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ARTICLE OPEN Entropy generation in Gaussian quantum transformations: applying the replica method to continuous-variable quantum information theory

Christos N Gagatsos1, Alexandros I Karanikas2, Georgios Kordas2 and Nicolas J Cerf1

In spite of their simple description in terms of rotations or symplectic transformations in phase space, quadratic Hamiltonians such as those modelling the most common Gaussian operations on bosonic modes remain poorly understood in terms of entropy production. For instance, determining the quantum entropy generated by a Bogoliubov transformation is notably a hard problem, with generally no known analytical solution, while it is vital to the characterisation of quantum communication via bosonic channels. Here we overcome this difficulty by adapting the replica method, a tool borrowed from statistical physics and quantum field theory. We exhibit a first application of this method to continuous-variable quantum information theory, where it enables accessing entropies in an optical parametric amplifier. As an illustration, we determine the entropy generated by amplifying a binary superposition of the vacuum and a Fock state, which yields a surprisingly simple, yet unknown analytical expression.

npj Quantum Information (2015) 2, 15008; doi:10.1038/npjqi.2015.8; published online 16 February 2016

INTRODUCTION Gaussian states (e.g., the vacuum state, resulting after amplifica- Gaussian transformations are ubiquitous in quantum physics, tion in a thermal state of well-known entropy), very few analytical 18 playing a major role in quantum optics, quantum field theory, results are available as of today. solid-state physics or black-hole physics.1 In particular, the In this article, we demonstrate that the replica method can be fi Bogoliubov transformations resulting from Hamiltonians that are successfully exploited to overcome this central problem and nd quadratic (bilinear) in mode operators are among the most an exact analytical expression for the entropy generated by significant Gaussian transformations, well known to model Gaussian processes acting on non-trivial bosonic states. The 2 replica method is known to be an invaluable tool in statistical superconductivity but also describing a much wider range of 19 physical situations, from squeezing or amplification in the context physics, especially with disordered systems, and in quantum – fi 20 fi of quantum optics3 6 to Unruh radiation in an accelerating eld theory. Here we rst apply it to continuous-variable 7–9 (Gaussian) quantum information theory and show that it enables frame or even Hawking radiation as emitted by a black fi hole.10–12 In the present article, we focus on Gaussian bosonic accessing the entropy generated by a quantum optical ampli er, transformations, which are at the heart of so-called Gaussian opening a new way towards the entropic characterisation of 13 Gaussian channels as considered in quantum communication quantum information theory. These transformations encompass theory. To illustrate the power of this approach in an interesting the passive coupling between modes of the electromagnetic field and experimentally relevant case, we investigate the output as effected by a beam splitter in bulk optics or an optical coupler entropy when amplifying a binary superposition of the vacuum in fibre optics, as well as the active transformations resulting from and an arbitrary Fock state, which yields a surprisingly simple parametric downconversion in a nonlinear optical medium, which analytical expression. are traditionally used as a source of quantum entanglement. Although they are common, quantum Gaussian processes are poorly understood in terms of entropy generation. Indeed, the RESULTS symplectic formalism in phase-space representation is not suited Gaussian bosonic transformations to calculate von Neumann entropies as this requires diagonalising 14 The linear coupling between two modes of the electromagnetic density operators in state space. When amplifying an optical fi state using parametric downconversion, for example, the output eld with, e.g., a beam splitter, is modelled by the (passive) ^y y^ ^ state suffers from quantum noise, which is an increasing function quadratic Hamiltonian H ¼ ia^b - ia^ b, where a^ and b are bosonic of the amplification gain.15 Characterising this noise in terms of mode operators. This operation can be shown to preserve entropy is indispensable for determining the capacity of Gaussian the Gaussian character of a quantum state, or more precisely bosonic channels.16–18 However, the output entropy is not the quadratic exponential form of its characteristic function. The accessible for an arbitrary input state because it is hard—usually corresponding transformation in phase space is the ^ ^ ^ 2 impossible—to diagonalise the corresponding output state in a^- cos y a^ þ sin y b and b- cos y b - sin y a^, where cos θ is infinite-dimensional Fock space. With the notable exception of the transmittance. Another generic Gaussian transformation

