Practical Framework for Conditional Non-Gaussian Quantum State Preparation Mattia Walschaers, Valentina Parigi, Nicolas Treps

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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

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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[ρêi 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 observables, 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 measurement outcomes for xˆ and pˆ.Itiscom- multimode state ρˆ, mon practice to represent a given state ρˆ by means of its measurement statistics for xˆ and pˆ on this optical χρˆ(α) = tr[ρˆ exp{iqˆ(α) }](4) phase space, as in statistical physics. However, because α xˆ and pˆ are complementary observables, they cannot be for an arbitrary point in phase space. The multimode measured simultaneously, and thus, a priori, we cannot Wigner function of the state is then obtained as the Fourier transform of the characteristic function construct a joint probability distribution of phase space that reproduces the correct marginals to describe the measure- 1 −iαTx W(x) = χρˆ(α) e dα,(5) 2m ment statistics of the quadratures. Therefore, the phase- (2π) R2m space representation of quantum states are quasiprobability distributions. The quasiprobability distribution that where x ∈ R2m can, again, be any point in the multimode reproduces the measurement statistics of the quadrature phase space, and the coordinates of x represent possible observables as its marginals is known as the Wigner measurement outcomes for xˆj and pˆj .

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The Wigner function can be used to represent and char- III. CONDITIONAL OPERATIONS IN PHASE acterise an arbitrary quantum state of the multimode sys- SPACE tem. In the same spirit, we can also define the phase-space ˆ In quantum optics, we associate a Hilbert space (more representation of an arbitrary observable A as precisely, a Fock space) to each of these modes. The Hilbert space H of the entire system can then be structured 1 ˆ −iαTx W ˆ(x) = tr[A exp{iqˆ(α) }]e dα,(6) as H = H ⊗ H , where H (H ) describes the quantum A ( π)2m f g f g 2 R2m states of the set of orthogonal modes f (g). Formally, the state of our full m-mode system is then described by a such that we can fully describe the measurement statis- density matrix ρˆ that acts on H. tics of an arbitrary quantum observable on phase space, by Within this manuscript, we perform a conditional oper- invoking the identity ation in the set of modes g, which we describe through a ˆ (not necessarily normalised) set of Kraus operators [65] Xj ˆ m ρˆ = ( π) ˆ() () tr[ A] 4 WA x W x dx (7) that act on Hg [66]: R2m to evaluate expectation values. In practice, it is often chal- ˆ ˆ † Xj ρˆX lenging to obtain Wigner functions for arbitrary states or ρˆ → j j . (11) ˆ † ˆ ρˆ observables, but in some cases they can take convenient tr[ j Xj Xj ] forms. A particular class of convenient states are Gaussian Such a conditional operation naturally arises as a post- () states, where the Wigner function W x is a Gaussian. As measurement state, when Xˆ is a projector, or when Aˆ = a consequence, the Wigner function is positive, and can j Xˆ †Xˆ is a more general positive operator-valued mea- thus be interpreted as a probability distribution. This Gaus- j j j sure (POVM) element, as represented in the sketch in sian distribution is completely determined by a covariance ˆ matrix V, and mean field ξ, such that the Wigner function Fig. 1. The positive semidefinite operator A is useful to takes the form express the reduced state of the set of modes f:

T −1 e−(1/2)(x−ξ) V (x−ξ) ˆρˆ () = √ trg[A ] W x .(8) ρˆ | ˆ = . (12) (2π)m det V f A tr[Aˆρˆ] This forms the basis of our preparation procedure for non- H Gaussian states as we assume that our initial multimode Here trg denotes the partial trace of the Hilbert space g system is prepared in such a Gaussian state. associated with the set of mode g. Our general goal is to ρˆ To perform the conditional state preparation, we divide understand the properties of the state f|Aˆ. the m-mode system into two subsets of orthogonal modes, As we are interested in the Wigner function for the state of the subset of modes f, we translate Eq. (12) to its phase- f ={f1, ..., fl} and g ={g1, ..., gl } with l + l = m,and perform a measurement on the modes in g. We can then space representation. We initialize the total system in a () describe the subsystems of modes f and g by phase spaces Gaussian state with Wigner function W x . Subsequently, ( ) R2l and R2l , respectively. As such, the joint phase space we also define the Wigner function WAˆ xg of the positive ˆ can be mathematically decomposed as R2m = R2l ⊕ R2l . operator A, which is a function that is defined according A general point x in the multimode phase space R2m to Eq. (6) on the phase space that describes the subset of can thus be decomposed as x =xf ⊕xg, where xf and xg modes g. As such, we find that describe the phase-space coordinates associated with the sets of modes f and g, respectively. In particular, xf can ( ) () R2l WAˆ xg W x dxg ... = W ˆ(x ) = . (13) be expanded in a particular modes basis f1, , fl as xf f|A f ( ) () ( ... ) R2m WAˆ xg W x dx xf1 , pf1 , , xfl , pfl , where the coordinates xfj and pfj are obtained as Because Aˆ is a positive semidefinite operator, the denomi- =T xfj x fj ,(9)nator is a positive constant. ( ) As presented in Eq. (13), the Wigner function Wf|Aˆ xf is impractical to use and its properties are not appar- T pf =x fj . (10) ent. Hence, we now introduce some mathematical tools to j ( ) obtain a more insightful expression for Wf|Aˆ xf .First,we A completely analogous treatment is possible for the coor- use the fact that, for Gaussian states, W(x) is a probabil- dinates associated with the set of modes g. ity distribution on phase space, such that we can define the

