A neural processing approach to quantum state discrimination
Saeed A. Khan Fangjun Hu Gerasimos Angelatos Michael Hatridge Hakan E. Türeci



]A neural processing approach to quantum state discrimination






             



Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544


Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544


Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544


Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh



Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544

§ ABSTRACT 
Although linear quantum amplification has proven essential to the processing of weak quantum signals, extracting higher-order quantum features such as correlations in principle demands nonlinear operations. However, nonlinear processing of quantum signals is often associated with non-idealities and excess noise, and absent a general framework to harness nonlinearity, such regimes are typically avoided. Here we present a framework to uncover general quantum signal processing principles of a broad class of bosonic quantum nonlinear processors (), inspired by a remarkably analogous paradigm in nature: the processing of environmental stimuli by nonlinear, noisy neural ensembles, to enable perception. Using a quantum-coherent description of a  monitoring a quantum signal source, we show that quantum nonlinearity can be harnessed to calculate higher-order features of an incident quantum signal, concentrating them into linearly-measurable observables, a transduction not possible using linear amplifiers. Secondly,  provide coherent nonlinear control over quantum fluctuations including their own added noise, enabling noise suppression in an observable without suppressing transduced information, a paradigm that bears striking similarities to optimal neural codings that allow perception even under highly stochastic neural dynamics. Unlike the neural case, we show that -engineered noise distributions can exhibit non-classical correlations, providing a new means to harness resources such as entanglement. Finally, we show that even simple  in realistic measurement chains can provide enhancements of signal-to-noise ratio for practical tasks such as quantum state discrimination. Our work provides pathways to utilize nonlinear quantum systems as general computation devices, and enables a new paradigm for nonlinear quantum information processing. 











[
    Hakan E. Türeci
    September 9, 2024
=====================






§ INTRODUCTION


While engineering quantum systems hinges on isolating quantum components from their noisy classical environment, the observation of any such system necessitates extracting its emitted weak quantum signals back to the classical world. It is therefore essential to bolster quantum signals prior to their interaction with noisy classical modes and the readout noise they introduce, using signal processors that are themselves quantum-mechanical <cit.>. The most widely-used such processors are quantum amplifiers <cit.>, which provide linear gain to an input signal and have been critical to quantum signal processing applications such as high-fidelity quantum state readout <cit.> and generation of non-classical light <cit.>. In spite of this success, the linearity of quantum amplifiers means they are only able to reveal a subset of the features of a complex quantum signal. A well-known limitation is the estimation of higher-order correlations. When only linear readout is available, the nonlinearity demanded by correlation calculations must be provided by classical post-processing: this process amplifies readout noise, leading to a signal-to-noise ratio that degrades exponentially with the order of the desired correlation function <cit.>. For signals that exhibit non-classical, higher-order quantum correlations <cit.>, linear processing restricts our ability to fully resolve the quantum domain, thereby limiting our capacity to control it.

 











Interestingly, even though a rich expanse of quantum devices can be realized beyond the confines of linearity <cit.>, the use of nonlinear quantum systems as general purpose quantum signal processors has been limited. Partly, this is because there are no established general principles of quantum information processing with nonlinear quantum systems.  

The very fact that nonlinear systems are a priori not bound by constraints of a linear dependence between input and output - which could hold the key to new ways of processing quantum signals - makes their operation much more complicated to analyze. More importantly, a complete description of the operation of any general quantum processor must be constrained by quantum mechanics, namely the uncertainty principle <cit.>. For linear quantum amplifiers, this translates to well-known limits on noise added during the amplification process <cit.>. Analogous limits are much less explored for nonlinear amplifiers (with some exceptions <cit.>), and nonlinear regimes are often even associated with excess noise <cit.>. As a result, in spite of some measurement schemes effectively utilizing nonlinear systems <cit.>, nonlinearity is often considered an inconvenience in quantum signal processing, being the source of gain saturation <cit.> and operational instabilities <cit.>, which has led to significant efforts to mitigate nonlinear effects <cit.>.

Remarkably, although the processing of weak stochastic signals using nonlinear devices is still not thoroughly explored in the quantum domain, this paradigm is fundamental to perhaps the most well-known information processor: the brain. It is understood that neurons, the signalling cells that constitute all nervous matter, function as part of large neural ensembles whose collective firing properties can be highly nonlinear <cit.> and exhibit great stochasticity <cit.>. Experiments recording the activity of neural ensembles in the visual cortex of animals have provided robust evidence that neuronal response exhibits significant variability even under repeated presentations of identical stimuli (see Fig. <ref>(a)), with the total noise power even increasing with the size of the neural ensemble <cit.>. Nevertheless, such noisy ensembles are able to successfully process distinct stimuli to enable perception <cit.>. Research over the past few decades has begun to establish an explanation <cit.>. Nature appears to prefer encoding cortical signals in the response of collective neuron modes, as opposed to highly-variable single neuron dynamics (referred to as population coding <cit.>). Furthermore, by exploiting the nonlinear response of neurons <cit.>, the encoding of sensory information in collective neuron modes (or coding directions) that have the largest noise power is avoided <cit.>. While neural noise still places limits on sensory signal discrimination <cit.>, this coding principle allows the perception of weaker stimuli than would be resolvable if the largest noise mode was used for coding. This mechanism provides an example of how Nature is able to harness nonlinearity to facilitate signal processing in the presence of seemingly overwhelming stochasticity. 









Motivated by the example of stochastic neural processing, we therefore ask the question: can nonlinear, fundamentally stochastic quantum systems provide an advantage in the detection and processing of weak quantum signals beyond the paradigm of linear amplification? If so, can any general principles be established? Answering this question in its most general setting requires a description of the joint dynamics of the quantum nonlinear processor (QNP) and the signal generating quantum system (QS), often excluded in descriptions of linear quantum amplifiers <cit.> (with exceptions e.g. Ref. <cit.>). Here we develop the necessary framework to model a  as an in situ processor of signals generated by a quantum system (QS) it is coherently linked to (see Fig. <ref>(b)); our approach thus accounts for the  nonlinearity (with some restrictions), the QS- interaction (including the possibility of their entanglement), but most importantly for not only the quantum fluctuations of the  itself, but also those due to the quantum nature of the input signals. Building on this framework, our analysis is able to identify two extremely general aspects of quantum information processing enabled by , which we summarize here. First, by characterizing their now nonlinear input-output map, we show that  enable higher-order transduction: the mapping of higher-order observables, such as the correlation of quantum signals, to lower-order observables such as quadrature means, making them accessible using much simpler linear readout protocols. Secondly,  are able to coherently manipulate incident quantum fluctuations so as to direct amplified noise to observables that do not encode the transduced signal. This feature, which is particularly reminiscent of population coding in noisy neural ensembles, relies on both the  nonlinearity and its deployment as a quantum processor. 











To demonstrate the above results, we show  at work for a practical quantum information processing task inspired by Gaussian state tomography <cit.>: the binary discrimination of states for which the quadratures of constituent QS modes possess identical mean values, but differ only in their quantum fluctuation signatures (Fig. <ref>(b)). These signatures may be evident in the quantum fluctuations of signals emanating from a single QS mode, or may only be revealed in non-classical noise correlations across distinct modes, such as for entangled states. When signals from a QS in such states are processed using standard linear amplifiers and heterodyne readout, the limitations of a linear input-output map ensure that obtained output distributions exhibit no difference in their mean values (Fig. <ref>(c), top). As a result, the distributions are impossible to distinguish perfectly using only a linear decision boundary. 

In contrast, we show that  are able to transduce the quantum fluctuation signatures of incident signals into displacements observable via heterodyne measurement, effectively functioning as quantum `cross-correlators' across multiple QS modes. As a result the obtained output distributions (Fig. <ref>(c), bottom) exhibit a nonzero mean separation δμ, enabling a linear classifier to separate them perfectly. Most importantly,  enable all the nonlinear processing demanded by this discrimination task to be performed on pristine quantum signals, prior to their corruption by classical readout noise n̅_ cl. Consequently, the signal-to-noise ratio for -enabled discrimination degrades only as n̅_ cl^-1. In contrast, with standard linear amplifiers a nonlinear post-processing step must be performed on the final classical signals; this worsens readout noise, furnishing a signal-to-noise ratio that scales as n̅^-2_ cl. We note that the concept of transduction using nonlinear systems has been explored previously <cit.>, often via a lossless Hamiltonian description; here we consider a much more general setting, and account for practical constraints of losses and measurement. Secondly, here  are deployed as nonlinear processors of stochastic quantum signals; this paradigm is therefore distinct from the more routinely explored use of quantum systems to compute nonlinear functions of deterministic classical signals.

Finally, we show how  can control quantum fluctuations for information processing in ways unavailable to their linear counterparts. Linear amplifiers are restricted to perform the same transformation on an incident signal and its noise; for a fixed input signal, therefore, a decrease in the magnitude of the amplifier output noise is accompanied by a decrease in the magnitude of the output signal, and vice versa. Unconstrained by linearity,  enable a much more complex interplay between the transduced signal and the output noise. In particular, we show how tuning  parameters alone, the output noise in the signal carrying quadrature P_δμ (the unique quadrature parallel to δμ) can be decreased while the transduced signal magnitude ||δμ|| remains fixed (Fig. <ref>(c), bottom). Most strikingly, a tuning condition can be reached such that the signal carrying quadrature is parallel to the quadrature with minimal output noise, and thus orthogonal to all other quadrature combinations which possess larger noise, a scenario that appears very similar to optimal coding in neural circuits. Crucially, for  processing quantum input signals, the large noise modes include quantum fluctuations generated by the upstream QS, but also fluctuations amplified by the  during processing itself, referred to as `added noise'. The minimum noise mode can even reach non-classical (i.e. sub-vacuum) levels, in the presence of squeezing or entanglement. Therefore the ability to control noise using  provides a practical means to harness non-classical correlations for quantum state discrimination.























A quantum-coherent description of a QS and  in the same measurement chain, valid for multi-mode systems, and across a range of excitation conditions and interaction types, demands multiple theoretical techniques. Our key analytic results are built upon a nonlinear van Kampen expansion <cit.> of the Fokker-Planck equation associated with the quantum state of the complete measurement chain. At the expense of being restricted to weakly nonlinear , this approach has the advantage of applying to arbitrarily-multimode bosonic quantum systems. As such, we expect the uncovered operating principles to hold for general  under heterodyne monitoring, especially relevant to cQED. These results are numerically verified and extended to stronger nonlinearities using a truncated cumulants approach that simulates measurement-conditioned dynamics of the measurement chain, with a complexity that is only quadratic in the number of total modes. Supplementing these results via full (stochastic) master equation simulations and exact results for select few-mode, multi-system measurement chains <cit.>, we are able to provide a comprehensive description of quantum information processing using .






