Quantum Simulations Confirm Stochastic Noise in EPR Correlations

Teh and colleagues at Swinburne University have shown that the stochastic term governing measured homodyne current aligns with the Stratonovich interpretation of stochastic noise, confirming experimental findings. The work clarifies the role of noise in quantum systems and offers insights into potential errors within developing quantum technologies. By modelling trajectories from a stochastic Schrödinger equation, the team also propose a new adaptation of Schrödinger’s thought experiment, exploring simultaneous measurement of position and momentum through direct and indirect methods.

Stochastic modelling reveals EPR correlation sensitivity to noise implementation and detector

Measurements of quantum outputs following amplification are increasingly realised. Stochastic Schrödinger equation (SSE) simulations of Einstein, Podolsky and Rosen (EPR) correlations are carried out. Initially, EPR considered spatially separated particles, but later realisations used homodyne measurements on fields, which were subsequently carried out experimentally. The simulations of the moments measured in such an experiment utilise an SSE that can model the output noise of the homodyne measurements.

The analysis shows that predicted current correlations change sharply with the stochastic method used. Ito stochastic noise gives incorrect equal-time correlations, with or without a time-shift, because only integrals of Ito noise have physical meaning. Including detector bandwidth effects resolves this, leading to EPR correlations between time-integrated observables. However, obtaining the correct unfiltered current, analogous to correlated positions x1 and x2, and anti-correlated momenta p1 and p2, requires Stratonovich noise at large bandwidth.

SSE Methods can simulate many quantum technology experiments, including Bose-Einstein condensates, superconducting quantum circuits, the coherent Ising machine (CIM) quantum computer and, in future, LIGO gravitational wave detectors. Large-scale quantum devices often utilise many modes and measurements displaying correlations and entanglement. The correct choice of stochastic term is essential for understanding wideband continuous output measurements with multimode correlations.

As an example, this approach is applied to a strongly interacting CIM model used to solve NP-hard problems. Simulations show that in the deep quantum limit, changing the detector bandwidth alters errors due to shot noise, significantly affecting the success rate. The SSE simulations also give insight into the assumptions behind the EPR argument, which attempted to demonstrate the incompleteness of quantum mechanics. EPR’s argument was based on two premises: a criterion for an “element of reality” and no action-at-a-distance.

For correlated spin states, these premises implied local hidden variable descriptions, subsequently negated by Bell’s work. However, EPR’s assumptions do not break down at the macroscopic level of the detector currents, where no-signalling is valid. Analysing the trajectories as measurement settings θ are changed, the simulations show consistency with a set of weakened EPR premises, applying to the currents and not negated by Bell’s theorem. Schrödinger proposed one could indirectly measure p1 of one particle by measuring p2 of the correlated, spatially-separated system, simultaneously measuring a particle’s position and momentum, “one by direct, the other by indirect measurement”. The validity of this statement is analysed by simulating Schrödinger’s gedanken experiment for two-mode EPR fields, proposing an experiment in which a wave-function measurement is interrupted and changed mid-collapse.

The simulations start by treating bosonic quadrature measurements on M orthogonal modes. The master equation for dissipation for a quantum density matrix ρ with mode operators ak and a Hamiltonian H with units such that ħ= 1 is ∂ρ/∂t = −i[ H, ρ] + Σk=1M γk ak ρa†k −1/2 [nk ρ + ρnk]. Here γk is the number decay rate, the number operators are nk = a†kak, and operators are denoted as O to distinguish them from measured results. The respective input and output fields bin and bout external to the system are related by the input-output relation bout = √γa + bin, where a = (a1, a2). For simplicity, equal damping is supposed such that γk = γ, with dimensionless times τ = γt, and dimensionless Hamiltonian H = H/γ. Detector quantum efficiency is ignored, although this can be readily included.

Two bosonic modes k = 1, 2 in a cavity or interferometer are treated, each decaying to an external detector, with corresponding external fields bk, which are also made dimensionless. In the case of a prepared photonic state in a vacuum, the quantum stochastic operator equations can be solved exactly, using dimensionless times and fields bout(τ) = e−τ a(τ) + ∫0τ eτ′ bin(τ′) dτ′ + bin(τ). An internal vector quadrature operator x = (x1, x2) = a + a† is defined, with input and output quadrature operators Xin = ( Xin,1, Xin,2) and Xout = ( Xout,1, Xout,2), where Xin(out) = bin(out) + b†in (out). Hence, Xout,k = xk + Xin,k, if the input field is in a vacuum state with ⟨Xin(τ)⟩Q = 0, mean values are obtained: ⟨ Xout⟩Q = e−τ/2⟨x⟩Q = ⟨x(τ)⟩Q, where ⟨.⟩ Q is a quantum ensemble average. The relationship of external field Xout,k and the measured current operator Jk depends on the local oscillator and the detector gain.