020305-3 WALSCHAERS, PARIGI, and TREPS PRX QUANTUM 1, 020305 (2020) conditional probability distribution through W(xg |xf). Similarly, we can use Eq. (7) to introduce the () notation W x W(xg |xf) = , (14) ( ) Wf xf ˆ = ˆρˆ = ( π)l ( ) () A tr[A ] 4 WAˆ xg W x dx (22) R2m where Wf(xf) is the reduced Gaussian state for the set of modes f, ˆ for the expectation value of A in the state ρ. Finally, we can use Eqs. (21) and (22) to recast Eq. (20) in the form W (x ) = W(x) dx . (15) f f g R2l ˆ A g|xf Because W(x) is a Gaussian probability distribution, the W | ˆ(xf) = Wf(xf). (23) f A Aˆ conditional probability distribution W(xg |xf) is also a Gaussian distribution [67] given by ˆ The major advantage of this formulation is that A g|xf rep- T −1 exp[−(1/2)(xg − ξg|x ) V | (xg − ξg|x )] resents the average with respect to a Gaussian probability ( |) = f g xf f W xg xf distribution, such that one can use several computational (2π)l det V | g xf techniques that are well known for Gaussian integrals. A (16) notable property is the factorisation of higher moments ˆ with covariance matrix in multivariate Gaussian distributions, such that A g|xf can generally be expressed algebraically in terms of the −1 T Vg|x = Vg − VgfV V , (17) ξ f f gf components of Vg|xf and g|xf (for more details, see the Appendix). where Vg and Vf are the covariance matrices describ- ˆ ˆ Finally, we remark that A | = A in the absence of ing the subsets of modes g and f in the initial state, g xf correlations between the set of modes g that are condi- whereas Vgf describes all the initial Gaussian correlations between those subsets. Note that this covariance matrix tioned upon and the set of modes f for which we construct 2l the reduced state. This result is directly responsible for is the same for all points xf ∈ R , which is a particular property of Gaussian conditional probability distributions. the previously obtained results related to the spread of non-Gaussian features in cluster states [68]. Furthermore, the distribution W(xg |xf) also contains a displacement IV. EINSTEIN-PODOLSKY-ROSEN STEERING ξ = ξ + −1( − ξ ) g|xf g VgfVf xf f , (18) AND WIGNER NEGATIVITY When two systems are connected through a quantum where ξg and ξf describe the displacements of the initial state in the sets of modes g and f, respectively. correlation, one can, in some cases, perform quantum X Generally, the phase-space probability distribution steering [28]. Colloquially, we say that a subsystem Y W(x |x ) is not a valid Wigner function of a well-defined can steer a subsystem when measurements of certain g f X quantum state, in the sense that it would violate the Heisen- observables in can influence the conditional measure- Y berg inequality. However, it does remain a well-defined ment statistics of observables in beyond what is possible probability distribution, i.e., it is normalised and positive. with classical correlations. Ultimately, in quantum steer- Thus, it still has interesting properties that we can exploit ing one studies properties of conditional quantum states as to formulate a general expression for W ˆ(x ).Letusfirst compared to a local hidden variable model for any observ- f|A f X Y use Eq. (14) to recast Eq. (13) in the following form: ables X and Y, acting on and , respectively. Contrary to the case of Bell nonlocality, quantum steering considers ( ) ( |) ( ) R2l WAˆ xg W xg xf Wf xf dxg an asymmetric local hidden variable model: W ˆ(x ) = (19) f|A f ( ) () R2m WAˆ xg W x dx P(X = x, Y = y) = P(λ)P(X = x | λ)PQ(Y = y | λ). λ [ W ˆ(x )W(x |x ) dx ]W (x ) = R2l A g g f g f f (24) ( ) () . (20) R2m WAˆ xg W x dx Here one assumes that the probability distributions PQ(Y = Subsequently, we can define | λ) Y y of steered party follow the laws of quantum mechanics. For the party X , which performs the steering, ˆ l A | = (4π) W ˆ(x )W(x |x ) dx , (21) g xf A g g f g no such assumption is made and any probability distribu- R2l tion is allowed. Such a local hidden variable model can which is the expectation value of the phase-space repre- typically be falsified, either by brute force computational sentation of Aˆ with respect to the probability distribution methods [69] or via witnesses [70]. These methods have