The remainder of this paper is organized as follows. In Sec. <ref>, we introduce our model for processing quantum signals using a , provide a summary of the key results of our theoretical approach, and introduce the particular classification tasks we analyze. Sec. <ref> then analyzes quantum state classification using a single-mode , based on a model that can be easily realized in cQED experiments. We highlight the crucial role of  nonlinearity and discuss how  parameters enable control over quantum fluctuations, including the ability to harness non-classical correlations. In Sec. <ref> we provide a comparison of quantum state measurement using a  against standard linear quantum amplifiers, finding improved robustness to added classical noise for the -based scheme. Sec. <ref> then considers quantum state classification tasks that require the non-local processing of quantum fluctuations, where a multi-mode  becomes essential. Here, we demonstrate how output field entanglement can be engineered using a  to produce sub-vacuum noise in a multi-mode quadrature. The paper concludes with a discussion of possible applications of  in the quantum information processing landscape.




§ QUANTUM NONLINEAR PROCESSORS FOR QUANTUM SIGNALS





 §.§ Quantum measurement chain for processing quantum signals


The complete measurement chain we consider is depicted in Fig. <ref>(b). The conditional evolution of the quantum state  of the chain under continuous measurement is formally described by the stochastic master equation (SME)

    d =  ( +  + ) dt +∑_k𝒮[√()b̂_k].

The superoperator  describes a quantum system (QS) that is the source of quantum signals to be processed for a given quantum information processing task. We consider the form

    ρ̂   =  -i[ℋ̂^(l)_ QS,ρ̂] + ∑_mκ_m 𝒟[â_m]ρ̂,

where ℋ̂_ QS is a linear Hamiltonian governing the dynamics of M bosonic modes â_m, m ∈ [1,M], and experiencing damping {κ_m}. The superscript (l) indexes distinct states of the QS, to be distinguished in a classification task. 







This QS is coupled to the quantum nonlinear processor () in the same measurement chain (via a non-reciprocal coupling described by , see Appendix <ref>), enabling the  to monitor the QS as an in situ measurement apparatus. The  is governed by , 

    = -i[ ℋ̂_ + 𝒩̂_ ,],

where ℋ̂_ is a general linear Hamiltonian governing the dynamics of K bosonic modes b̂_k, k ∈ [1,K]. The  nonlinearity can then be furnished by any general bosonic nonlinear interaction; for simplicity we consider a Hamiltonian describing K Kerr modes with common nonlinearity strength Λ, 𝒩̂_ = -∑_k Λ/2kkkk.

The chosen measurement scheme is of the standard heterodyne type: it comprises continuous monitoring of the  decay channels (with unit efficiency). The resulting measurement-conditioned evolution of the complete measurement chain in accordance with quantum measurement theory <cit.> is governed by the stochastic heterodyne measurement superoperator 𝒮[√()b̂_k], with a rate  assumed equal for all modes (see Appendix <ref>). This yields heterodyne records ℐ_k(t),𝒬_k(t) for each monitored  mode <cit.>:

    ℐ_k(t)    = ξ_ℐ_k(t) +√(γ_ H)[ X̂_k +  ξ_ℐ_k^ qm(t)] + √(n̅_ cl) ξ^ cl_ℐ_k(t), 
    𝒬_k(t)    = ξ_𝒬_k(t) + √(γ_ H)[P̂_k + ξ_𝒬_k^ qm(t)] + √(n̅_ cl) ξ^ cl_𝒬_k(t).

Heterodyne monitoring probes the canonically-conjugate  quadratures X̂_k = 1/√(2)(k+k), P̂_k = -i/√(2)(k-k), but a given measurement record also contains contributions from multiple noise sources ξ. The terms ξ_ℐ_k,ξ_𝒬_k describe vacuum noise that would be present even if no signal was emanating from the monitored quantum modes (i.e. → 0). The much more interesting terms are marked `qm': these describe noise contributions of a quantum origin, such as non-classical correlations due to squeezing or entanglement, or added noise by quantum dynamics, and are contingent on the measurement superoperator 𝒮 (see Appendix <ref>). In contrast, the terms `cl'  define classical readout noise in the measurement chain; these are not associated with a stochastic measurement superoperator and hence have no backaction on the quantum measurement chain. Equivalently, n̅_ cl quantifies noise added after the so-called Heisenberg-von Neumann cut <cit.>. 

These stochastic records are thus often filtered to obtained heterodyne quadratures (I_k,Q_k) ≡∫_t_0^t_0+𝒯 dτ 𝒦(τ)×(ℐ_k(τ),𝒬_k(τ)), where 𝒦(τ) is the filter function (we assume a boxcar filter 𝒦(τ) = 1/√(2𝒯) ∀ τ) over a window of length 𝒯 starting from an initial time t_0. The quadratures can be compactly represented via the vector x = (I_1,Q_1,…,I_K,Q_K)^T ∈ℝ^2K. For any quantum information processing task, the data x is typically further processed to obtain a vector y of output features, a step we define generally via y = x, which can include ensemble averaging over distinct shots, but also nonlinear processing to be clarified in due course. The complete measurement chain thus described is designed to measure QS properties via the obtained outputs y; note that our description subsumes the standard paradigm of linear quantum amplification plus heterodyne measurement if 𝒩̂_→ 0 and an appropriate choice of ℋ̂_ is made.


















We can now specialize to the consideration of binary quantum state discrimination (although  can be deployed for more general processing tasks as well). Binary discrimination has a simple objective: to distinguish QS state ρ_ QS^(l) from ρ_ QS^(p), where ρ_ QS^(l)≡ tr_[^(l)], based on the corresponding outputs y^(l) and y^(p) from the measurement chain. To this end, we introduce a standard measure of the distinguishability of the two measured distributions of y^(l) and y^(p): Fisher's discriminant , defined as

    (y) = δμ^T ·𝐕^-1·δμ,

where δμ = μ^(l)-μ^(p) is the difference of means of the two measured distributions, while 𝐕 = 1/2(Σ^(l)+Σ^(p)) is a measure of their combined variance:

    μ^(l) = 𝔼[ y^(l)], Σ^(l) = 𝔼[y^(l)y^(l)T] - 𝔼[y^(l)]𝔼[y^(l)]^T.

Note that  has the intuitive form of a generalized signal-to-noise ratio (SNR). Furthermore, the fidelity 𝒞 of classifying two Gaussian distributions with identical covariance matrices Σ^(1)=Σ^(2), is simply 𝒞 = 1/2(1 +  erf/2√(2)), where erf z is the standard Gaussian error function; as →∞, 𝒞→ 1. For binary QS state discrimination, the  of measured distributions is correlated with the fidelity of discriminating the two QS states that give rise to these distributions. 

Using such binary classification tasks, where the concepts of signal δμ and noise 𝐕 are precisely defined, we will demonstrate how  can process quantum signals in situ for practical quantum information processing applications.



 §.§ Approximate input-output map for quantum nonlinear processors



In the context of binary state discrimination, the role of  can be characterized by evaluating the central quantities δμ and 𝐕 identified in the previous section. However, the nonlinearity introduced by the  - central to its operation - generally excludes exact theoretical treatments that would be valid if the measurement chain comprised only linear bosonic modes and interactions. Furthermore, allowing the QS and  to comprise arbitrary numbers of modes renders exact numerical integration of Eq. (<ref>) unfeasible for all but the lowest excitation numbers.

Our solution begins by introducing an alternate set of dynamical variables to describe the quantum state  of the entire nonlinear measurement chain: quantum cumulants (see Appendix <ref>). Formally, cumulants are an infinite set of dynamical variables parameterizing , indexed by an integer order n_ ord∈ℤ^+. Crucially, however, we show that retaining cumulants only up to a certain finite order n_ ord≤ n_ trunc can very accurately describe the nonlinear measurement chains we consider, contingent on the nonlinearity strength. In this work, we show that n_ trunc = 2 is an excellent approximation provided certain well-understood and achievable constraints on the nonlinearity strength are met. In this case, defining b̂≡ (b̂_1,b̂_1^†,…,b̂_K,b̂_K^†)^T and â analogously for QS operators, the retained first-order cumulants are simply single-operator expectation values [  â; b̂ ], while second-order cumulants are normal-ordered covariances, 𝐂 = ⟨:[  â; b̂ ][  â^T b̂^T ]:⟩ - [  â; b̂ ][  â^T b̂^T ]. This constitutes a truncated cumulants ansatz for : an efficient description of multimode nonlinear quantum dynamics, with the number of retained cumulants scaling only quadratically, and not exponentially, with the total number of modes =K+M for n_ trunc=2.

We then develop an approximation that allows the truncated cumulants to be solved for analytically, formally employing a nonlinear van Kampen (NVK) expansion <cit.> in the  nonlinearity (also referred to as a `system size' expansion, see full details in SI Sec. <ref>). Under the NVK approximation, first-order cumulants are written as [  â; b̂ ]= Λ^-1/2[  â; b̂ ] + [  δâ; δb̂ ] where we have introduced the dimensionless nonlinearity Λ = Λ/γ, and γ, the  mode decay rate, serves as a normalization factor. Λ→ 0 describes the classical (large occupation) limit, where â, b̂ become the dominant contributions to first-order cumulants; these define the expansion point and satisfy the equations of motion:

    
    d_t â   = 𝐋^(l)_a â + Λ^1/2η⃗^(l)
    
        d_t b̂   = 𝐋_b b̂ + N⃗_b(b̂) - η⃗_b^(l)


These equations immediately provide useful insight. Being linear, the QS response is governed entirely by 𝐋_a and possible coherent drives η⃗^(l), as dictated by ℋ̂_ QS^(l) (and dissipative terms). The non-reciprocal QS→ interaction defined by  ensures that the QS drives the  via the coupling Γ and not vice-versa, leading to an effective QS state-dependent drive η⃗_b^(l)≡Γâ^(l) on the . The  dynamics, in contrast, contain both a linear contribution 𝐋_b as well as a nonlinear contribution N⃗_b(b̂), the latter determined by the nonlinear Hamiltonian 𝒩̂_. Thus Eq. (<ref>) is nonlinear in b̂.



The full quantum state further requires specifying the deviation δb̂ (we can show that δâ→ 0) and the second-order cumulants 𝐂. Starting with the latter and introducing the block form 𝐂 = [    𝐂_a   𝐂_ab; 𝐂_ab^T    𝐂_b ], we show that second-order cumulants satisfy the Lyapunov differential equation:

    d_t 𝐂 =  
        [ 𝐂[ 𝐋_a   0;  -Γ 𝐉_b ]
        + m.t.] +
        [ 𝐃_a^(l)       0;       0     𝐃_b ],

where m.t. is the (unconjugated) matrix transpose. Here 𝐉_b is the Jacobian matrix of the , [𝐉_b]_ij∝ [𝐋_b]_ij + ∂ [N⃗_b]_i/∂b̂_j, with [·]_ij indicating tensor notation, and (·) specifying evaluation at the expansion point b̂. Then 𝐃_a,  𝐃_b are diffusion matrices for the QS and  respectively; these describe incident fluctuations - quantum or classical - beyond vacuum fluctuations, and must be nonvanishing to yield nontrivial 𝐂.

Finally, we introduce the most important dynamical equation in the NVK approximation, for δb̂:

    d_t δb̂ = 𝐉_b δb̂ + Λ^1/2𝐇_b : 𝐂_b^(l).