1Quantum Information and Communication, Ecole polytechnique de Bruxelles, Université libre de Bruxelles, Brussels, Belgium and 2Nuclear and Particle Physics Section, Physics Department, University of Athens, Athens, Greece. Correspondence: NJ Cerf ([email protected]) Received 17 March 2015; revised 18 June 2015; accepted 1 August 2015

© 2015 University of New South Wales/Macmillan Publishers Limited Entropy generation in Gaussian quantum transformations CN Gagatsos et al 2 results from parametric downconversion in a nonlinear medium, consider a two-mode squeezer of parameter ξ ¼ jjξ eiϕ, applying which is modelled by the (active) quadratic Hamiltonian the unitary transformation ^ y^y H ¼ ia^b - ia^ b . It effects the Bogoliubov transformation - ξ^y^y ξ ^^ U^ ¼ e a b þ ab ð6Þ ^y ^ ^ y 2 a^-cosh r a^ þ sinh r b and b-cosh r b þ sinh r a^ , where cosh r is the parametric amplification gain. More generally, the set of all on the initial state jim aji0 b (subscript a refers to the signal mode, ρ^ linear canonical transformations effected by quadratic bosonic while b refers to the idler mode). The reduced output state m Hamiltonians, also referred to as symplectic transformations, can of the signal mode is diagonal in the Fock basis, with a fi vector of eigenvalues given by18,21 easily be characterised in terms of af ne transformations in phase space (e.g., rotations and -preserving squeeze mapping). m mþ1 k þ m pðÞ¼ 1 - jjτ 2 jjτ 2k; k Aℕ ð7Þ k k Accessing the generated entropy Calculating the von Neumann entropy SðÞ¼ρ^ - trðÞρ^lnρ^ of a where τ ¼ tanhξ, from which we find ^ bosonic mode that is found in state ρ at the output of a Gaussian X1 n nmðÞþ1 k þ m transformation is often an intractable task because it requires tr ρ^n ¼ 1 - jjτ 2 jjτ 2nk: ð8Þ m k turning to state space and finding the infinite-size vector of k¼0 eigenvalues of ρ^. The above symplectic formalism for Gaussian Equation (8) can be re-expressed in a closed form as transformations is useless in this respect, albeit for Gaussian states. nmðÞþ1 However, this problem can be circumvented by adapting the 1 - jjτ 2 ρ^n ρ^n ðÞm τ 2n : replica method, which relies on expressing trðÞas a function of tr m ¼ 2n Li - n jj ð9Þ the number of replicas nAℕ and taking the derivative at n =1, jjτ ρ^ thereby avoiding the need to diagonalise (see Materials and where Methods). As we shall show, dealing with a quadratic Hamiltonian X1 n makes the replica method a perfect tool to access von Neumann k þ m LiðÞm ðÞζ ζkþ1 ð10Þ entropies because it involves Gaussian integrations, or else some - n k k 0 tricks can be used in order to bring trðÞρ^n to a calculable form. ¼ ðÞ1 22 First, we illustrate this principle with a generic zero-mean and Li - nðÞζ denotes the polylogarithm of order − n. Applying rotation- Gaussian state, namely a thermal state ρ^ ¼ the replica method to Equation (9) and taking into account P 0 LiðÞm ðÞ¼ζ ζ=ðÞ1 - ζ mþ1, we obtain the entropy 1 - jjτ 