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been applied in a variety of contexts to experimentally g if and only if Vg|xf violates the Heisenberg inequal- observe quantum steering [31,32,71–77]. ity [29,30]. The crucial consequence is that W(xg |xf),as A paradigmatic example is found when performing given by Eq. (16), is itself a well-defined Gaussian quan- homodyne measurements on the EPR state [78]: when the tum state when the modes in f cannot steer the modes g. entanglement in the system is sufficiently strong, one can For all possible xf, we can thus associate this Gaussian ˆ ˆ Y ρˆ condition the x and p quadrature measurements in on quantum state with a density matrix g|xf . the outcome of the same quadrature measurement in X . ˆ The crucial observation is that A g|xf , as defined in The obtained conditional probability distributions for the Eq. (21), is the expectation value of Aˆ in a well-defined quadrature measurements in Y can violate the Heisenberg ˆ quantum state ρ | for any x . Because A is a positive inequality, even when averaged over all measurement out- g xf f semidefinite operator, we directly find that comes in X . The violations of such a conditional inequality are impossible with classical correlations, but are a ˆ ˆ 2l A | = tr[ρ | A] 0 for all x ∈ R . (25) hallmark of quantum steering. g xf g xf f Quantum steering can occur in all types of quantum Therefore, the overall conditional Wigner function W ˆ(x ) states, with all kinds of measurements. In CV quantum f|A f ˆ < physics, one often refers to the particular case of Gaussian in Eq. (23) is non-negative. We can only achieve A g|xf ∈ R2l states that can be steered through Gaussian measurements 0 for certain points xf when Vg|xf violates the Heisen- as EPR steering. Recently, other forms of steering for berg inequality. This concludes that in absence of EPR ( ) Gaussian states have been developed under the name of steering Wf|Aˆ xf 0. nonclassical steering [79]. In this approach, one checks X Note that the steps in this proof rely heavily on the fact whether Gaussian measurements in can induce a non- that the initial state is Gaussian. For other types of quantum classical conditional state in Y. Throughout this work, the X Y states, we cannot directly relate quantum steering to the focus lies on EPR steering, where the systems and are properties of W(x |x ). the sets of modes f and g, respectively. g f In previous work, we showed that EPR steering is a nec- V. EXAMPLES essary prerequisite to remotely generate Wigner negativity through photon subtraction [80]. More precisely, when a A. Heralding photon is subtracted in a mode g, the reduced state Wigner In the first example, we consider a scenario where a function of a correlated mode f can only be nonpositive photon-number-resolving measurement is performed on if mode f is able to steer mode g. When one allows for one of the output modes, which can be considered a special an additional Gaussian transformation on mode g prior to case of the situation considered in Ref. [47]. Herald- photon subtraction, we found that EPR steering from f to ing is ubiquitous in quantum optics, as it is one of the g is also a sufficient condition to reach Wigner negativity most common tools to generate single-photon Fock states in mode f . [49–51]. The formalism that is developed in the previous section To study heralding, we use Eq. (23) where a measure- allows us to generalize this previous result to arbitrary ment of the number of photons n in a single mode g is conditional operations on an arbitrary number of modes. performed. We assume that this measurement is optimal, and, thus, that we project on a Fock state |n . In this case, Theorem 1. For any initial Gaussian state ρˆ and any con- we set Aˆ =|n n|, and therefore we obtain ditional operation AinEq.ˆ (12), EPR steering between the n 2 set of modes f and the set of modes g is necessary to induce (− )n+k 2k −(1/2)xg ( ) = n 1 xg e Wigner negativity in W ˆ(x ). W ˆ xg , (26) f|A f A k k! 2π k=0

Proof. Gaussian EPR steering is generally quantiﬁed where we used the closed form of the Laguerre polyno- through the properties of Vg|xf . In particular, one can show mial. Hence, we can now use this expression to calculate that the set of modes in f can jointly steer the set of modes | | n n g|xf . In this calculation, we must evaluate

− n n+k 2k −( / )( − ξ )T 1 ( − ξ ) − 2 / n (−1) x exp[ 1 2 xg g|x V | xg g|x xg ] 2 ( ) ( |) = g f g xf f WAˆ xg W xg xf , (27) k k! 4π 2 det V | k=0 g xf

020305-5 WALSCHAERS, PARIGI, and TREPS PRX QUANTUM 1, 020305 (2020) and we can recast