Crucially, the change in first-order  cumulants depends on the second-order cumulants, via the final term. This dependence is quantified by the Hessian tensor 𝐇_b of the , defined as [𝐇_b]_ijk∝∂ [N⃗_b]_i/∂b̂_j ∂b̂_k. The Hessian operates on matrices - here on 𝐂_b - to return a vector, via the tensor double contraction (:) over pairs of indices. The Hessian contribution is often neglected in standard linearization schemes; we will see its crucial role in quantum nonlinear processing. Note that if the  is linear, N⃗_b → 0, and the Hessian vanishes.



The NVK approximation, combined with input-output theory <cit.>, allows us to analytically obtain μ^(l) and Σ^(l) as defined in Eq. (<ref>), and eventually . The quadrature means after a filtering time 𝒯 are given by 

    μ^(l) = √(𝒯/2) 𝐔_K( Λ^-1/2b̂^(l) - Λ^1/2𝐉_b^-1𝐇_b :  𝐂_b^(l)),

where 𝐔_K is the K-mode quadrature change-of-basis matrix. The second term demonstrates transduction: the sensitivity of linearly-measurable  quadratures to its higher-order (here, second-order) quantum cumulants. 

The covariance matrix in the long-filter limit (γ𝒯≫ 1, see SI Sec. <ref>) is given by:





    Σ^(l)γ𝒯≫ 1=σ_ vac^2(n̅_ cl + 1)𝐈_K 
       +σ_ vac^2/γ𝐔_K 𝐉_b^-1[ 𝐃_b + Γ𝐋_a^-1𝐃_a^(l) (𝐋_a^-1)^TΓ^T ](𝐉_b^-1)^T𝐔_K^T.

The first term describes output vacuum fluctuations σ_ vac^2 = 1/2 and classical readout noise ∝n̅_ cl, the latter also in units of σ_ vac^2 (𝐈_K ∈ℝ^2K is the identity matrix). The second line then describes all other contributions from quantum noise. The term ∝𝐃_b describes noise added by the  itself. The term ∝𝐃_a^(l), on the other hand, arises due to our quantum-coherent description of both the  and QS: it describes noise originating from the QS, which arrives at the  via the coupling Γ, after undergoing QS evolution via 𝐋_a^-1. Both noise terms are processed by the , as indicated by the appearance of the Jacobian 𝐉_b^-1. 

The above expressions can be used to understand quantum state classification using general weakly-nonlinear bosonic quantum systems. First, we consider the limit of ideal linear quantum amplifiers, processing input drive signals (𝐃_a → 0). In this case 𝐇_b → 0, 𝐉_b^-1→𝐋_b^-1, and Λ^-1/2b̂→ -𝐋_b^-1(𝐋_a^(l))^-1η⃗^(l); hence both the mean and covariance are determined entirely by the matrix 𝐋_b, as must be the case for a linear system. The resulting expressions can be used to obtain the standard quantum limits <cit.> on amplification (see SI Sec. <ref>). 

Practical quantum amplifiers, in contrast, can exhibit nonlinear behaviour for sufficiently strong input signals. Then, the leading contribution b̂ to μ is determined by the nonlinear Eq. (<ref>), while the covariance is determined by 𝐉_b, an indicator of the difference in response of nonlinear quantum systems to signal and noise. However, if the aim is still to process input drives η⃗^(l), the large drives typically needed to reach such regimes lead to high signal-to-noise ratios where this difference is often ignored, with more attention instead paid to mitigating the nonlinear response of b̂.

However, in other quantum information processing tasks, noise takes center stage. A simple case arises when discriminating quantum states l and p such that â^(l) = â^(p), so that the leading term ∝b̂ in μ makes no contribution to δμ. Then, quantum state classification becomes the task of processing quantum fluctuations encoded in 𝐃_a^(l), and consequently in 𝐂_b^(l). The  nonlinearity enables such processing in situ, via the Hessian tensor. Furthermore, while the covariance is still determined by 𝐉_b^-1 alone, δμ is now determined both by 𝐉_b^-1 and by the Hessian tensor 𝐇_b. This enables a complex interplay of signal and noise only possible using nonlinear quantum systems. We demonstrate the implications of this result using various examples in this paper.  

As these expressions are derived using the NVK approximation, one may question their validity beyond the lowest order in  nonlinearity. Importantly, the truncated cumulants ansatz holds past the NVK regime; we use this to develop a computational approach, the Stochastic Truncated Equations of Motion (STEOMs), that can be used to simulate measurement-conditioned dynamics of the nonlinear multimode measurement chain beyond the NVK approximation (see SI Sec. <ref>). The STEOMs allow us to account for classification under finite sampling as in real experiments, numerically verify NVK results, and analyze performance in strongly-nonlinear  regimes. Finally, we provide select comparisons using exact (S)ME integration to qualitatively verify our results without any assumptions on nonlinearity strength (see SI Sec. <ref>).




 §.§ Quantum state discrimination tasks



We begin by defining the QS of Eq. (<ref>) whose states we wish to classify. Precisely, we consider the QS to be a general M-mode linear quantum system under coherent driving at frequencies {ω_dm}; written in the interaction picture and assuming resonant driving, it is described by the Hamiltonian

    ℋ̂_ QS^(l)   = ( 1/2∑_m G_m^(l)e^-iϕ_m^(l)â_m^2 + ∑_n≠ m G_nm^(l)e^-iϕ_nm^(l)â_nâ_m + h.c. ) 
           +∑_mη_m^(l)(-iâ_m+iâ_m^†).

The integer l then indexes the states we wish to classify, generated by distinct choices of parameters also indexed by l. Such a Hamiltonian can be easily realized in the cQED architecture using tunable parametric drives <cit.>.

We are ultimately interested in coupling the QS directly to the  - a nonlinear quantum device - for quantum signal processing. However, to provide a benchmark for later comparisons, we begin by introducing the classification tasks and how they may be performed in a simpler, more usual context: heterodyne readout of the QS using linear quantum amplifiers. Recall that this simplified measurement chain, depicted in Fig. <ref>(a) for M=2, is still described by the same general form of SME, Eq. (<ref>), provided 𝒩̂_→ 0 and ℋ̂_→ℋ̂_ PP, where the latter Hamiltonian specifically describes the phase-preserving (PP) style of linear amplifier (for full model, see Appendix <ref>).



We can finally define the binary quantum state discrimination tasks we wish to analyze: our work considers the discrimination of QS states which possess the unique feature that any monitored modes (i.e. modes that are coupled to the downstream processor) have identical steady-state quadrature expectation values for both states. More precisely, for two distinct states indexed by l and p, â_m^(l)=â_m^(p) =  ∀ m  s.t. Γ_m ≠ 0, where  defines a common amplitude of monitored modes. Such states can then only be distinguished on the basis of their quantum fluctuation statistics, if these are distinct.

For Task I of this type, we consider distinguishing a single-mode squeezed state of mode â_1 (l=1) from a two-mode squeezed state (l=2), by monitoring only mode â_1 (Γ_1 ≠ 0, Γ_2 = 0). The QS parameters used to realize these states are summarized in Table <ref>. The top plot in Fig. <ref>(b) shows the steady-state quadrature expectation values of the monitored mode â_1, which are equal by construction. In contrast, the depicted full QS quadrature covariance matrices, related to the second-order QS cumulants 𝐂_a^(l) via a simple change-of-basis (see SI), clearly indicate the differences in quantum fluctuations of the two QS states. The discrimination task requires extracting these differences from measurement records x, typically following some processing to obtain readout features y=x. If we restrict this processing to only linear operations, the obtained distributions of features y = x = (I_1,Q_1) (simulated using the STEOMs) are shown in the first panel of Fig. <ref>(c) for single-shot (S=1) readout of different realizations of each QS state. Clearly, the mean values of both measured feature distributions overlap; note that this mean would be unchanged by averaging over repeated shots (S>1). Hence ||δμ||→ 0 and therefore → 0, leading to the conclusion that the two states cannot be distinguished in the space of (I_1,Q_1). 















However, the distributions of measured quadratures are visibly distinct, just not in their mean values, but instead in their second-order moments or covariances. To estimate such second-order moments (routinely required for example for Gaussian state tomography <cit.>), the standard approach is to obtain S shots and estimate the variance of the measurements over the dataset yielding readout features y = x = 1/S∑_s^S(I^2_1,Q^2_1) (shot s dependence of quadratures I_k,Q_k is implied).  For Task I, we plot distributions of these nonlinear readout features in the second panel of Fig. <ref>(c). The mean values of these features are now estimators of the monitored mode covariances; as these are distinct, the centers of the distributions no longer coincide, rendering them linearly separable. Without this nonlinear processing step, it is impossible to perfectly distinguish the QS states in Task I under QS readout using only linear amplifiers.




For Task II we consider a more complex classification task, which requires both nonlinear and nonlocal information processing. Specifically, we wish to distinguish a pair of two-mode squeezed states l ∈ (3,4) that experience an identical amount of joint squeezing, but whose two-mode squeezed quadratures are mutually orthogonal. Such states can be generated using two-mode squeezing interactions of equal strength but opposite phase (see Table <ref>). The full QS covariance matrices (see Fig. <ref>(d)) then differ only in the sign of cross-correlations between the two QS modes (off-diagonal blocks); all other covariance metrics are identical. 

Measuring such cross-correlations necessitates monitoring both QS modes (Γ_1, Γ_2 ≠ 0). We again first show measured features under linear processing alone, now for two distinct modes, y = x = (I_1,I_2), in the left panel of  Fig. <ref>(e). Clearly, the distributions have anisotropic profiles that differ in the axis of minimal fluctuations, indicative of two-mode squeezing of distinct joint quadratures. However, the centers of measured distributions overlap as before, due to their equal mean values. Also as before, nonlinear processing to estimate second-order moments provides the solution, but with an important caveat: all local second-order moments are insensitive to the sign of cross-correlations of the QS modes. Instead, the estimation of nonlocal second-order moments, y = x = 1/S∑_s^S(I_1I_2,Q_1Q_2), is necessary to obtain linearly separable distributions for Task II.

Therefore, both our considered classification tasks - instances of the broad family of tasks that require measuring correlations between observables to distinguish quantum states - demand nonlinear processing of heterodyne records at room temperature. However, such processing can be particularly sensitive to high-temperature noise ∝n̅_ cl in the measurement chain. A key result of our work is that using  to process signals from the QS in the same cryogenic environment where they are generated can circumvent the need for nonlinear processing at room temperature. This renders classification schemes incorporating  much more robust to excess noise in the measurement chain (see Sec. <ref>).










§ QUANTUM STATE DISCRIMINATION USING A SINGLE-MODE 












We now demonstrate the paradigm of in situ nonlinear processing of quantum signals enabled by the  in the context of quantum state discrimination. In particular, we will show that such processing would not be possible using standard linear amplifiers in the same measurement configuration. Our analysis begins with the simpler Task I; Task II is analyzed in Sec. <ref>. 