2 1 jjτ 2kjik hjk characterised by a mean photon number - 1 k¼0 ∂ τ 2= - τ 2 ρ^ n N jj 1 jj .As 0 is in a diagonal form, it is of course S ¼ - tr ρ^ 9 m ∂n m n¼1 straightforward to calculate its entropy, giving the well-known ρ^ - 2 mþ1 formula SðÞ¼0 gNðÞðÞN þ 1 log ðÞN þ 1 N log N. However, τ 2 - τ ∂ jj 1 jj ðÞm 2n we may also start with its non-diagonal representation in the ¼ ln- Li - jjτ 9 : ð11Þ 2 mþ1 τ 2 ∂n n n¼1 coherent-state basis fgjiα , where α is a complex number, namely 1 - jjτ jj Z 0 2 2 ðÞ - τ - 1 - jjτ α 2 Using Li - nðÞ¼ζ ζ=ðÞ1 - ζ , it is easy to check that Equation (11) 1 1 jj 2 2 jj ρ^ ¼ d α e jjτ jiα hjα ð1Þ 0 π 2 gives the correct value for S0, that is, the entropy of a thermal state jjτ ρ^ ðÞ1 ζ 0 as in Equation (5). Interestingly, as the polylogarithm Li - nðÞis By making the change of variable α-jjτ α and using a well-studied function, a closed expression can also be found for - α 2 β 2 - αβ = αβ ðÞjjþjj 2 2 S1 in terms of Eulerian numbers (Supplementary Information). For h ji¼ e , we can write 4 ðÞm ζ Z Z m 1, the function Li - n ðÞassumes a summation form, which is 2 n 1 - jjτ - αy α convergent and differentiable with respect to n, yielding an tr ρ^n ¼ d2α ¼ d2α e M ð2Þ 0 πn 1 n analytical expression for Sm (Supplementary Information). α α ; ¼ α T where ¼ ðÞ1 n is a column vector and M is the n × n Amplifying a superposition of Fock states circulant 0 1 Now, we show that the same procedure makes it possible to express 1 - jjτ 2 0 ¼ 0 the entropy in situations where no diagonal form is available for the B - τ 2 ¼ C output state, so the replica method becomes essential. As an illus- B 01jj 0 C: M ¼ @ ^^^& ^ A ð3Þ tration, consider the amplification of a binary superposition of the type - jjτ 2 00¼ 1 ji0 þ zmji jiψ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð12Þ Equation (2) is a simple Gaussian integral, which, using the 1 þ z2 2n determinant det M¼ 1 - jjτ , can be expressed as where we take z Aℝ without loss of generality. By using the Baker– n Campbell–Hausdorff relation, the unitary transformation (6) can be 1 - jjτ 2 tr ρ^n ¼ ð4Þ rewritten in the form 0 - τ 2n 1 jj y - ν - τ^y ^y - ν ^y ^ ^ ^ τ ^^ U^ ¼ e e a b e a aþb b e ab ð13Þ Applying the replica method, we readily find that ξ where ν ¼ ln cosh jjξ and τ ¼ tanh jjξ , so that the joint output ∂ τ 2 jjξ n 1 jj 1 Ψ - tr ρ^ 9 ¼ ln þ ln ð5Þ state jiof the two modes can be expressed in the double-coherent- ∂ 0 n¼1 - τ 2 - τ 2 τ 2 α β n 1 jj 1 jj jj state basis jiajib,namely ρ^ which coincides with the above expression SðÞ¼0 gNðÞfor the ^ ^ entropy of a thermal state, as expected. hα; βjiΨ ¼ pffiffiffiffiffiffiffiffi1 α; β U 0; 0 þ z α; β U m; 0 1þz2 0 1 mþ1 2 Amplifying a Fock state 1 1 1 - jjτ 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi@ 1 - jjτ 2 2 þ z pffiffiffiffiffiffi A Next, we consider the problem of expressing the entropy Sm 1 þ z2 m! generated by amplifying an arbitrary Fock state jim . For this, we