After a substitution in the integral, we then ﬁnd that

σ = (1 + ) where we defined Vg|xf Vg|xf , which is now as g. In the limiting case where the initial thermal noise the covariance matrix of a new Gaussian probability dis- vanishes, we recover the well-known EPR state that mani- tribution. The final expression is then determined by the fests perfect photon-number correlations between modes f moments of the Gaussian distribution with covariance and g. In this case, it is clear that a detection of n photons σ ξ | matrix and displacement g|xf . Even though this expres- in mode g will herald the state n in mode f . However, by sion is relatively elegant, it can be remarkably tedious to introducing thermal noise, the photon-number correlations compute for larger values of n. fade and the properties of the heralded state in mode f First, let us focus on the experimentally relevant case are less clear. Thermal noise will also gradually reduce the where n = 1 as an illustration. The evaluation of Eq. (29) EPR steering in the system, such that the Wigner negativ- is than conducted by calculating the second moments of a ity in mode f will vanish when the thermal noise becomes Gaussian distribution, such that we ultimately find that too strong. Hence, with this example we can study the interplay between Wigner negativity and EPR steering in a controlled setting. Wf| |1 1|(xf) The squeezed thermal state is characterised by a covari-

−1 2 −1 ance matrix V = diag[δ/s, δs], where δ denotes the amount = (1 + V | ) ξ | + tr[(1 + V | ) V | ] − 1 g xf g xf g xf g xf of initial thermal noise, and s is the squeezing parame- −( / )ξT 1+ −1ξ 1/2 1 2 | [ Vg|x ] g|x det(1 + V ) e g xf f f ter. We initially start with two copies of such a state, and × g W (x ), (1 + )1/2 (1 + )−1 − f f det Vg|xf tr[ Vg Vg] 1 (30) f rˆ f|Aˆ where we set ξg = 0, thus assuming that there is no mean field in mode g. We note that this function reaches nega- (1 + )−1 < tive values if and only if tr[ Vg|xf Vg|xf ] 1. Using Conditioning Williamson’s decomposition, as we did in Ref. [80], it can EPR steering be shown that this condition can only be fulfilled when the rˆ set of modes f can perform EPR steering in mode g,or,in other words, when Vg|xf violates the Heisenberg inequality. This is exactly what we can expect from our general result Aˆ in Sec. IV. g In general, we know that the Wigner function (23) can only be negative when Vg|xf is not a covariance matrix of a well-defined quantum state. However, determining the FIG. 1. Sketch of the conditional state-preparation scenario: a existence of zeroes of this Wigner function is a cumber- multimode quantum state with density matrix ρˆ is separated over some task. For heralding with n > 1, we therefore restrict two subsets of modes, f and g. A measurement is performed on to numerical simulations using a specific initial state. the modes in g, yielding a result associated with a POVM element This specific initial state is generated by mixing two Aˆ. Conditioning on this measurement outcome “projects” the ρˆ squeezed thermal states on a balance beam splitter, where subset of mode f into a state f|Aˆ . The directional EPR steering, one of the output modes will serve as f , and the other discussed in Sec. IV, is highlighted.

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(a)

(b) (c)

FIG. 2. Photon heralding with a particular Gaussian input state, generated by mixing two equal squeezed thermal states (b) on a balanced beam splitter (a). On one of the outputs of the beam splitter, a projective measurement is performed on the Fock state |n , which heralds a non-Gaussian state in the other mode. The Wigner functions of this non-Gaussian state are shown for the case where n = 5, with varying degrees of EPR steering μ, controlled by varying the thermal noise δ for a ﬁxed squeezing s = 5 dB. The Wigner negativity, measured by N given in Eq. (32), is shown in (c) for varying degrees of EPR steering and a varying number of measured photons n. rotate the phase of one of them by π/2 (see Fig. 2). When [81–83] both modes are mixed on a beam splitter, the resulting state δ manifests EPR steering depending on parameters and s, N = | ( )| − Wf |Aˆ xf dxf 1. (32) which can be quantiﬁed through [30] R2

μ = When the state is pure (here for 0.55), a detection 1 | μ = max 0, − log det Vg|x , (31) of n photons in one mode herald a Fock state n in the 2 f other mode and the Wigner negativity thus increases with n. However, once the state is no longer pure and the steering decreases, we observe the existence of an optimal value where we explicitly use the fact that V | is a two- g xf n for which the maximal amount of Wigner negativity is dimensional matrix. When we then postselect on the num- obtained. For very weak EPR steering (e.g., μ = 0.08 in ber of photons, n, measured in one output mode, we herald this calculation), this optimal value is obtained for n = 1. a conditional non-Gaussian state in the other mode. In Fig. This numerical study shows the fruitfulness of our 2, we show the resulting Wigner functions for the case presented framework to study a very concrete herald- where the detected number of photons is n = 5. When the μ = ing scheme. Furthermore, the example confirms the rela- amount of EPR steering is varied (note that 0.55 cor- tionship between Gaussian EPR steering and Wigner responds to the pure state), we see that the resulting Wigner negativity. function rapidly loses Wigner negativity. In full agreement with our general result of the previous section, we also find that the Wigner negativity vanishes when there is no EPR B. Photon-added and -subtracted states steering. Ideal photon addition and subtraction are defined by act- A more quantitative study of the Wigner negativity can ing with a creation operator aˆ† or annihilation operator aˆ, be found in panel (c) of Fig. 2, where we vary both the respectively, on the quantum state. In practice, these oper- amount of steering μ and the number of detected pho- ations are often realised by using some form of heralding tons n. The Wigner negativity is measured by the quantity [52], which we treated in the previous example. However,