Recalling that only a single QS mode is read out in Task I, we consider a K=1  to monitor this QS mode, which proves sufficient. Our chosen  is defined by a single Kerr-nonlinear mode with frequency ω_1 and nonlinearity Λ (for details, see Appendix <ref>). The resulting quantum measurement chain is then depicted in Fig. <ref>(a). When coupled, the quantum state of the  is determined by the state of the QS, as desired: this dependence is shown in Fig. <ref>(b) via the covariance matrices of the complete measurement chain for the two QS states to be disntinguished. In the NVK approximation, the dependence is quantified by the linear Lyapunov system, Eq. (<ref>), although the exact relationship can be more complex. From Eq. (<ref>), it is also clear that this inter-dependence itself does not require the  to be nonlinear; it only requires a nonzero coupling Γ.

Crucially, we require this dependence to be transduced to  readout features to enable successful QS state discrimination by linear readout of the  alone. It is here that the role of  nonlinearity becomes clear. To demonstrate this, we analyze  readout distributions for features obtained under linear processing only, y = x = (I_1,Q_1), as a function of nonlinearity. Our STEOMs framework enables simulating individual quantum trajectories of the QS and  (crucially accounting for the latter's nonlinearity), providing the resulting heterodyne measurement records defined by Eq. (<ref>) that are used to construct y; typical examples are shown in Fig. <ref>(c), (d). By repeating over several measurement chain initializations for each QS state, here l=1,2, we obtain distributions of measured features shown in Fig. <ref>(e), (f). Then Fisher's discriminant  computed for the two distributions determines the fidelity of classifying the QS states.

Recall that  depends on the mean separation δμ of the distributions. The NVK approximation provides a very useful form of δμ, Eqs. (<ref>), which specialized to the single-mode Kerr  takes the form:

    δμ = √(𝒯/2)𝐔_K [  √(Λ/γ) 𝐉_b^-1𝐇_b : ( 𝐂_b^(l)-𝐂_b^(p)) ],

where γ =  + Γ_1 is the total  damping rate. The nonlinear dependence is associated with the Hessian 𝐇_b, which vanishes for linear systems, so that ||δμ|| = 0. This is exactly what is observed for readout with linear amplifiers in Sec. <ref>, as well as with vanishing  nonlinearity (Λ = 0), shown in Fig. <ref>(e). In this case, increasing the filtering time 𝒯 also has no effect on ||δμ||, and consequently on . 





The measured distributions change qualitatively when the  is nonlinear (Λ≠ 0), as shown in Fig. <ref>(f). Now ||δμ||≠ 0, and the mean separation increases with 𝒯 as predicted by Eq. (<ref>). The  is able to transduce information encoded in  fluctuations 𝐂_b^(l), which in turn depend on the QS state, to  quadrature mean values, an operation governed by the Hessian 𝐇_b. Now even with only linear processing · of measured quadratures, ≠ 0: a suitable  is thus able to circumvent the need for nonlinear post-processing of measurement records, and the resulting classification accuracy shown in Fig. <ref>(f) approaches unity. In contrast, for vanishing nonlinearity, perfect classification is impossible using only ·.



This simple example highlights the extremely general principle of transduction using . However, making successful use of this principle requires asking some more complex questions, each of which we address in this paper. First, one may ask whether any  works for a given quantum state discrimination task. Unsurprisingly, this is not the case; instead,  optimization is important, and features a landscape far richer than that of linear quantum amplifiers. The reason for this is simple: the very nonlinearity that enables a nonzero `signal' ||δμ|| also determines the fluctuations 𝐂_b^(l) that appear in the `noise', characterized by the measured covariance matrix Σ^(l). This leads to a complex interplay between signal and noise for classification using , where one cannot be optimized without considering the other. Further complicating this interplay is the noise added by the  itself, which must also be accounted for in the optimization. While the limits of added noise have been well-established for linear quantum amplifiers <cit.>, much less is known about added noise in the context of nonlinear quantum systems, and even less so for driven-dissipative cases relevant in practical settings. 

Our framework characterizes this nontrivial relationship between signal and noise in . Even more importantly, it also accounts for noise added by  during processing. The latter crucially includes the possibility of the noise being correlated, even non-classically, either due to correlations in the incident quantum signals arriving from the QS, or even due to correlations generated by the  dynamics itself. Crucially, we find that the nonlinear nature of  implies that the signal and noise contributions exhibit different relative functional dependencies, a freedom not available to linear quantum systems. This distinction provides the ability to manipulate the readout noise for  so as to limit its impact on the signal, by tuning parameters of the  alone. This capability proves crucial to the optimization of a  for learning, as is discussed in the next subsection.

We emphasize that a precise understanding of the noise physics and optimal operating regimes of  is not merely an academic question, but one that must be addressed to understand the ultimate limits of information processing with such devices. For example, for classification with minimal resources (e.g. filtering time 𝒯), information processing must be carried out in regimes where the signal ||δμ|| is of comparable magnitude to the standard deviation of measured distributions. Such a case is depicted in Fig. <ref>(f). From the observed non-isotropic distributions, it is clear that  readout can exhibit correlated noise statistics. If the correlated nature of fluctuations was ignored (in other words, if the added noise was assumed to be uncorrelated and isotropic), the decision boundary would simply be the bisector of the mean separation ||δμ||, depicted by the dashed purple line. Unless 𝒯 is very large so that ||δμ|| dominates over noise (a significant increase in measurement resources), the resulting classification accuracy (purple) is substantially lower than the optimal. Therefore the correlation properties of noise in  readout - including of noise added by the  itself - must be accounted for, as we do here and analyze in detail next.

































































 §.§ Coherent control of quantum fluctuations using a 



We have seen that the  nonlinearity is essential to learn QS states using linear readout. However, not all  will be equally effective at learning: performance metrics such as  for a fixed 𝒯 are strongly dependent on  operating parameters. In this section, we address how  can be optimized for learning. Our analysis centers around understanding how Fisher's discriminant  for -enabled classification can be maximised, to which end analytic expressions obtained using the NVK approximation are particularly powerful. In particular, while our analysis is presented for Task I using a K=1 mode , the NVK approximation allows us to uncover general principles for  operation that apply to more complex tasks with larger , as we show in Sec. <ref> with Task II.

A nonzero Fisher's discriminant  requires a non-vanishing δμ. Within the NVK approximation, Eq. (<ref>), it is clear that the magnitude ||δμ|| must be determined by the inverse of the Jacobian 𝐉_b^-1 describing the . This is unsurprising, as the Jacobian typically determines the linearized response of a nonlinear system, with the dimensionless susceptibility |χ_b| given by the largest eigenvalue of 𝐉_b^-1. As discussed in Appendix <ref>, for a K=1 mode Kerr-type  this takes the form

    |χ_b| =  max{| eig 𝐉_b^-1|} = γ/γ/2 - √((2n_1)^2- (Δ_1+4n_1)^2  ),

where n_1 = |b_1|^2, obtained by solving Eq. (<ref>). Importantly, it can be shown that |χ_b| depends on the  nonlinearity only via the dimensionless effective parameter ,

    = √(Λ/γ)·Γ_1/γâ_1^(l) = √(Λ/γ)·Γ_1/γ

which is independent of l due to our choice of classification tasks. The above form of |χ_b| is ubiquitous in studies of the linearized response of a variety of Kerr-based systems, from parametric amplifiers <cit.> to frequency comb generators <cit.>. The susceptibility can be made large by suitable parameter choices, including ; typically this is determined by the strength of a separately applied pump tone. Our use of Kerr-based  exhibits a slight difference:  is instead set by the amplitude  of signals incident from the QS upstream. The QS state to be measured hence serves to `pump' the very  being used to measure it (although a distinct  pump tone could equivalently be applied). 

This difference notwithstanding, suitable choices of (,Δ_1) can similarly cause |χ_b| to become large, increasing the magnitude of  response to any input, including to quantum fluctuations from the QS. In Fig. <ref>(a), we plot |χ_b| as a function of Δ_1/γ and  (also Λ/γ for ·Γ_1/γ=80/9). The orange region, where |χ_b| diverges, marks the well-known classical bistability of the single Kerr oscillator <cit.>, here brought about by the mean amplitude  of incident QS signals. Operating near this bistability - and more generally, any instability of the Jacobian 𝐉_b - will increase |χ_b|. 

We therefore consider  near the bistability, more precisely for the fixed detuning Δ_1/γ = -0.67 and for varying  across the vertical dashed line in Fig. <ref>(a). Here the susceptibility |χ_b| is plotted in the inset of Fig. <ref>(b), which exhibits a Lorentzian-like profile with a single maximum. Note that the important quantity for Fisher's discriminant is instead the magnitude of measured mean separation ||δμ||; this is also shown in Fig. <ref>(b), using both the NVK approximation and STEOMs integration. Interestingly, while ||δμ|| does increase in conjunction with |χ_b|, it also displays a double-peak structure that is manifestly distinct. Thus, while |χ_b| is clearly important, it does not completely define the physics of QS state learning using a . Here, the difference can be attributed to the fact that ||δμ|| is not simply the linearized response to an input field, but the response to quantum fluctuations encoded in 𝐂_b^(l), which drive the  via the Hessian tensor 𝐇_b (which is absent in standard linearization approaches). 

The other factor determining  is the noise in measured quadratures, quantified by the covariance matrices Σ^(l). To aid our analysis of noise properties, we first introduce the minimum and maximum noise eigenvalues and corresponding eigenvectors of the 2K-by-2K covariance matrix Σ^(l) as {σ_ min^2(l),v_ min^(l)} and {σ_ max^2(l),v_ max^(l)} respectively. These eigenpairs denote combinations of measured  quadratures with minimal and maximal noise, and for K=1, completely define Σ^(l). We next note that binary classification in an arbitrary-dimensional measured space can be cast into a two-dimensional subspace for visualization. We introduce a vector v_∥ = 1/||δμ||δμ parallel to δμ, which is unique upto normalization, and a vector v_⊥ orthogonal to v_∥, namely v_∥^Tv_⊥ = 0, of which there are 2K-1 choices. These vectors allow us to a define a measured  quadrature P_δμ that is parallel to δμ, and R_⊥ as one of 2K-1 measured quadratures orthogonal to δμ,

    P_δμ = v_∥^Ty, R_⊥ = v_⊥^Ty.

The quadrature P_δμ has the property that 𝔼[P_δμ^(l)-P_δμ^(p)] = ||δμ||, as may be readily verified. For isotropic noise distributions where only the mean separation determines distinguishability, P_δμ then defines the only feature that need be measured for classification. Of course, noise in our situation is far from isotropic. We therefore also introduce the noise projected along δμ, 

    σ^2(l)_δμ = v_∥^T·Σ^(l)·v_∥.

Note that the noise in the P_δμ quadrature is exactly σ^2(l)_δμ. 