npj Quantum Information (2015) 15008 © 2015 University of New South Wales/Macmillan Publishers Limited Entropy generation in Gaussian quantum transformations CN Gagatsos et al 3 - α 2 β 2 = - ταβ ταβ: ´ e ðÞjjþjj 2 þ ð14Þ the entropy generated by other superpositions, such asji 1 þ zji2 or ji0 þ zji1 þ z0ji2 . Another striking case is the amplification of a From Equation (14), we can easily write the reduced output state ρ^ coherent state: although jiα is an infinite superposition of Fock states, obtained by tracing jiΨ over the idler mode and paying attention to the resulting entropy is S , just as for the vacuum state 0 . the non-orthogonality of coherent states. Using again the notation 0 ji T α ¼ ðÞα1; ¼ αn ,weget Z DISCUSSION 2 n Yn 1 - jjτ 2 - αy α trðÞ¼ρ^n d2α 1 þ cαm e M ð15Þ We have demonstrated that the replica method, a tool borrowed πn 1 z2 n j j ðÞþ j¼1 from other of physics, provides a new of attack to where the matrix M is definedasinEquation(3)and access the quantum entropies generated by Gaussian bosonic transformations, especially in cases where the symplectic formal- z m=2 c ¼ pffiffiffiffiffiffi 1 - jjτ 2 : ð16Þ ism is useless. As a matter of fact, it did require considerable effort m! simply to show that the vacuum state minimises the entropy In order to bring this back to a Gaussian integral, we use the ‘sources’ produced by Gaussian bosonic channels, as recently proven in 23 m - x2 ∂m - x2þλx ref. 24. The difficulty behind this proof was precisely that no trick, exploiting the identity x e ¼ m e jλ .Then, ∂λ ¼0 diagonal representation of the output state ρ^ is available when Equation (15) becomes non-Gaussian input states are considered. A similar problem is also n Z 1 - jjτ 2 Y Yn at the heart of other—yet unproven—Gaussian entropic con- trðÞ¼ρ^n ðÞn d2α πn 2 n ∂λ j jectures for bosonic channels. For example, determining the ðÞ1 þ z j 1 ¼ capacity of a multiple-access or broadcast Gaussian bosonic y ´ exp - αyMα þ αyλ þ λ α ð17Þ channel is pending on being able to access entropies when non- λ¼0 – Q Q Gaussian mixed input states are considered, see, e.g., refs 25 27. n 9 ∂m=∂λm 2 The replica method holds the promise to unblock these situations where ∂λðÞn j¼1 1 þ c j j is a differential operator in the as it provides a generic trick to overcome the difficulty: trðÞρ^n is variables λ λ ; ¼ λ T .Noteherethatλ and λ are treated as ¼ ðÞ1 n j j expressed for n replicas of state ρ^ by solving Gaussian integrals, independent variables, instead of their real and imaginary parts. The without ever accessing the eigenvalues of ρ^. λ’ derivatives with respect to all s have been pushed in front of the We have illustrated this procedure for the amplification of a integrals in Equation (17), so that we get a Gaussian integral that is superposition state of the formji 0 þ zmji, which allowed us to immediately calculable, resulting in unveil a remarkably simple behaviour for the entropy of the  tr ρ^n Y amplified state. The replica method thus clearly extends the limits ρ^n 0 λy λ ; trðÞ¼ n ∂λðÞn exp N λ ð18Þ of exactly computable entropies in comparison with the usual ðÞ1 þ z2 ¼0 methods. Yet, one may question whether this method is ρ^n fi where tr 0 is given by Equation (4) and corresponds to a vacuum suf ciently robust to address more complicated situations, such input state (z = 0). We have defined here the circulant matrix as arbitrary input states or general Gaussian transformations m N ¼ 1 - jjτ 2n M - 1,with (other than the Bogoliubov transformation considered here). Even 0 1 if a closed expression for trðÞρ^n can be found, taking its derivative 2 2ðÞn - 1 1 jjτ ¼ jjτ with respect to n is a task that may become impractical. Although B 2ðÞn - 1 2ðÞn - 2 C - 1 1 B jjτ 1 ¼ jjτ C it is uncertain whether we can apply the replica method in such M ¼ @ A ð19Þ 1 - jjτ 2n ^^& ^ cases, we nevertheless believe that this method remains valuable jjτ 2 jjτ 4 ¼ 1 if the goal is to find a bound on the entropy (instead of its exact expression). Take, for example, a general input state Equation (18) can finally be re-expressed in a form that is suitable to P jiψ ¼ c jik in Fock space. Our current calculations have shown the replica method (see Materials and Methods), which yields a nice k k that applying a Bogoliubov transformation yields an output expression for the output entropy ρ^n ρ^n state whose moments can be written as trðÞ¼tr 0 T ψðÞn , 1 z2 where ρ^ is the output state that results from a vacuum input SzðÞ¼ S þ S ð20Þ 0 1 þ z2 0 1 þ z2 m Intriguingly, the entropy of the amplifiedstateisthusaconvex combination of the extremal points S0 and Sm with the exact same weights as if we had lost coherence between the componentsji 0 and 2.0 jim of the input superposition. This is schematically pictured in Figure 1. We have also numerically verified this behaviour, which, to our knowledge, has never been observed before. This is illustrated in 1.5 Figure 2, where we show that the entropy is a monotonically