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ˆ it tends to be more convenient to use the idealised model, for which we must evaluate ng g|xf . To this end, we eval- based on creation and annihilation operators, and it has uate the Wigner function of the number operator, which is been shown experimentally that this model is highly accu- given by rate. This model also fits the conditional state framework ˆ 1 of Eq. (11), where we set Xj to be a creation or annihilation ( ) = ( 2 − ) Wnˆg xg xg 2 , (35) operator. 16π In multimode systems, photon addition and subtraction such that we directly find that have been considered for their entanglement properties, 1 2 ˆ | = ( | +ξ | − ) which sprouted a range of theoretical [84–91] and exper- ng g xf 4 trVg xf g xf 2 , (36) imental [92–94] results. Many of the obtained theoretical ξ results rely on the purity of the initial Gaussian state, and where the dependence on xf comes from g|xf . Thus, we are hard to generalise to arbitrary Gaussian states. In recent find that, for the photon-subtracted state, years, there has been some progress in developing analyt- +ξ 2 − − trVg|xf g|xf 2 ical tools to describe general photon subtracted states [91, W |ˆ (xf) = Wf(xf). (37) f ng 2 95], but it remains challenging to use these techniques to trVg +ξg − 2 evaluate entanglement measures. Therefore, related ques- From this result, we immediately observe that the poten- tions have been investigated, such as, for example, the tial Wigner negativity of these states depends on whether spread of non-Gaussian features in multimode systems or not trV | < 2. In Ref. [80] it was shown through [68,80,96,97]. g xf The framework presented in this manuscript is partic- the Williamson decomposition that trVg|xf 2 det Vg|xf . ularly fruitful to investigate the spread of non-Gaussian This directly implies that EPR steering (31) is a necessary features through photon addition or subtraction. We first condition to reach Wigner negativity. It is instructive to show how the results of Ref. [80] can be recovered via emphasise that Eq. (23). Then, we use the present framework to provide ξ 2 =ξ + −1( − ξ )2 analytical results for the states that can be obtained by g|xf g VgfVf xf f , (38) subtracting multiple photons in a multimode system. from which one ultimately retrieves the expression

{ξ + −1( − ξ )2 + − } 1. Adding or subtracting a single photon − g VgfVf xf f trVg|xf 2 W |ˆ (xf) = Wf(xf), f ng 2 We start by studying the addition and subtraction of a trVg +ξg − 2 single photon. The scenario for photon-subtracted states was studied in detail in Ref. [80] and our goal in this exam- which is the result that was derived in Ref. [80]. ple is to show how these previous results can be obtained For the photon-added state, we can perform a com- in the context of our present framework. Furthermore, we pletely analogous computation with also study photon addition, which has not yet been con- 1 ( ) = ( 2 + ) sidered in the context of the remote generation of Wigner Wnˆg+1 xg xg 2 , (39) 16π negativity. Creation and annihilation operators are by construction from which we find that operators that act on a single mode g. In the single-photon +ξ 2 + + trVg|xf g|xf 2 scenario, we find the photon-subtracted state W |ˆ (xf) = Wf(xf). (40) f ng 2 trVg +ξg + 2 † aˆgρâˆg This result immediately shows that this Wigner function ρˆ− = (33) tr[nˆgρˆ] is always positive, which implies that it is impossible to remotely create Wigner negativity through photon addi- and the photon-added state tion. In previous work, we highlighted that photon addition always creates Wigner negativity in the mode where the aˆ†ρâˆ ρˆ = g g photon is added [91]. What we observe in Eq. (40) can + (ˆ + 1)ρˆ . (34) tr[ ng ] be understood as the complementary picture for the other modes. This result also highlights an operational difference These states clearly fit the framework of Eq. (11). In the between photon subtraction and addition: photon addition context of Eq. (12), the reduced state of the set of modes f is a more powerful tool to locally create Wigner negativ- ˆ ˆ is obtained by choosing A =ˆng and A =ˆng + 1 for photon ity, whereas photon subtraction has the potential to create subtraction and addition, respectively. We can then use Eq. Wigner negativity nonlocally (i.e., in modes that can steer (23) to obtain the Wigner function in the subset of modes f, the mode in which the photon is subtracted).