In Fig. <ref>(c), we now plot this quantity, together with the minimum and maximum noise eigenvalues, for l=1 and for the same parameters as Fig. <ref>(b). Finally,  and  are plotted over the same parameter range in Fig. <ref>(d). Clearly, σ_ max^2(1) increases following |χ_b|, describing the amplification of fluctuations by the . In contrast, σ_ min^2(1), which should represent squeezing, does not undergo a corresponding dip; this is because the Kerr-based  does not generate ideal squeezed states. Regardless, we still have σ_ min^2(1) < σ_ vac^2, so the measured distributions exhibit squeezing below vacuum. The projected noise σ_δμ^2(1) varies in between these maximum and minimum noise eigenvalues, with some important features that we now describe.

To illustrate the interplay of mean separation and noise, we consider three  defined by nonlinearity values marked (i)-(iii) in Fig. <ref>. As seen in Fig. <ref>(b), measured output distributions from  (i) have the largest mean separation. However, the projected noise plotted in Fig. <ref>(c) is far from minimal. From the resulting measured distributions depicted in Fig. <ref>(i), it is visually clear that noise in the direction of δμ is not minimized.

 (ii) ends up being the most interesting operating point. While the mean separation of outputs from  (ii) is lower than (i), the projected noise is minimized, σ_δμ^2(1) = σ_ min^2(1), as seen in Fig. <ref>(ii). From the measured distributions, we see that δμ is now aligned with the direction of minimum noise, defined by the noise eigenvector v^(1)_ min. The resulting Fisher's discriminant  and  are in fact larger than for  (i), even though ||δμ|| is slightly smaller.

Finally,  (iii) is specifically chosen so that its output ||δμ|| is equal to that for  (ii); however, the projected noise is much larger. From Fig. <ref>(iii), it is visually clear that the measured distributions from  (iii) are the least distinguishable of the three  considered, leading to the smallest value of  and hence .





This analysis has several important conclusions. First, it emphasizes the necessity of accounting for quantum fluctuations for optimal classification. This is the only factor that distinguishes  (ii) and (iii). Secondly, we are able to clarify the role of -mediated squeezing of quantum fluctuations in classification. As seen from Fig. <ref>(d), the measured outputs from all three  exhibit distributions with squeezing below vacuum (σ_ min^2(1) < σ_ vac^2). However, this squeezing does not optimally aid classification unless the specific measured  quadrature defined by P_δμ is the one undergoing squeezing, here  (ii). 

Finally, and most remarkably, even a simple, single-mode Kerr  provides the ability to manipulate quantum fluctuations separately from mean values, so that the aforementioned optimal squeezing scenario can actually be engineered using suitable parameter choices. This capability is emphasized by  (ii) and (iii), which exhibit the same `signal' ||δμ|| but very different projected noise properties, a scenario not possible in, for example, linear amplifiers. The remarkable freedom in adjusting the projected noise independently of the mean separation is demonstrated via the surface plot of σ_ min^2(1) in Fig. <ref>(e); here the white curve, which defines operating points with fixed ||δμ||, traverses through regions of both very large projected noise, as well as minimal projected noise that is sub-vacuum. Clearly, optimal performance requires operating in regimes where the projected noise can be lowered (not necessarily always exactly minimized, as  also depends on ||δμ||).







Before proceeding, we make two further observations. First, it is clear that ||δμ|| (and hence ) exhibits a complex dependence on the  nonlinearity, in contrast to the monotonic dependence alluded to by Eq. (<ref>). This is because simply varying Λ also modifies  via Eq. (<ref>) if  is fixed, thus changing the  operating conditions in a nontrivial fashion. To explore the performance improvement with Λ hinted at by Eq. (<ref>) a more careful analysis is needed; this is presented in Appendix <ref>. Secondly, note that all quantities calculated using the NVK approximation agree well with their counterparts obtained using exact integration of the STEOMs, especially in qualitative terms. This highlights the utility and validity of the NVK approximation in analyzing  and, more generally, networks of coupled nonlinear quantum systems.






§  ADVANTAGE FOR QUANTUM SIGNALS



Thus far we have demonstrated that  enable processing of quantum signals in a way that is distinct from linear quantum amplifiers. However, it is not yet clear that this difference will lead to a practical advantage for quantum information processing. We now address this key question.

To do so, we will compare readout using the  against standard linear quantum amplifiers for the task of quantum state discrimination. Note that both these schemes have already been discussed separately in Sec. <ref> and Sec. <ref> respectively, albeit the latter only for phase-preserving amplification. Now, we take advantage of the sufficiently general and comprehensive description of quantum measurement chains enabled by our framework to consider the configuration in Fig. <ref>, where a general quantum processor follows the QS to be read out. The processor can be the , or one of a phase-preserving (PP) or phase-sensitive (PS) linear quantum amplifier (see Appendix <ref> for exact models). In this way, the individual couplings, losses, and readout rates of modes constituting the quantum processor can be held fixed from one model to the next, ensuring a direct comparison. Secondly, we now also include the effect of excess classical noise ∝n̅_ cl, which necessitates the use of quantum processors in the first place. We can then identify what practical advantages (if any) are enabled by the .







We briefly note that if one considers only binary quantum state discrimination tasks, provably optimal measurement protocols are known, namely Helstrom measurements <cit.>. While a sufficiently broad definition of a  would also incorporate such schemes, such measurements are specific to the particular states to be distinguished, and are not always straightforward to perform as they typically require non-Gaussian operations or strong constraints on coherence. We envision the  considered in this paper to be akin to linear quantum amplifiers in this respect: they may not necessarily be the optimal choice for a single task, but can provide general quantum processing for quantum signals for much more general tasks. Nevertheless, we will still compare the  and linear amplifiers against bounds set by optimal discrimination schemes, to be discussed shortly.

Even for processors in the same measurement chain, one last degree of freedom remains: the operating parameters. For linear PS and PP amplifiers, it is straightforward to identify equivalent operating regimes: we can simply choose them to operate at the same gain, or equivalently the same susceptibility |χ_b|. The PS amplifier has only one additional degree of freedom, namely the phase of the quadrature to be amplified. Unlike these amplifiers, the  does not provide only linear amplification, and so its operation is not determined entirely by |χ_b|, as shown earlier. Nevertheless, we choose to operate the  at the same fixed |χ_b| as the linear amplifiers to ensure comparable operating points. These operating points are defined by `isogain' contours of |χ_b| in Λ-Δ_1 space; considered examples are shown in Fig. <ref>(b). 



 §.§ Quantum Chernoff Bound and optimal discrimination


In assessing the relative performance of the various processors at quantum state discrimination, we first note that the upper bound on the discrimination accuracy must depend on the fundamental distinguishability of the two reduced QS quantum states _ QS^(1),_ QS^(2), independent of the specific processing or measurement scheme. One measure of this distinguishability is the Quantum Chernoff Bound (QCB) ζ. The QCB bounds the discrimination accuracy according to 1 - 𝒞_ max∼exp ( -N ζ ) where access to N identical copies of each state is assumed, all of which may be measured <cit.>. Clearly, the larger the value of the QCB ζ, the more easily distinguishable the two quantum states are for fixed N. While strictly speaking a bound that holds in the asymptotic limit of many copies N, we use the QCB here due to the ease of its calculation for Gaussian states. We then define ζ_ QS as the QCB for the QS states _ QS^(1),_ QS^(2). Similarly, when such states are processed using a downstream , the QCB ζ can also be computed to quantify the distinguishability of the  states _^(1),_^(2), where _^(l) =  tr_ QS[^(l)]. We will consider Task I in what follows, so that the QCB is computed for the single mode to be read out (see Appendix <ref> for full details). 

We plot the ratio of these two QCB values, ζ/ζ_ QS, for three different |χ_b| values for the different quantum processors in Fig. <ref>(c). For the PS amplifier the optimal QCB is chosen by varying the PS amplification phase. For the , the QCB is plotted along the isogain contours in Fig. <ref>(b), using both TEOMs and within the NVK approximation. Immediately, we note that ζ/ζ_ QS < 1: the use of a quantum processor will generally introduce loss channels that are independent of the states to be distinguished and therefore hinder classification. We also note that for the present task, PS amplification enables a larger QCB than PP amplification: even though the PS amplifier can only amplify information in a single quadrature of the quantum input signal, it does so without adding any noise. Interestingly, we see that the  can approach and at certain operating points even exceed the QCB obtainable using linear quantum amplifiers in the same measurement chain. We emphasize that this was not a priori guaranteed in the presence of the added noise of the ; hence this observation is promising for the use of  for quantum state discrimination. 





 §.§  robustness to readout noise


Given the reduction in the QCB when using a downstream quantum processor, we reiterate why such processors are necessary in the first place. Excess classical noise n̅_ cl in the measurement chain can swamp weak quantum signals, such as those emanating from the QS directly, as they are extracted to the classical observer. The QCB is a discrimination bound that assumes optimal measurements, and therefore does not account for the limitations on readout imposed by n̅_ cl. In practical measurements subject to readout noise, quantum processors such as linear amplifiers then provide the pre-amplification necessary to overcome this noise and enable visibility of weak quantum signals. 

To quantify the impact of this practical constraint, we instead compute the classification accuracy for the different quantum processors under heterodyne readout and in the presence of n̅_ cl. We consider the operating point labelled (d) in Fig. <ref>(c), where ζ/ζ_ QS agrees well between the NVK and TEOMs. We also deliberately consider an operating point where the PS amplifier enables a larger QCB ζ than the ; as we will show, the particular  advantage we wish to highlight prevails in spite of this. Finally, we define readout features obtained using linear (·) and nonlinear (·) post-processing of measurement records over multiple shots, y = 1/S∑_s^S(I_1,Q_1,I_1^2,Q_1^2,I_1Q_1). This enables a fair comparison of the  against linear amplifiers, as the latter can only perform this discrimination task when nonlinear post-processing is allowed. 

We plot the obtained 𝒞_ max using the different quantum processors as a function of n̅_ cl in Fig. <ref>(d). For low enough n̅_ cl, both  and PS readout exhibit similar performance while the PP amplifier is worse, consistent with the obtained QCB values. With increasing n̅_ cl, the performance of all readout schemes degrades as expected. However  readout is clearly the most robust, exhibiting the smallest reduction. The red shaded region marks typical n̅_ cl values for cQED measurement chains, attributed primarily to noisy HEMT amplifiers (for calibration and mapping to equivalent noise temperatures as shown, see SI Sec. <ref>).  readout therefore provides enhanced robustness against experimentally relevant levels of excess classical noise.

The observed advantage stems from a simple principle:  enable nonlinear processing of quantum signals from the QS in the same quantum environment where they are generated, and - crucially - prior to corruption by excess readout noise n̅_ cl. In contrast, linear quantum amplifiers provide only linear gain to quantum signals, so that nonlinear processing of the extracted noisy signals is still required. By reducing or even eliminating the need for nonlinear post-processing,  therefore purvey a fundamentally different scaling of the signal-to-noise ratio with readout noise n̅_ cl. As shown in the inset of Fig. <ref>(d), Fisher's discriminant (which defines the signal-to-noise ratio for binary discrimination) degrades only linearly with n̅_ cl for the . This is markedly different to linear amplifiers, in which case  degrades quadratically with n̅_ cl, as the post-processing required to compute second-order nonlinear features necessary for this task must also amplify the added readout noise. Most importantly, our results emphasize that this  processing advantage does not demand large-scale or highly-coherent quantum systems as : few- and even single-mode nonlinear quantum systems suffice, deployed in lossy quantum measurement chains with minimal additional components. 