increasing function of the superposition parameter z for fixed values S(z) 1.0 of the squeezing parameter ξ. It must be stressed that this analytical 1 result is highly non-trivial as we do not expect similar expressions for 0.5 0.75 0.5 0.0 0 1 2 3 4 5 z Figure 1. Generated entropy as a linear function over a one- dimensional convex polytope. The von Neumann entropy S(z), Figure 2. Amplification of a superposition of the vacuum and a function of the superposition parameter z, is pictured by a point single-photon state. Plot of the von Neumann entropy S(z)asa belonging to a one-dimensional convex polytope. The two extremal function of the superposition parameter z for m = 1 and several points correspond to the entropies S0 and Sm, obtained by values of the squeezing parameter ξ.AsS14S0, Equation (20) implies amplifying states ji0 and jim , respectively. that the curve S(z) always lies above S0 for a given ξ.

© 2015 University of New South Wales/Macmillan Publishers Limited npj Quantum Information (2015) 15008 Entropy generation in Gaussian quantum transformations CN Gagatsos et al 4 state and TψðÞn is some analytical function of n, can also be seen by taking the limit z → ∞ in Equation (18). If we put all which encapsulates the actual input state jiψ . Similarly as in pieces together, we may re-express Equation (18) as Equation (17), the function T ψðÞn can be written by applying ("# ! ! Q m n mn n 2 ρ^n - τ 2 - τ 2 α n tr 2 1 jj 2n 1 jj some differential operator to the function j¼1 f j with trðÞ¼ρ^ 0 1 þ z - z P pffiffiffiffi 2 n 2n 2n α 1 αk= ! ρ^n ðÞ1 þ z 1 - jjτ 1 - jjτ fðÞ¼ k¼0 ck k . The above expression for trðÞ, written in terms of tr ρ^n , is especially relevant in the context of  0 trρ^n Gaussian entropic conjectures. It can, for example, be linked to þFðÞm ðÞþn z2n m ð24Þ trρ^n the property that the vacuum input minimises the output entropy of 0 the Bogoliubov transformation, as shown in a forthcoming work. ρ^ fi where m is the reduced output state resulting from the ampli cation of Another possible extension of the present work would be to jim , and FðmÞðÞn is defined in the Supplementary Information. Now, consider more general Gaussian transformations as well as mixed applying the replica method (Equation (21)) to Equation (24), we get input states. A possibility is to use the Glauber–Sudarshan P representation of the input and output mixed states (instead 1 z2 ∂ SzðÞ¼ S þ S - FðÞm ðÞn 9 : ð25Þ of their non-diagonal coherent-state basis decompositions), which 1 þ z2 0 1 þ z2 m ∂n n¼1 may lead to a simpler way to express trðÞρ^n . Our preliminary results indicate that the replica method is again promising here, Finally, we prove in the Supplementary Information that the last term of for example, to compute (or lower bound) the output entropy of a the right-hand side of Equation (25) vanishes, so that it simplifies into beam splitter if the input state is mixed. This avenue will be further Equation (20). explored in a future work. In conclusion, we anticipate that the replica method will ACKNOWLEDGEMENTS become quite a valuable tool in order to reach a complete We thank R. García-Patrón, J. Schäfer and O. Oreshkov for useful discussions. This entropic characterisation of Gaussian bosonic transformations, or work was supported by the FRS-FNRS under Project No. T.0199.13 and by the Belgian perhaps even solve pending conjectures on Gaussian bosonic Federal IAP programme under Project No. P7/35. CNG acknowledges financial channels. 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