After rescaling to a dimensionless form, the ideal output current in the wideband limit is simply J = Xout, where J = ( J1, J2). Since the measured current has a finite bandwidth, this can only hold over a restricted bandwidth. To analyse EPR correlations, this is generalised to a balanced homodyne scheme for measuring the internal quadrature xφk k = xk cos φk + pk sin φk, by combining the output field with a macroscopic local oscillator (LO) field, each with an independent phase-shift φk for each mode k. To treat this, ak = ake−iφk is defined, and the rotated quadrature as xk = ak + a†k, for measurements with a fixed local oscillator phase. There is an SSE equivalent to the above equation, which scales linearly in the Hilbert space dimension, giving a lower complexity than the master equation.

In its simplest form it is a stochastic differential equation (SDE) following Ito calculus: d |Ψ⟩I/dτ = ( −i H + ⟨ x⟩I − a† · a/2 −⟨ x⟩2I/8 + ∆ a · ξ ) |Ψ⟩. Here |Ψ⟩I is a state conditioned on a pseudo-current jI = (jI1, jI2) with an Ito noise ξI, the fluctuation operator is ∆ a ≡ a −1/2 ⟨ x⟩S and jI(τ) = ⟨ x(τ)⟩I + ξI (τ). The real noise ξI is defined such that ξI k (τ) ξI j (τ′) = δkjδ (τ −τ′), where ξI k is the noise for mode k, and ⟨x(τ)⟩I = ⟨Ψ| x |Ψ⟩I. Several proposals exist for interpreting jI(τ) as a realistic sample of measured output currents J, whose ensemble averages and correlations must match the quantum predictions. Since Ito noise has unusual mathematical properties, it is important to clarify this interpretation as it becomes more relevant to modern quantum experiments. The simulations show that an Ito interpretation gives no initial EPR correlations, completely different to the quantum prediction, because Ito calculus requires corrections that do not occur for correlations of physical measurements.

Evaluating the Ito noise at an earlier time to the wave-function, so that jd = ⟨ x(τ)⟩I + ξI (τ −dτ), also gives incorrect correlations. Another approach is to derive a Stratonovich SDE, which is the wideband limit of a finite bandwidth stochastic equation following standard calculus. Several forms exist, but an SSE equivalent to the above equation, explained in Appendix A, is used, giving: d |Ψ⟩S/dτ = −i H + ∆ a · jS + x2 −M/4 − x a/2. |Ψ⟩S. Here the fluctuation operator is ∆ a ≡ a −1/2 ⟨ x⟩S and the Stratonovich pseudo-current jS = (jS1, jS2) is jS(τ) = ⟨ x(τ)⟩S + ξS(τ). This is defined using Stratonovich noise ξS with the same correlations as before.

The physical wideband current is simply J(τ) = jS(τ), which gives correct wideband EPR correlations. The proof is based on more rigorous characteristic functional methods, explained in Appendix B, which show that only the time-integral of an Ito noise is physical. To obtain the physical current, a finite bandwidth detector is assumed to give a second coupled SDE for the detected photocurrent J = (J1, J2): dJ/dτ = −κ (J −j). In this approach, the current J has a finite bandwidth κ, and follows standard calculus.

Since Ito equations can be transformed to Stratonovich equations using known rules, this resolves the Ito vs. Stratonovich ambiguity. In the wideband detector limit of κ →∞, one can adiabatically eliminate the ’fast’ variable J, so that J → jS. Hence, the physical current in the limit of a wideband detector is the Stratonovich current. To explain the physical EPR argument, two interferometers prepared in a two-mode squeezed state are considered, which is a method for implementing the quantum correlations proposed in the original EPR gedanken-experiment.

