pure)exp S ( C | M ) ( Φ ⊗ I M )(ˆ ρ AM ) n ≥ η exp S ( A | M ) ˆ ρ AM n + | − η | exp S = η exp − S (ˆ ρ A ) n + | − η | exp S ≥ η exp( − g ( E )) + | − η | exp S , (137)where in the last step we have used Lemma 9 again. Finally, I ( C : M ) ( Φ ⊗ I M )(ˆ ρ AM ) = S ( Φ (ˆ ρ A )) − S ( C | M ) ( Φ ⊗ I M )(ˆ ρ AM ) ≤ n g (cid:18) η E + | − η | E + η + | − η | − (cid:19) − n ln (cid:16) η e − g ( E ) + | − η | e S (cid:17) , (138) so that C ea ( Φ ) ≤ n g (cid:18) η E + | − η | E + η + | − η | − (cid:19) − n ln (cid:16) η e − g ( E ) + | − η | e S (cid:17) . (139)
9. Conclusions
We have proven the conditional Entropy Power Inequality for Gaussian quan-tum systems, which are the most promising platform for quantum communica-tion and quantum key distribution. This fundamental inequality determines theminimum quantum conditional entropy of the output of the beam-splitter or ofthe squeezing among all the quantum input states where the two inputs are con-ditionally independent given the memory and have given quantum conditionalentropies. This inequality is optimal, since it is asymptotically saturated by asuitable sequence of quantum Gaussian input states. In the unconditioned case,the optimal inequality is still an open challenging conjecture [15] that is turningout to be very hard to prove [7–12]. The quantum conditional Entropy PowerInequality instead definitively settles the problem in the conditioned case.
Acknowledgements.
GdP acknowledges financial support from the European Research Council(ERC Grant Agreements no 337603 and 321029), the Danish Council for Independent Research(Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No.10059).
A. Proof of the classical conditional Entropy Power Inequality
We have from the definition of the classical conditional entropy S ( C | M ) = Z M S ( C | M = m ) d p M ( m ) , (140)where p M is the probability distribution of M . From the classical Entropy PowerInequality (3) we have for any mS ( C | M = m ) ≥ k (cid:18) η exp 2 S ( A | M = m ) k + | − η | exp 2 S ( B | M = m ) k (cid:19) . (141)Finally, since the function( a, b ) k (cid:18) η exp 2 ak + | − η | exp 2 bk (cid:19) , a, b ∈ R (142)is convex, we have from (140), (141) and Jensen’s inequality S ( C | M ) ≥ Z M k (cid:16) η e k S ( A | M = m ) + | − η | e k S ( B | M = m ) (cid:17) d p M ( m ) ≥ k (cid:16) η e k R M S ( A | M = m )d p M ( m ) + | − η | e k R M S ( B | M = m )d p M ( m ) (cid:17) = k (cid:16) η e k S ( A | M ) + | − η | e k S ( B | M ) (cid:17) . (143) he conditional Entropy Power Inequality for bosonic quantum systems 23 The conditional Entropy Power Inequality (9) is saturated iff all the inequal-ities in (143) are equalities. The first inequality is saturated iff, conditioningon any value m of M , A and B are independent Gaussian random variableswith proportional covariance matrices. The second inequality is saturated iff S ( A | M = m ) − S ( B | M = m ) does not depend on m . For A and B as above, thisis equivalent to having the proportionality constant between their covariancematrices independent on m . B.Lemma 8 ( [16], Lemma 12.25).
Let ˆ ρ be a quantum state on a Gaussianquantum system with finite average energy, and let ˆ γ be the Gaussian quantumstate with the same first and second moments. Then S (ˆ γ ) = S (ˆ ρ ) + S (ˆ ρ k ˆ γ ) . (144) Lemma 9.
Let ˆ ρ be a quantum state on a Gaussian quantum system with finiteaverage energy, and let ˆ ω be the thermal Gaussian quantum state with the sameaverage energy. Then S (ˆ ω ) ≥ S (ˆ ρ ) . (145) Proof.
Let β > ω = e − β ˆ H Tr e − β ˆ H . (146)We then have S (ˆ ω ) = S (ˆ ρ ) + S (ˆ ρ k ˆ ω ) + β Tr h ˆ H (ˆ ω − ˆ ρ ) i ≥ S (ˆ ρ ) . (147) Lemma 10.
Let A and B be Gaussian quantum systems with m and n modes,respectively, and Φ : A → B a Gaussian quantum channel. Let ˆ ρ A be a quantumstate on A , and p X a probability measure on R m . We define ˆ σ A := Z R m ˆ D A ( x ) ˆ ρ A ˆ D A ( x ) † d p X ( x ) . (148) Then, S ( Φ (ˆ ρ A )) − S (ˆ ρ A ) ≥ S ( Φ (ˆ σ A )) − S (ˆ σ A ) . (149) Proof.
Let ˆ σ AX be the joint state on AX such that its marginal on X is p X ,and for any x ∈ R m ˆ σ A | X = x = ˆ D A ( x ) ˆ ρ A ˆ D A ( x ) † . (150)We notice that the marginal of ˆ σ AX on A is ˆ σ A , and S ( A | X ) ˆ σ AX = Z R m S (cid:0) ˆ σ A | X = x (cid:1) d p X ( x ) = S (ˆ ρ A ) , (151) S ( B | X ) ( Φ ⊗ I X )(ˆ σ AX ) = Z R m S (cid:0) Φ (ˆ σ A | X = x ) (cid:1) d p X ( x ) = S ( Φ (ˆ ρ A )) , (152) where in the last step we used that for any x ∈ R m Φ (cid:16) ˆ D A ( x ) ˆ ρ A ˆ D A ( x ) † (cid:17) = ˆ D B ( K x ) Φ (ˆ ρ A ) ˆ D B ( K x ) † (153)for some matrix K : R m → R n . We then have S (ˆ σ A ) − S (ˆ ρ A ) = I ( A : X ) ˆ σ AX , (154) S ( Φ (ˆ σ A )) − S ( Φ (ˆ ρ A )) = I ( B : X ) ( Φ ⊗ I X )(ˆ σ AX ) , (155)and the claim follows from the data-processing inequality for the mutual infor-mation. References
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