We conclude with two practical observations. First, while other choices of readout features y (e.g. for different temporal filters 𝒦(τ)) could influence the quantitative performance of either type of processor, the qualitative difference in  scaling with n̅_ cl, which relies on the in situ nonlinear processing paradigm enabled by , will remain. 

Secondly, in the interest of a fair comparison, here we have considered  as a direct replacement for linear quantum amplifiers, demanding  to offer both transduction and amplification to overcome readout noise. In this configuration, we are thus not able to take full advantage of the 's ability to engineer minimal projected noise σ^2_δμ = σ^2_ min, as this minimal noise can be swamped by the following readout noise n̅_ cl≫σ^2_ min. Using the  to control quantum fluctuations is still important, but to instead operate in a regime where σ^2_δμ≳n̅_ cl (to the extent possible for  a given |χ_b|). Alternatively,  can be deployed in conjunction with linear amplifiers. Then, a  would first be used to engineer a transduced signal quadrature P_δμ with minimal projected noise σ^2_ min. Next, a phase-sensitive amplifier would provide noise-free gain √(𝒢_ PS) to this quadrature, yielding ||δμ||^2 = 𝒢_ PS(𝔼[P^(l)_δμ-P^(p)_δμ])^2 and σ^2_δμ = 𝒢_ PSσ^2_ min + (n̅_ cl+1)σ_ vac^2. Provided 𝒢_ PS is large enough to satisfy 𝒢_ PSσ^2_ min≫n̅_ cl, the output  becomes ≃ (𝔼[P^(l)_δμ-P^(p)_δμ])^2/σ^2_ min, taking advantage of both transduction using  (which determines the `signal'), and their ability to engineer non-classical noise distributions (which determines the `noise').








































































§ MULTI-MODE 



In this final section, we analyze the use of a  to perform Task II: distinguishing a pair of two mode squeezed states. Here, we see that the need for multi-mode  naturally arises, which then allows us to identify the role of entanglement in quantum information processing tasks using .



 §.§ Role of coupling


As shown in Sec. <ref>, successfully performing Task II under direct QS readout requires monitoring of both QS modes, and computation of cross-correlations. For QS modes with disparate frequencies |ν_1 - ν_2| ≫κ (as required to realize non-degenerate squeezed states), quantum signals from the QS modes at ω_dn∼ν_n are also at widely separated frequencies. A single  mode with similar bandwidth γ_1+Γ_1 ≃κ_1+Γ_1, will thus not exhibit a strong response to two signals with such a wide spectral separation. Note that increasing the  bandwidth relative to the QS requires γ_1 ≫Γ_1; however, this undercouples the QS to the , reducing the influence of input QS signals and hence the  response (see Eq. (<ref>)). 

The processing of quantum signals from multiple non-degenerate quantum modes is therefore a situation where the need for a multi-mode  naturally emerges. We consider the measurement chain shown in Fig. <ref>(a) incorporating a K=2 mode , and where each QS mode is monitored by a  mode via the non-reciprocal interaction defined by Γ. A tunable (e.g. parametric) coupling _12 between the  modes allows us to clearly identify the role played by the multi-mode nature of the .








The full measurement chain covariance matrices are shown in Fig. <ref>(b) for the two QS states l=3,4 constituting Task II, first for _12 = 0. Only matrix elements that are different between the two states are shown in color, to aid visibility. In particular, when _12 = 0, the local  mode covariances are independent of l; this is because the two QS states also have identical local covariances. Instead, even if the  modes are uncoupled, non-trivial correlations that vary with the QS state are established between the  modes. While this may appear surprising at first glance, it is merely a consequence of the fact that the two uncoupled  modes are nevertheless driven by a correlated quantum input signal originating from the QS, which means the two  modes' states must also be correlated. However, it is only when the coupling is turned on, Fig. <ref>(c), that these cross-correlations can be transduced to differences in the local  mode covariances. We will see how, in contrast to Task I, this means that now both the nonlinearity and coupling of  modes is required for success at Task II.










Unfortunately, with increasing  size, the number of tunable parameters and the complexity of their interplay grows, making it a priori difficult to isolate the importance of a particular parameter, such as the coupling _12. Here, our analysis of  optimization for Task I in Sec. <ref> proves its generality, and hence its value. For a given , two principles were found: the importance of |χ_b| (the largest eigenvalue of the  Jacobian) in enhancing the response of the  to any input, and the control of quantum fluctuations to minimize projected noise σ^2(l)_δμ. Our strategy to isolate the role of coupling is to therefore fix other parameters such that |χ_b| can be held constant and σ^2(l)_δμ is simultaneously minimized.

The two-mode  model we consider is a realization of the Kerr or Bose-Hubbard dimer that has been analyzed in prior work, but only under coherent driving <cit.>, not with both modes driven by quantum signals from a QS upstream as illustrated schematically in Fig. <ref>(a). By analyzing the Jacobian of this two-mode , we identify regions of parameter space where the susceptibility |χ_b| grows and diverges, namely near the classical bistability of the Kerr dimer, marked as the orange region in the phase diagram of Fig. <ref>(a). Then, using the NVK approximation, we identify a trajectory through this parameter space near the classical bistability - labelled the optimal noise trajectory (gray curve) - where |χ_b| = 9.0 at every coupling strength, and simultaneously the projected noise is minimized, σ^2(3)_δμ≃σ^2(3)_ min. An example of how this choice constraints parameters is shown for point (ii) on this trajectory in the lower panel of Fig. <ref>(a).

We again consider single-shot readout features obtained under linear processing only, now from both  modes, y = x = (I_1,Q_1,I_2,Q_2). Performing Task II for parameters along this trajectory, we obtain  as a function of coupling in Fig. <ref>(c), which follows a smooth curve. We also plot the corresponding mean separation ||δμ|| and projected noise in Fig. <ref>(c). For _12→ 0, we see that ||δμ||→ 0, even though 𝐂_b^(3)≠𝐂_b^(4) as seen in Fig. <ref>(b) and the nonlinearity is nonzero. This is because the Hessian tensor in Eq. (<ref>) is still local, as the nonlinearity is on-site, so that δμ is only sensitive to local  mode covariances which are independent of l when _12 = 0. This observation is the result of a much simpler fact: since each  mode is coupled to only one QS mode, and the difference between QS states l=3,4 is present in correlations of different QS modes, it is necessary for the two  modes to `communicate' (hence, be coupled) to be able to distinguish the inputs they receive. If this communication is not performed in situ using _12, it will have to be performed in post-processing, by computing correlations of measured  quadratures for distinct  modes explicitly, which constitutes a nonlinear processing step. Turning on _12 allows these correlations to be computed via the  dynamics, allowing the difference in QS states to translate to a nonzero ||δμ|| of measured  quadratures, enabling → 1.





 §.§ Engineering entanglement for classification


We are now well-placed to explore the role of entanglement in the processing of quantum signals using a . Importantly, we are interested in output field entanglement, which requires non-classical correlations between measured  quadratures. For the two-mode , the 4-by-4 measured covariance matrix can be expressed in the block form

    Σ = 
        [   Σ_11   Σ_12; Σ_12^T   Σ_22 ]

where Σ_jk is the contribution from covariances between  modes j and k (see Appendix <ref>), and we have suppressed the superscript (l) indicating the corresponding QS state. The metric we then use to quantify the degree of entanglement in  outputs is the logarithmic negativity E_ N <cit.>, a standard entanglement monotone. In terms of Eq. (<ref>), E_ N can be defined as

    E_ N   =  max{0,-ln 2ν^-}, ν^- = √(d-√(d^2-4Σ)/2), 
    
        d    = Σ_11+Σ_22-2Σ_12

In  Fig. <ref>(b), we then plot E_ N as a function of coupling _12 along the optimal noise trajectory, computed using the NVK approximation as well as using simulated covariance matrices via the STEOMs. At first glance, the role of entanglement appears straightforward: increasing coupling coincides with an increase in the degree of entanglement between measured  quadratures, and leads to improved classification performance.

However, this observation does not indicate whether entanglement is always useful, or if it provides an advantage that goes beyond non-classical correlations of only single-mode variables. The context of binary quantum state discrimination allows us to probe these questions directly. We once again consider the reduced quadrature description introduced in Sec. <ref>. In Fig. <ref>(d), we then plot measured  distributions in the projected space spanned by (P_δμ,R_⊥) for three different operating parameters (i)-(iii), labelled in Fig. <ref>(a). 

By construction, the separation of distribution means for all  (i)-(iii) is entirely confined to the P_δμ quadrature.  (i) and (ii) fall along the optimal noise trajectory and are thus engineered to minimize noise in the P_δμ quadrature, as is clearly visible in the distributions. Fig. <ref>(d). In the top panel of Fig. <ref>(d), we also show the content of the P_δμ quadrature. For (i), P_δμ is predominantly comprised of quadratures of only a single  mode (here k=2). Hence, the sub-vacuum noise in P_δμ is the result of quantum correlations between outputs of only a single  mode, indicative of single-mode squeezing. In this weak coupling regime, entanglement is not necessary to obtain sub-vacuum noise; particular, the measured  outputs do not exhibit entanglement as E_ N=0.0.

However, Task II benefits from increasing coupling between the  modes, as observed previously; this not only increases the mean separation ||δμ||, but also generates output entanglement, as seen for  (ii) where E_ N = 0.11. Now P_δμ is comprised of quadratures of distinct  modes, and thus is a non-local quadrature. Sub-vacuum noise in P_δμ therefore must arise due to non-classical correlations amongst these non-local quantum modes, namely entanglement. Here, entanglement of measured quadratures directly improves classification performance by ensuring that non-local quadratures that carry useful information (encoded in ||δμ||) are also non-classically correlated, such that their fluctuations are reduced below the vacuum limit.

Perhaps most interestingly, just the presence of output field entanglement is not always guaranteed to be beneficial for classification. A simple counterexample is  (iii), which demonstrates a nonzero logarithmic negativity E_ N = 0.10 like  (ii), but is not on the optimal noise trajectory as seen from the lower panel of Fig. <ref>(a). P_δμ is still a non-local quadrature, but its noise is clearly not minimal, and is in fact above vacuum. The presence of output field entanglement means that there is some set of non-local quadratures that exhibit non-classical correlations that lead to sub-vacuum noise; however, these quadratures are not always guaranteed to carry useful information encoded in ||δμ||, and hence such non-classical correlations may not always be useful for the task at hand. 

Crucially, the use of nonlinear  provides the ability to control quantum fluctuations, such that the situation in (ii) can be engineered, and non-classical correlations can be manipulated for computational benefit. For binary classification, this entails finding operating points where the unique quadrature with maximal signal P_δμ can simultaneously exhibit minimal, non-classical (e.g. sub-vacuum) fluctuations. The ability to engineer this unique quadrature also opens up avenues to only amplify and measure said quadrature using noiseless phase-sensitive amplification, instead of phase-preserving amplification of all 2K quadratures for a K-mode , which necessarily adds noise.




