Practical Framework for Conditional Non-Gaussian Quantum State Preparation Mattia Walschaers, Valentina Parigi, Nicolas Treps To cite this version: Mattia Walschaers, Valentina Parigi, Nicolas Treps. Practical Framework for Conditional Non- Gaussian Quantum State Preparation. PRX Quantum, APS Physics, 2020, 1 (2), 10.1103/PRXQuan- tum.1.020305. hal-03037761 HAL Id: hal-03037761 https://hal.archives-ouvertes.fr/hal-03037761 Submitted on 3 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PRX QUANTUM 1, 020305 (2020) Practical Framework for Conditional Non-Gaussian Quantum State Preparation Mattia Walschaers ,* Valentina Parigi, and Nicolas Treps Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, 4 place Jussieu, Paris F-75252, France (Received 24 August 2020; accepted 5 October 2020; published 22 October 2020) We develop a general formalism, based on the Wigner function representation of continuous-variable quantum states, to describe the action of an arbitrary conditional operation on a multimode Gaussian state. We apply this formalism to several examples, thus showing its potential as an elegant analytical tool for simulating quantum optics experiments. Furthermore, we also use it to prove that Einstein-Podolsky- Rosen steering is a necessary requirement to remotely prepare a Wigner-negative state. DOI: 10.1103/PRXQuantum.1.020305 I. INTRODUCTION sensing. They have been shown to be easily simulated on classical devices [33], and in particular Wigner negativ- In continuous-variable (CV) quantum physics, Gaussian ity is known to be a necessary resource for reaching a states have long been a fruitful topic of research [1–10]. quantum computation advantage [34]. However, it should They appear naturally as the ground states of systems of be stressed that recent work has found large classes of many noninteracting particles in the form of thermal states Wigner negative states that can also be simulated easily [11], or as the coherent states that describe the light emitted [35]. In other words, Wigner negativity is necessary but by a laser [3]. Through nonlinear processes, it is possible not sufficient to reach a quantum computation advantage to reduce the noise beyond the shot noise limit (at the price [36]. of increased noise in a complementary observable), and In the particular case of CV quantum computation, create squeezed states [12–17]. For the purpose of metrol- Gaussian states play an essential role in the measurement- ogy, such squeezed states are often enough to obtain a based approach [37]. In this paradigm, one establishes significant boost in performance [18–21]. large Gaussian entangled states, known as cluster states, On theoretical grounds, Gaussian states are relatively which form the backbone of the desired quantum routine easy to handle [8,9]. The quantum statistics of the [38]. Several recent breakthroughs have led to the experi- continuous-variable observables (e.g., the quadratures in mental realisation of such states [39–43]. Nevertheless, to quantum optics) are described by Gaussian Wigner func- execute quantum algorithms that cannot be simulated effi- tions. All interesting quantum features can be deduced ciently, one must induce Wigner negativity. In the spirit from the covariance matrix that characterises this Gaussian of measurement-based quantum computation, this feature distribution on phase space. Hence, whenever the num- is induced by measuring non-Gaussian observables, e.g., ber of modes remains finite, the techniques of symplectic the number of photons, on a subset of modes [44–47]. matrix analysis are sufficient to study Gaussian quantum Such a measurement then projects the remainder of the states. This has generated an extensive understanding of system into a non-Gaussian state. The exact properties of the entanglement properties of Gaussian states [22–27], the resulting state depend strongly on the result of the and recently it has also led to the development of a mea- measurement. sure for quantum steering (see [28]) of Gaussian states The conditional preparation of non-Gaussian quantum with Gaussian measurements [29–32], which we refer to states is common procedure in quantum optics experiments as Einstein-Podolsky-Rosen (EPR) steering. [48]. Basic examples include the heralding of single- Even though they have many advantages, Gaussian photon Fock states after parametric down-conversion [49– states are of limited use to quantum technologies beyond 51], photon addition