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2. Subtracting multiple photons and

When multiple photons are added or subtracted, or † 1 T T T â aˆg | = [g Vg|x g2 +g Vg|x g2 when we chain combinations of addition and subtrac- g1 2 g xf 4 1 f 1 f ˆ tion operations, the evaluation of A | (x ) will rapidly + (T −TT ) g xf f i g1 Vg|xf g2 g1 Vg|xf g2 ]. (46) become more complicated. A general strategy to approach ( ) this problem avoids the explicit evaluation of WAˆ xg , The computation required to obtain the final result is but rather uses standard techniques for the evaluation of tedious but straightforward. We find that moments of multivariate Gaussian distributions. This ulti- ˆ ˆ mately boils down to applying Wick’s theorem [98]and ng1 ng2 g|xf summing over all matchings (see the Appendix for details). 1 2 = [(trV | +ξ | − 2)(trV | Even though this task can be implemented numerically, 16 g1 xf g1 xf g2 xf 2 T T the corresponding analytical expressions quickly become +ξ | − 2) + 2tr(C C) + 4ξ Cξ | ], (47) g2 xf g1|xf g2 xf intractable. To illustrate this method, we consider the multimode where we have defined the submartix C as the off-diagonal scenario where two photons are subtracted in different block of Vg|xf via orthogonal modes, g1 and g2, which implies that the | conditioning implements the map = Vg1 xf C Vg|xf T . (48) C Vg2|xf aˆ aˆ ρâˆ† aˆ† ρ → g1 g2 g2 g1 Nonzero entries in the block C can occur due to various ˆ ˆ ρˆ . (41) tr[ng1 ng2 ] causes. First, they can be due to a correlation between the modes g1 and g2 in the initial Gaussian state [as seen from This implies that we must apply our formalism with Aˆ = the term Vg in Eq. (17)]. However, nontrivial entries in C ˆ ˆ also arise when modes g1 and g2 are both correlated to the ng1 ng2 . To treat this problem with the technique of match- −1 T ings, we use the Gaussian identity (note that we do not same modes in f, which is induced by the term VgfVf Vgf explicitly write the dependence on x to simplify notation) in Eq. (17). f Result (47) directly shows the appearance of a triv- ( +ξ 2 − )( +ξ 2 − ) ial term, trVg1|xf g1|xf 2 trVg1|xf g1|xf 2 , ˆng nˆg g|x 1 2 f which multiplies the effect of photon subtraction in g1 2 2 2 = ˆ ˆ + ˆ ˆ with that of photon subtraction in g2. However, when both ag1 g|xf ag2 g|xf ng1 g|x ag2 g|xf f modes are sufficiently “close” to each other, we find the 2 † † T T + ˆn â | + â aˆ â | â | additional terms 2tr(C C) + 4ξ | Cξg |x , which can be g2 g|xf g1 g xf g1 g2 g|xf g2 g xf g1 g xf g1 xf 2 f † † † † interpreted as some form of interference between the two + â aˆ | â aˆ | + â aˆ | â aˆ | g1 g2 g xf g2 g1 g xf g1 g2 g xf g1 g2 g xf photon subtractions. † † In Fig. 3 we provide an illustration, where we inject + ˆng | ˆng | + â aˆg | a g|x âg g|x 1 g xf 2 g xf g2 1 g xf g1 f 2 f three pure squeezed vacuum states into a series of beam † † + â aˆ â | â | g1 g2 g|xf g1 g xf g2 g xf splitters to generate an entangled three-mode state from † † which we subtract two photons. The first two squeezed + âg aˆg | â g|x â g|x , (42) 1 2 g xf g1 f g2 f vacuum states have 5 dB squeezing in opposite quadratures and are mixed on the beam splitter with 75% transmittance. where · denotes the nondisplaced version of the distri- g|xf One of the output ports will serve as mode g1, whereas bution. We can immediately identify the other is injected into a section beam splitter of 25% transmittance. In the other input port of this beam splitter, 1 T T ˆ | = (ξ + ξ ) we inject the third squeezed vacuum state, which is also ag1 g xf 2 g|x g1 i g|x g1 ; (43) f f squeezed by 5 dB. One of the output ports of the 25% trans- subsequently, from Eq. (36), we obtain mittance beam splitter serves as mode g2, and in the other output port we find mode f , which is the mode for which

1 we reconstruct the output Wigner function using Eq. (47). ˆng | = (trVg |x − 2), (44) 1 g xf 4 1 f Photon subtraction is represented by a highly transmitting beam splitter that sends a small amount of light to a photon and finally we find new types of terms that are given by detector. Two-photon subtraction then happens when both detectors click at the same time, and we can condition the † † 1 T T T â aˆ = [g V | g −g V | g state in mode f upon this detection outcome. This posts- g1 g2 g|xf 4 1 g xf 2 1 g xf 2 election scheme effectively implements the operators aˆ − (T +TT ) g1 i g1 Vg|xf g2 g1 Vg|xf g2 ] (45) ˆ and ag2 on modes g1 and g2, respectively.