The initial state is |TMSS⟩, defined as |TMSS⟩= 1/cosh r Nc Σn=0 (tanh r)n|n⟩1|n⟩2, where r is the squeezing parameter, |n⟩k is a number state for mode k and Nc is the photon cutoff, taken as Nc →∞ in the idealized case. Analysis of trajectories generated by the stochastic Schrödinger equation (SSE) reveals that the time evolution of the internal quantum correlation ⟨x1(τ)x2(τ)⟩Q from the operator equations has an analytical solution: ⟨x1 (τ) x2 (τ)⟩Q = e−τ sinh(2r). Similarly, defining pk = (a −a†)/i, it is found that ⟨p1 (τ) p2 (τ)⟩Q = −e−τ sinh(2r), and also, ⟨xi (τ) xi (τ)⟩Q = ⟨pi (τ) pi (τ)⟩Q = e−τ cosh(2r). For a stochastic Einstein-Podolsky-Rosen (SSE) current jk to be valid, it should satisfy the same correlations at equal times: i.e. ⟨j1(τ)j2(τ)⟩= ⟨J1(τ) J2(τ)⟩Q. This correlation is measurable and directly reflects that between the two internal cavity modes, which leads to an EPR paradox. Theories of the measured homodyne current generated by a stochastic Schrödinger equation (SSE) can be tested in a simulation of the Einstein-Podolsky-Rosen (EPR) correlations for a two-mode squeezed state.

A simulation was carried out to determine the correct stochastic term for the measured current in the broad-band limit. Stratonovich stochastic noise, rather than Ito noise, agrees with experimental results. This distinction is relevant to measurement noise and errors in quantum technologies. By analysing SSE trajectories as measurement settings are changed, a modern version of Schrödinger’s gedanken experiment is proposed, where one measures position and momenta simultaneously, “one by direct, the other by indirect measurement”. Trajectories generated by the SSE were originally proposed as models for state reduction in quantum measurement and are now used to simulate homodyne output measurements, which are increasingly important in quantum technology and foundational experiments.

Most applications have been limited to cases where the trajectory models a single mode quantum system. The value of the trajectory is interpreted as providing a realization of the outcome of the measurement once quantum outputs are amplified. This simulation of EPR correlations considered spatially separated particles, realised using homodyne measurements on fields. Analysis shows that predicted current correlations change drastically with the stochastic method used.

Modelling measurements with Ito stochastic noise gives incorrect equal-time correlations, either with or without a time-shift, because only integrals of Ito noise have physical meaning. This can be resolved by including detector bandwidth effects, which leads to EPR correlations between time-integrated observables. However, to obtain the correct unfiltered current satisfying an EPR correlation, analogous to correlated positions and anti-correlated momenta, it is proven that one must use a Stratonovich noise at large bandwidth. The correct choice of stochastic term is essential for understanding wideband continuous output measurements with multimode correlations.

Stratonovich methods validate quantum measurement simulation with fidelity-noise trade-offs

Accurate simulation of quantum systems is important for building functional quantum technologies, yet faithfully capturing the impact of measurement remains a significant hurdle. The Stratonovich approach is superior for modelling broad-band homodyne current, despite a trade-off between signal fidelity and noise. Higher bandwidths demand increased sampling to manage amplified noise. This presents a practical challenge for scaling simulations to complex, multimode systems, as computational cost rises with the demand for finer resolution.

Acknowledging the necessary compromise between signal clarity and noise reduction is important, and this delivers a strong validation of simulation techniques for quantum systems. Demonstrating the Stratonovich method accurately models measurement processes offers a pathway to more reliable designs for quantum technologies. Confirming the appropriate stochastic modelling of quantum measurement represents a vital advancement for simulating complex systems. This establishes that Stratonovich noise, rather than the commonly used Ito noise, accurately represents measured current in broad-band quantum experiments utilising a two-mode squeezed state, ensuring simulations reflect observed physical behaviour. By validating this approach through simulations of Einstein-Podolsky-Rosen correlations, scientists provide a more reliable foundation for designing and analysing future quantum technologies, including devices employing continuous variables.

The research confirmed that the Stratonovich method accurately simulates broad-band homodyne current generated by a stochastic Schrödinger equation, unlike the Ito approach. This is important because accurate simulation of quantum measurement is crucial for developing functional quantum technologies. Scientists demonstrated this by successfully modelling Einstein-Podolsky-Rosen correlations using a two-mode squeezed state, revealing a trade-off between signal fidelity and noise at higher bandwidths. The authors suggest this work provides a more reliable basis for the design and analysis of future quantum devices employing continuous variables.