and subtraction [52–57], and known schemes to prepare more exotic states such as Schrödinger- *[email protected] cat [58,59] or Gottesman-Kitaev-Preskill states [60]. It should be noted that conditioning on the measurement Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Fur- of Gaussian observables can also be relevant in certain ther distribution of this work must maintain attribution to the protocols [61]. Remarkably, though, a practical frame- author(s) and the published article’s title, journal citation, and work to describe the effect of arbitrary conditional oper- DOI. ations on arbitrary Gaussian states is still lacking. Notable 2691-3399/20/1(2)/020305(14) 020305-1 Published by the American Physical Society WALSCHAERS, PARIGI, and TREPS PRX QUANTUM 1, 020305 (2020) exceptions where one does study arbitrary initial states function [62–64] usually rely on specific choices for the conditional mea- 1 (α ˆ+α ˆ) − (α +α ) surement. W(x, p) = tr[ρˆei 1x 2p ]e i 1x 2p dα dα . 2 1 2 Here, in Sec. III, we introduce a practical framework (2π) R2 to describe the resulting Wigner function for a quantum (1) state that is conditionally prepared by measuring a subset For some quantum states, this function has the peculiar of modes of a Gaussian multimode state. The techniques property of reaching negative values. This Wigner nega- used in this work are largely based on classical multivariate tivity is a genuine hallmark of quantum physics, and it is probability theory and provide a conceptually new under- understood to be crucial in reaching a quantum computa- standing of these conditioned states. In Sec. IV, we unveil tional advantage. the most striking consequence of this new framework: we Here, we consider a multimode system compris- can formally prove that EPR steering in the initial Gaus- ing m modes. Every mode comes with its own sian state is a necessary requirement for the conditional infinite-dimensional Hilbert space, associated to a two- preparation of Wigner-negative states, regardless of the dimensional phase space, and observables xˆ and pˆ . measurement upon which we condition. This solidifies a j j The total optical phase space is, thus, a real space R2m previously conjectured general connection between EPR with a symplectic structure = ω, where the two- steering and Wigner negativity. As shown in Sec. V, our m dimensional matrix ω is given by framework reproduces a range of known state-preparation schemes and can be used to treat more advanced scenarios, 0 −1 ω = .(2) which could thus far not be addressed by other analyti- 10 cal methods. First, however, we review the phase-space description of multimode CV systems in Sec. II. Therefore, has the properties 2 =−1 and T =−. Any normalised vector f ∈ R2m defines a single optical mode with an associated phase space span{f, f} (i.e., when f generates the phase-space axis associated with II. PHASE-SPACE DESCRIPTION OF MULTIMODE CONTINUOUS-VARIABLE the amplitude quadrature of this mode, f generates the SYSTEMS axis for the associated phase quadrature). Henceforth, we refer to the subsystem associated with the phase-space The CV approach studies quantum systems with an span{f, f} as “the mode f ”. Every point α ∈ R2m can infinite-dimensional Hilbert space H based on observ- ˆ ˆ also be associated with a generalised quadrature observ- ables, x and p, that have a continuous spectrum and obey able the canonical commutation relation [xˆ, pˆ] = 2i (the factor 2 is chosen to normalise the vacuum noise to 1). Common m ˆ(α) = (α ˆ + α ˆ ) examples include the position and momentum operators in q 2k−1xk 2kpk .(3) mechanical systems, or the amplitude and phase quadra- k=1 tures in quantum optics. In this work, we use quantum These observables satisfy the general canonical commu- optics terminology, but the results equally apply to any tation relation [qˆ(α) , qˆ(β) ] =−iαTβ. Physically, such other system that is described by the algebra of canonical observable qˆ(α) can be measured with a homodyne detec- commutation relations (i.e., any bosonic system). tor by selecting the mode that is determined by the direc- In a single-mode system, the quadrature observables xˆ tion of α, and multiplying the detector outcome by α.In and pˆ determine the optical phase space. The latter is a our theoretical treatment, such generalised quadratures are two-dimensional real space, where the axes denote the useful to define the quantum characteristic function of any possible