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the problem with the correlation function methods of Ref. [91], even though this method is highly successful for single-photon subtraction in multimode states. These methods can in principle be extended to deal with higher numbers of added and/or subtracted photons in various modes. However, it must be emphasised that one will quickly encounter practical boundaries as finding all possible matchings is a computationally hard problem [99]. Finding an exact description of the Wigner function that is obtained by subtracting a large number of photons from a subset of an entangled Gaussian state seems to be a computationally hard problem that has its roots in ( ) FIG. 3. Conditional state Wigner function Wf |Aˆ xf , obtained graph theory. The problem of finding all matchings also by subtracting a photon in two of the three modes in a three- lies at the basis of Gaussian boson sampling [100,101], mode entangled state. This entangled state is generated by mixing and it is not expected to be easy to overcome. The prob- three squeezed vacuum states in a sequence of beam splitters lem of Gaussian boson sampling can in turn also be related with transmittances of 75% (left) and 25% (right). Two of the to CV sampling from photon-added or -subtracted states squeezed vacuum states are squeezed by 5 dB in the x quadrature [102]. (left, right) and one is squeezed by 5 dB along the p quadrature (middle). The photon subtraction is represented by highly transmitting beam splitters that send a small fraction of light to a ˆ photon detector, which effectively implements the operators ag1 VI. CONCLUSIONS ˆ and ag2 on the modes g1 and g2, respectively. We present a general framework that describes the Wigner function that is obtained by applying an arbitrary ( ) ˆ = We observe that the conditional state Wf |Aˆ xf with A operation on a subset of modes of a multimode Gaussian ˆ ˆ state, and conditioning the remaining modes on this oper- ng1 ng2 reaches negative values in two distinct regions of phase space. Indeed, with the Williamson decomposition ation. The most natural way of interpreting this scenario is by considering this operation to be a measurement, such of Vg|xf we can quantify [30] the strength of EPR steering from mode f to the set of modes g to be μ = 0.548. Fur- that the state of the remaining modes is obtained by post- thermore, the fact that there are two negativity regions is a selecting on a specific measurement outcome, as is the hallmark of the subtraction of two photons. This example case for heralding. However, this framework can also be shows that our framework is a highly versatile tool for CV used to study the nonlocal effects of photon addition and quantum state engineering. subtraction. Finally, we consider the complementary scenario where Our framework relies heavily on classical probability two photons are subtracted from one mode. In this case, theory, and in particular on properties of conditional prob- we can still use the perfect matching technique (42), when ability distributions (14). We use the fact that Gaussian creation and annihilation operators are in normal ordering. states have positive Wigner functions, such that associ- ˆ † † In this case, we obtain A =âgaˆgaˆgaˆg, and analogously to ated conditional probability distributions on phase space Eq. (42), we find that are well defined as probability distributions (but not necessarily as quantum states, because they can violate the † † 1 2 2 2 Heisenberg inequality). In this regard, our general results â aˆ aˆgaˆg g|x = [(trVg|x +ξg|x ) + 2tr(V | ) g g f 16 f f g xf (21)–(22) are valid for all initial states with a positive T + 4ξ (V | − 21)ξ | − 8trV | + 8]. g|xf g xf g xf g xf Wigner function. (49) Gaussian states are not only the most relevant initial states from an experimental point of view, they also have This result can then be directly inserted into Eq. (23) to the theoretical advantage of leading to a Gaussian condi- obtain the final conditional state for the set of modes f tional probability distribution. The latter is an enormous ˆ when two photons are subtracted in mode g. As expected, advantage for evaluating the crucial quantity A g|xf ,as the subtraction of two photons can induce Wigner negativ- defined in Eq. (21). On a more fundamental level, we note ity only when there is EPR steering from the modes f to that the covariance matrix (17) of this Gaussian condi- mode g. tional probability distribution is essential in the theory of As such, we have shown that our framework allows us EPR steering. This observation allows us to directly prove to analytically describe conditional non-Gaussian states in that EPR steering is a necessary prerequisite for the con- a regime that is highly challenging for many other meth- ditional preparation of Wigner negativity, regardless of the ods. For example, it is highly challenging to approach conditional operation that is performed.

