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Entropy Generation in Gaussian Quantum Transformations: Applying the Replica Method to Continuous-Variable Quantum Information Theory

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Entropy Generation in Gaussian Quantum Transformations: Applying the Replica Method to Continuous-Variable Quantum Information Theory www.nature.com/npjqi All rights reserved 2056-6387/15 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 rotation ^ ^ ^ 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 ξà ^^ 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 area-preserving squeeze mapping). ÀÁ 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 ÀÁ ÀÁX1 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. ÀÁnmðÞþ1 However, this problem can be circumvented by adapting the ÀÁ 1 - jjτ 2 ÀÁ 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 nAℕà 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-invariant Gaussian state, namely a thermal state ρ^ ¼ the replica method to Equation (9) and taking into account ÀÁP 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 ÀÁk¼0 ∂ ÀÁ 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 ÀÁ ρ^ - 2 mþ1 formula SðÞ¼0 gNðÞðÞN þ 1 log ðÞN þ 1 N log N. However, τ 2 - τ ∂ ÀÁ jj 1 jj ðÞm 2n we may also start with its non-diagonal representation in the ¼ lnÀÁ- 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 - αÃβ = αβ ðÞjjþjj 2 2 S1 in terms of Eulerian numbers (Supplementary Information). For h ji¼ e , we can write 4 ðÞm ζ ÀÁZ Z m 1, the function Li - n ðÞassumes a summation form, which is ÀÁ 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 matrix 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.
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