§ CONCLUSIONS AND OUTLOOK


Nonlinearity is an essential component of signal processing, playing a key role from digital processing to neural circuits. However, the role of general, fundamentally stochastic quantum nonlinear devices in processing quantum signals - beneficial or otherwise - is much less explored. In this paper, we have addressed this limitation by identifying key general principles of quantum information processing enabled by a broad class of nonlinear bosonic quantum systems, which we refer to as .

Our two main results, which hold beyond the quantum state discrimination tasks we have chosen to demonstrate them, and which make explicit use of  being both quantum and nonlinear, can be simply stated. We show that  can be efficient information transducers (cf. Eq. (<ref>)): by processing quantum signals in situ,  can render nonlinear properties of quantum signals such as correlation functions accessible via linear readout schemes like heterodyne monitoring. Secondly, by harnessing a nonlinear input-output map,  can coherently manipulate quantum fluctuations in a manner unavailable to linear amplifiers that are linearly coupled to a quantum signal source: without suppressing the transduced signal magnitude,  can modify its output noise properties. This requires  to control not only quantum fluctuations emanating from the quantum system the  is monitoring, but also those resulting from the  operation itself, referred to as `added noise' for linear amplifiers (cf. Eq. (<ref>)). Remarkably, both these capabilities can be accessed using small-scale , making them relevant for implementations in current experiments. In fact, by analyzing realistic measurement chains, we show that even single-mode  can provide robustness against classical readout noise. 

On the other hand, the key theoretical tools these results are built on - the analytic NVK approximation of nonlinear measurement chains and the numerical STEOMs framework to efficiently simulate their conditional dynamics, both verified against exact master equation methods - provide the means to analyze quantum information processing using very general, arbitrarily-multimode nonlinear bosonic quantum systems. As such they can be used to study the processing of signals from many-body quantum systems and non-classical correlations or entanglement across several modes, as we study in Task II. Furthermore, they can enable the exploration of more general paradigms of quantum information processing. One example pointed out in the conclusion of Sec. <ref> is the use of measurement chains employing both  and phase-sensitive amplifiers. Related paradigms enabled by our framework include the possibility of entangling the  with the QS, via for example non-reciprocal entangling operations <cit.>. 



Taking an even broader view, our work has direct applications to general information processing and computation paradigms such as quantum machine learning and quantum sensing. Nonlinearity is considered essential to the expressive capacity of physical neural networks including quantum systems <cit.>. However, several popular bosonic quantum machine learning platforms are linear <cit.>, instead enabling nonlinear processing by careful use of nonlinear input encoding schemes. Our work provides both tools and possible directions to explore the utility of multi-mode quantum nonlinear devices for learning applications; distinct from other approaches, our framework also describes learning on quantum inputs. In particular, it is expected that genuinely quantum advantages in such applications must make use of decidedly quantum properties, such as squeezing and entanglement.  harness nonlinearity to control quantum fluctuations such that non-classical correlations appear in desired observables only. Furthermore, we elucidate general principles that can enable  to operate in regimes to enable such processing. The resulting quantum mechanism to harness quantum correlations for the enhancement of classification accuracy could prove useful in extracting quantum advantages for quantum machine learning and quantum sensing.

Our work also invites exploration of more complex  beyond the weakly-nonlinear regime. Firstly, the scaling advantage with n̅_ cl can be expected to be even more significant with increasing complexity of the required nonlinear computation, such as the calculation of higher-order or many-body correlations <cit.>. Such computations will demand the analysis of  with higher-order nonlinearities, or operation in increasingly non-Gaussian regimes. Secondly, for the specific case of quantum state discrimination, we have shown that  can approach and even exceed the optimal discrimination bound, quantified here by the Quantum Chernoff bound (QCB), achievable using linear quantum processors. However, a study of the maximum attainable QCB limits using  requires going beyond the NVK approximation, and could quantify the ultimate constraints on quantum information processing using nonlinear systems. Both these directions are natural extensions that we leave for future work.

Finally, we return full circle to our original motivation: by extracting more information from the quantum domain, we hope  can ultimately improve our ability to control quantum systems. By enabling the efficient simulation of measurement-conditioned dynamics of measurement chains including , our STEOMs framework provides the necessary first step in the study of quantum feedback and control using . Such control is necessary for important quantum information processing tasks such as error correction <cit.>, either via continuous monitoring and feedback <cit.> or autonomous protocols via coherent quantum feedback <cit.>.





We would like to thank Dan Gauthier, Luke Govia, Peter McMahon, Ioan Pop, Graham Rowlands, Guilhem Ribeill, Tatsuhiro Onodera, Logan Wright, Ryan Kaufman, Boris Mesits, Shyam Shankar, Florian Marquardt, and Ryotatsu Yanagimoto for useful discussions. This work is supported by AFOSR under Grant No. FA9550-20-1-0177 and the Army Research Office under Grant No. W911NF18-1-0144. Simulations in this paper were performed using the Princeton Research Computing resources at Princeton University, which is a consortium of groups led by the Princeton Institute for Computational Science and Engineering (PICSciE) and Office of Information Technology's Research Computing.




    30mm1pt









§ MEASUREMENT CHAIN DESCRIPTION AND TABLE OF SYMBOLS



In this Appendix section we expand on details of the measurement chain considered described by Eq. (<ref>) that were omitted for brevity in the main text.

First, we define the interaction between the QS and  described by . We engineer this interaction to be non-reciprocal by balancing a coherent and a dissipative hopping interaction with an appropriately-chosen phase <cit.>,

    ℒ_cρ̂ = -i[i/2∑_mΓ_mâ_m^†b̂_m + h.c. ,ρ̂]+∑_mΓ_m𝒟[â_m + b̂_m]ρ̂.

Some features of this interaction are of note. Firstly, although the interaction has a dissipative component, it defines a coherent link, such that it allows for a non-separable joint quantum state of the QS and . Secondly, for simplicity we require that one mode of the QS couples to at most one mode of the , with strength Γ_m (although the coupling can also vanish if Γ_m = 0 for a given â_m). Finally, we note that Eq. (<ref>) describes a standard circulator <cit.>, and is therefore realized as a matter of course in cQED experiments.

For practical implementations in the cQED architecture, we consider for the  a model of K coupled Kerr nonlinear modes b̂_k, k∈[1,K], furnished by capacitively-shunted Josephson junctions <cit.>. The nonlinear modes have frequencies {ω_k} and nonlinearity strengths {Λ_k}, with linear coupling {_jk} between modes i and j. Then, in an appropriate interaction picture at frequencies {ω_dk} (close to {ω_k} respectively), the linear  Hamiltonian takes the form

    ℋ̂_ = -∑_kΔ_k kk + ∑_jk_jk(jk + jk),

with detunings Δ_k = ω_dk-ω_k, while the nonlinear component of the  Hamiltonian is given by

    𝒩̂_ = - ∑_k Λ_k/2kkkk.

For simplicity, we assume Λ_k ≡Λ ∀ k, although we emphasize this is not necessary for .



Y>X




Finally, the conditional dynamics of  under heterodyne monitoring are governed by the stochastic measurement superoperator 𝒮[√()b̂_k], given by:

    𝒮[√()b̂_k] = √(/2)( b̂_k  + b̂_k^† - b̂_k+b̂_k^†) dW_ℐ_k(t) 
             +√(/2)( -ib̂_k  + ib̂_k^† - -ib̂_k+ib̂_k^†) dW_𝒬_k(t) 
             +𝒟[b̂_k],

where ô indicates the conditional expectation value of an arbitrary operator ô with respect to the measurement-conditioned quantum state, ô =  tr{ô}. Here dW_ℐ_k, dW_𝒬_k are independent Wiener increments describing measurement noise, satisfying dW_ℐ_k=dW_𝒬_k=0, and dW_ℐ_kdW_𝒬_k' = 0, dW_ℐ_kdW_ℐ_k' = dW_𝒬_kdW_𝒬_k' = δ_k,k'dt. The Wiener increments are therefore related to the white noise terms introduced in Eq. (<ref>) via ξ_ℐ_k≡dW_ℐ_k/dt, ξ_𝒬_k≡dW_𝒬_k/dt. The classical readout noise terms in Eq. (<ref>) are also taken to obey white noise statistics.

On the other hand, the quantum noise contributions ξ^ qm_ℐ_k, ξ^ qm_𝒬_k introduced in Eq. (<ref>) generally do not obey white noise statistics. Quantum trajectories  are conditioned on the heterodyne measurement record via the stochastic measurement superoperator 𝒮. Quantum noise contributions depend on these quantum trajectories; more concretely, for example, ξ_ℐ_k^ qm(t) = X̂_k - X̂_k, and thus describes the stochastic deviation of an observable conditioned on a specific quantum trajectory from the ensemble average. Hence the quantum noise contributions inherit nontrivial correlation and noise properties from quantum trajectories. Lastly, monitoring of  modes opens them up to linear damping at the rate , described by the standard dissipative superoperator 𝒟[ô] = ôô^† - 1/2{ô^†ô,}. 





Definitions of all parameters characterizing the general measurement chain are summarized in Table <ref>. Specific parameter values used to generate the various figures in the main text are summarized in Table <ref> in the SI.











































§ TRUNCATED CUMULANTS APPROACH



For an arbitrary  describing a quantum measurement chain with = M+K modes, we can define its associated characteristic function χ(w⃗,w⃗^*) <cit.>,

    χ(w⃗,w⃗^*) = tr{∏_j=1^ e^iw_j^∗ô_j^†∏_l=1^ e^iw_lô_l}

where ô_j ∈{â_m,b̂_k} describes any mode of the complete quantum measurement chain, and w⃗ = (w_1,…,w_) are auxiliary variables. Then, normal-ordered cumulants of the density matrix are formally defined via the log of the characteristic function (sometimes called the generating function),

    C_o_1^† p_1⋯ o_^† p_ o_1^q_1⋯ o_^q_≡.  ∂^n_ ordlnχ(w⃗,w⃗^*)  /∏_j∂ (i w_j^∗)^p_j∏_l∂ (i w_l)^q_l|_w⃗=w⃗^∗=0

where n_ ord = p_1 + … + p_J + q_1 + … + q_J defines the order of cumulants. Notwithstanding the complex formal definition, cumulants can be transparently related to more familiar operator expectation values: for a general operator ô_j, first-order cumulants C_o_j≡ô_j are simply expectation values, while second-order cumulants C_o_jo_l = ô_jô_l - ô_jô_l are their covariances. Expressions for higher-order cumulants become increasingly more involved, but can be systematically obtained, as discussed in the SI <cit.>. 