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In previous work, we already showed that Gaussian EPR commutation relations [11]. A defining property of such steering is also a sufficient ingredient for the remote prepa- functionals is that truncated correlation functions [5]for ration of Wigner negativity, in the sense that there always any product of more than two creation and annihilation exists a combination of a Gaussian operation and photon operators vanish. This property is the direct analog of the subtraction in the modes g that induces Wigner negativity cumulants of a multivariate Gaussian distribution and it · in the modes f. We thus establish a fundamental relation implies that the functional g|xf is fully determined by † † ∗ † between Gaussian EPR steering and the ability to pre- the quantities âg aˆg = âg aˆg , âg aˆg ,and 1 2 g|xf 1 2 g|xf 1 2 g|xf pare a Wigner-negative state in correlated modes. This † ∗ âg g|x = âg | . Here the · | is the nondisplaced result is particularly important in the light of measurement- 1 f 1 g xf g xf based quantum computation, where large Gaussian cluster version of the functional, which is formally defined as states form the backbone for implementing a quantum # # # # # # â aˆ | = â aˆ g|x − â g|x â g|x , (A1) algorithm. The actual computation is then executed by per- g1 g2 g xf g1 g2 f g1 f g2 f forming measurements (or more general operations) on where aˆ# can be either a creation or an annihilation oper- some modes of the cluster, in order to project the remain- g1 der of the system in a desired quantum state. To claim ator. We can then write the following general property of that such a computation is universal, one must be able to the Gaussian functional: induce Wigner negativity. Our results therefore show that ˆ† ···ˆ† ˆ ···ˆ ag ag ag + agn+m g|x EPR steering is an essential figure of merit in these cluster 1 n n 1 f states in order to claim that a cluster state is suitable for # # # = â aˆ | â g|x . (A2) universal quantum computation. j1 j2 g xf k f M∈M {j ,j }∈M {k}∈M From a practical point of view, the examples in Sec. V 1 2 show that our framework is highly versatile. However, the Here M is the set of all “matchings” for the set boundaries of analytical treatments are also highlighted. {g1, ..., gn+m}. We use the term matching to refer to a par- ˆ { ... } Even though the obtained Eq. (21) for A g|xf is easy to tition of the set g1, , gn+m in subsets with either one or interpret conceptually, the actual evaluation can still be two elements. An example of such a possible matching is challenging. Regardless, we must emphasise that the ele- given by M = {{g1, g2}, ..., {gn−1, gn}, {gn}, ..., {gn+m}}. gance and simplicity of our framework does allow us to For each partition M ∈ M, we then evaluate the product obtain results with far greater ease than previously pos- of associated two-point and one-point functions, where any # # sible. Many of the methods known in the literature are pair {j1, j2}∈M is associated with â aˆ and {k}∈M j1 j2 g|xf either hard to generalise to arbitrary Gaussian initial states ˆ# = ... is associated with ak g|xf . Note that, for i g1, , gn, [44,59], focused on one particular measurement or oper- the operator aˆ# is a creation operator, whereas, for i = ation [47,91,95], or are just generally hard to interpret or i gn+1, ..., gn+m, it is an annihilation operator. use analytically. Our framework can be applied to any ini- The problem of finding all matchings is a well-known tial Gaussian state, and any conditional operation, provided problem in graph theory. To make the connection, we the Wigner function of Aˆ is known. can represent each element of the set {g1, ..., gn+m} as As such, our results provide a starting point for inves- a vertex in a full connected graph, and then consider tigating a wide range of new questions related to multi- the resulting partitions as the matchings of this graph mode conditional preparation of non-Gaussian states. By [99]. The number of terms in Eq. (A2) quickly explodes establishing a fundamental relation between EPR steer- as the number of creation and annihilation operators ing and Wigner negativity, we specifically highlight that increases, which ultimately makes the problem of evalu- this framework is also suited to obtain general analytical ˆ† ···ˆ† ˆ ···ˆ ating ag1 agn agn+1 agn+m g|xf computationally hard. results, which is often challenging in the study of states ˆ A subtle point in our treatment of A | is that · | that are both highly non-Gaussian and highly multimode. g xf g xf is not an expectation value of a Gaussian quantum state. Hence, it is legitimate to wonder up to what extent the tech- APPENDIX: MATCHINGS niques of Gaussian quantum states can be used to evaluate ˆ In Sec. 2, we refer to the method of perfect matchings A g|xf . From its definition in Eq. (21), it can be deduced ˆ that · | is a functional on the algebra of observables. It to evaluate A g|xf , which we present here with more rigour g xf and detail. directly inherits the Gaussian properties from W(xg |xf), The technique of perfect matchings is a common prac- such that it is a Gaussian functional. In particular, prop- tice to evaluate correlation functions in Gaussian states, erty (A2) can be directly traced back to the structure of which can be traced back to works such as Refs. [5,98]. the moments of the multivariate Gaussian probability dis- ( |) · In formal terms, we consider a Gaussian (also known as tribution W xg xf . The Gaussian functional g|xf is not “quasifree” in the mathematical physics literature) func- associated to a state because it is not a positive functional, · ˆ ˆ < tional g|xf on the algebra of observables for the canonical i.e., we can find positive operators A for which A g|xf 0.

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