 §.§ Numerical scheme: Stochastic Truncated Equations of Motion


The crucial advantage of cumulants as descriptors of a quantum state is that specific multimode quantum states admit particularly efficient representations when expressed in terms of cumulants. In particular, a quantum system in a product of coherent states is described entirely by its nonzero first-order cumulants; all cumulants with n_ ord > 1 vanish (see SI <cit.> for derivations). Multimode quantum states that are defined entirely by their first and second-order cumulants admit Gaussian phase-space representations, and are thus labelled Gaussian states. States with nonzero cumulants of third or higher-order are thus by definition non-Gaussian states <cit.>. 

Our numerical approach leverages this economy of representation by using normal-ordered cumulants as a set of dynamical variables for quantum modes of the measurement chain. For example, for specific systems such as coherently-driven linear bosonic systems initialized in a Gaussian state, cumulants of order n_ ord≤ 2 are sufficient, as such systems can be shown to persist in Gaussian states. If the  considered in this paper were also linear, our model would satisfy this requirement. However, our interest is precisely in the role of the  nonlinearity. In this case, the cumulants describing the measurement chain and their dependencies are shown schematically in Fig. <ref> for M=K=1. In particular, the nonlinearity generates states with nonzero cumulants of n_ ord> 2. In the most general case, the dynamical equations for these cumulants couple to all orders, forming an infinite hierarchy of equations that does not close.

To obtain a tractable numerical method, we consider an ansatz wherein the quantum state of the complete measurement is described entirely by cumulants up to a finite order n_ ord≤ n_ trunc; all cumulants of order n_ ord > n_ trunc are thus set to zero, truncating the hierarchy and yielding a closed set of equations for the retained nonzero cumulants. In this paper, we choose n_ trunc = 2 for a quantum measurement chain defined entirely by its first and second-order cumulants, although the truncation can similarly be carried out at higher order. The resulting Stochastic Truncated Equations Of Motion (STEOMs) form the basis of our numerical approach in this paper.

We emphasize that the truncated cumulants ansatz has some important differences when compared to standard linearization approaches. As seen in Fig. <ref>, note that both the nonlinearity and measurement terms couple second-order cumulants to first-order cumulants: this fact proves critical for the quantum state classification tasks we consider in this paper. 

The utility of the STEOMs is naturally determined by the validity of the truncated cumulants ansatz. Since we are specifically interested in nonlinear quantum systems, which can generate higher-order cumulants in dynamics, one must ask when such an ansatz may hold. In the SI <cit.>, we carry out a detailed analysis of the regimes of validity of this framework, together with benchmarking against exact solutions and full SME integration. We find that the STEOMs provide a very good approximation of full SME integration provided the strength of nonlinearity of  modes is weak relative to their loss rates; good agreement remains even up to Λ_k/γ∼ 0.1, which is around an order of magnitude larger than the strongest nonlinearity we consider in this paper.











 §.§ Semi-analytic scheme: Nonlinear van Kampen expansion based of the Fokker-Planck equation


In addition to enabling the (S)TEOMs as a practical numerical method for multimode measurement chains, the truncated cumulants approach is also central to our main analytic tool: a description of quantum dynamics that is perturbative in the nonlinearity of the measurement chain. This analysis is enabled by the close connection between normal-ordered cumulants and the positive-P representation via the characteristic function. More precisely, the positive-P representation is simply the Fourier transform of the characteristic function <cit.>, 

    𝒫(𝒪⃗,𝒪⃗^†) = 1/π^2J∫∏_j=1^ d^2 w_j d^2 w_j^* e^-iw_j^* 𝒪_j^†e^-iw_j 𝒪_jχ(w⃗,w⃗^*)

The dynamics of the positive-P distribution follow a Fokker-Planck equation that is equivalent to Eq. (<ref>), while also employing normal-ordered cumulants as its natural dynamical variable set. We use this connection to first obtain an approximate Fokker-Planck equation for dynamics of the measurement chain in powers of the  nonlinearity. This directly allows us to obtain semi-analytic solutions for the TEOMs, Eqs. (<ref>), (<ref>), from which classification metrics such as the Fisher's discriminant can be readily evaluated. Our analysis and its results, which are used throughout the main text, are detailed in the SI, Sec. <ref> and <ref>.



§ SCALING OF  PERFORMANCE WITH NONLINEARITY FOR FIXED 



In Sec. <ref>, we explored how  parameters can be optimized to improve classification performance. However, a plain reading of the expression for the mean separation δμ, Eq. (<ref>), would appear to suggest that an even simpler strategy may be to simply increase the  nonlinearity strength Λ, which monotonically scales δμ. However, in practice this dependence is much more complex: for a classification task defined by a given , varying Λ also varies  via Eq. (<ref>). This changes the operating conditions of the  and directly influences the mean separation and noise properties.







Therefore, to observe and understand the scaling of ||δμ|| with nonlinearity, we must keep  fixed while varying the nonlinearity. Practically, we do so by using  of varying nonlinearity to perform separate instances of Task I, characterized by different values of ·Γ_1/γ. For each instance, the achieved  as a function of Λ is plotted in Fig. <ref>. 

We immediately note that for decreasing ·Γ_1/γ, the required  nonlinearity to reach the optimal  increases, as required by the form of . The crosses indicate three specific , one for each instance of the considered task, with nonlinearity Λ such that  is the same. Comparing these , we now clearly see that the optimal  increases with increasing Λ. In the top panel of Fig. <ref>, we plot the mean separation amplitude ||δμ|| and the projected noise σ^2(l)_δμ for l=1 for each considered instance of Task I, under the NVK approximation and using integration of STEOMs. To lowest order in nonlinearity as captured by the NVK approximation, the projected noise properties remain the same for each instance. In contrast, the mean separation increases with nonlinearity, indicative of the sought-after scaling. Unchanged noise properties with increasing mean separation imply an increase in  and hence classification accuracy. 

Note that with increasing nonlinearity Λ, the agreement between the NVK approximation and exact integration of the STEOMs is reduced. In particular, the exact mean separation is lower than the NVK approximation, while the projected noise is larger. Both effects serve to reduce  and hence limit the improvement in  to below that predicted by the NVK approximation. These observations are an example of saturation that is higher-order in the  nonlinearity and hence not captured by the NVK approximation, and are commonplace in strongly-driven nonlinear quantum systems <cit.>. 









§ MODELS OF STANDARD LINEAR QUANTUM AMPLIFIERS



In this section we provide the models for both phase-preserving and phase-sensitive linear quantum amplifiers deployed for readout in cQED, and which we use as benchmarks in Sec. <ref> of the main text. Note that our description of the measurement chain, Eq. (<ref>), is general enough to include this standard paradigm, by neglecting the nonlinear contribution to the Hamiltonian, 𝒩̂_→ 0.

In particular, by taking ℋ̂_→ℋ̂_ PP where

    ℋ̂_ PP =  -∑_k Δ_kb̂_k^†b̂_k  + G_ PP(-ib̂_1b̂_2 + h.c.),

we are able to describe QS readout using linear phase-preserving quantum amplifiers.

For phase-sensitive (PS) amplifiers on the other hand, we can set ℋ̂_→ℋ̂_ PS, where

    ℋ̂_ PS = -∑_k Δ_kb̂_k^†b̂_k  + G_ PS(-ib̂_1^2 + h.c.).






§ QUANTUM CHERNOFF BOUND: ADDITIONAL DETAILS



In this Appendix section we provide some additional details of the calculation of the Quantum Chernoff bound (QCB) used in Sec. <ref> of the main text to determine the optimal discrimination bounds for quantum state discrimination tasks.

To define the QCB as used in the main text, we first introduce the quantity <cit.>:

    Q(^(l),^(p)) = - min_s ∈ [0,1]log tr[ (^(l))^s(^(p))^1-s]

We compute the QCB using only the quantum states of the modes that are intended to be monitored. For Task I, this is simply mode â_1 for the QS. We can then define the QCB for the QS ζ_ QS as:

    ζ_ QS≡ Q(  tr_â_2[^(1)_ QS],  tr_â_2[^(2)_ QS] )

where tr_ô[] is used to trace out the sector corresponding to mode ô from the density matrix . For the general quantum processor, only mode b̂_1 is monitored, so the QCB ζ analogously becomes:

    ζ≡ Q(  tr_b̂_2[ ^(1)_],  tr_b̂_2[^(2)_] )

Note that the  and the PS amplifier are both single-mode devices for Task I, so the trace operation has no effect in those cases.

The expressions in Eqs. (<ref>), (<ref>) can be computed straightforwardly if the quantum states are Gaussian. These simplified expressions are derived in Ref. <cit.>; we employ these results in plotting the QCB in the main text, Fig. <ref>.





§ NON-CLASSICAL NOISE DUE TO THE  ALONE: CLASSIFYING THERMAL STATES



In the main text, the considered Tasks I and II both require classification of states that exhibit some degree of quantum correlations, namely at least one state exhibits some degree of single or two-mode squeezing. As a result, the observed quantum (i.e. sub-vacuum) noise observed at the  output will in general contain a contribution attributable to the quantum correlations of QS signals, and cannot be attributed to the  alone.

In this Appendix section, we show that non-classical correlations can be generated solely by appropriate  models, and they can be manipulated for use in classification in the same way as has been demonstrated for both Task I and II. To this end, we first introduce the modified Liouvillian for the QS:

    = -i[ℋ̂^(l)_ QS,] + 
             ∑_m κ_m (n^ th(l)_m+1)𝒟[â_m] 
        + κ_m n^ th(l)_m 𝒟[â^†_m],

where in contrast to Eq. (<ref>) each QS mode is now coupled to a finite temperature bath. For simplicity, we further restrict ourselves to a single-mode QS being monitored by a single-mode , M=K=1; all QS parameters for mode m=2 are set to zero. Secondly, we turn off single-mode squeezing of the QS mode, G_1^(l) = 0 ∀ l, so that the QS states to be distinguished have no non-trivial quantum correlations. Instead, the states to be distinguished are characterized entirely by different thermal bath temperatures and hence occupation numbers, n^ th(l)_m. While not straightforward to generate in practice, this choice is ideal to test whether non-classical correlations can be generated by the  alone, and whether they can still be manipulated by the . All parameters characterizing Task III are summarized in Table <ref>.














It suffices to consider only the NVK approximation and to then plot both the the mean separation of measured  quadratures ||δμ|| and the projected noise in Fig. <ref>, as a function of  nonlinearity. We note that the qualitative features are as observed in Fig. <ref>, and therefore reiterate the main results in the paper: the ability to compute correlations in situ, and the ability to control noise in measured quadratures differently from their mean value. Crucially, projected noise below vacuum in Fig. <ref>(b) can now solely be attributed to the action of the , as the QS states are purely thermal and do not demonstrate such non-classical correlations; in fact, they exhibit fluctuations above the vacuum value due to the coupling to a finite temperature bath. This emphasizes that the  can provide the ability to not just manipulate existing quantum correlations, but also provide useful quantum correlations itself.

We emphasize that the  model we consider here is based on coherently-driven Kerr oscillators, which by themselves are not ideal squeezers. However, our general approach applies to much more general  models, where more non-classical correlations could be generated.









































































































































































































































































































































































































































    



