Quantum information · Theory
Distributed Inner-Product Estimation for Continuous-Variable Quantum States
arXiv:2610.40141v1 ↗PDF ↗Submitted 29 October 2026
- Question
- Two separate platforms may each prepare a complex multimode bosonic state with no tractable classical description, yet they cannot measure jointly. How can they compare the states, by estimating \(\operatorname{Tr}(\rho\sigma)\), with only local measurements and classical communication? Discrete-variable protocols do not carry over, because the state space is infinite-dimensional.
- Result
- A protocol using only homodyne detection at shared random phases gives an unbiased estimate of a truncated inner product, with explicit guarantees. Under a mean-energy bound, the number of copies scales as \(\epsilon^{-(2M+2)}\log(2/\delta)\) for \(M\) modes. For Gaussian states, this improves to \(\epsilon^{-2}[\log(2M/\epsilon)]^{2M}\log(2/\delta)\).
The paper in detail
Two platforms hold \(M\)-mode states \(\rho,\sigma\). With Weyl operators \(W(\mathbf u)=e^{i\mathbf u^{\mathsf T}\hat{\mathbf R}}\) and characteristic functions \(\chi_\tau(\mathbf u)=\operatorname{Tr}[\tau W(\mathbf u)]\), the Plancherel identity gives \[\operatorname{Tr}(\rho\sigma)=\frac1{(2\pi)^M}\int_{\mathbb R^{2M}}\chi_\rho(\mathbf u)\chi_\sigma(-\mathbf u)\,d\mathbf u.\] This writes the inner product as a phase-space integral of a product of local quantities, which is the opening needed for an LOCC protocol. The question is how to sample this integral with accessible measurements and how much it costs.
A homodyne measurement at a list of phases \( heta_1,\dots, heta_M\) gives direct access to a slice: the Fourier transform of its outcomes is \(\chi_\tau\) on the line through the origin at those phases. So if both platforms measure at the same random phases, the expectation of \(e^{i\mathbf s^{\mathsf T}(\mathbf x-\mathbf y)}\) is the integrand at the point \(\mathbf u\) with components \(s_\ell(\cos heta_\ell,\sin heta_\ell)\). The integral over all of phase space would have unbounded weights, so the authors truncate to a product of disks of radius \(R\), obtaining \(f_R\) and a bounded estimator.
The error splits into statistical error and truncation bias. Hoeffding's inequality handles the first, since the single-setting statistic is bounded by \(c_R=(R^2/2)^M\), so \(N_\theta\ge8c_R^2\log(2/\delta)/\epsilon^2\) suffices when the bias is at most \(\epsilon/2\). The bias is controlled by assumptions on the states: \((4ME+2M)/R^2\) under a mean-energy bound \(E\) per mode, and \(M\exp[-R^2/2(4ME+2M)]\) for Gaussian states.
Choosing \(R\) to make the bias \(\epsilon/2\) gives the two sufficient setting counts, and the total number of copies is \(2nN_\theta\). A complementary Bernstein bound brings in the dependence on \(n\), the shots per setting. The sections we read do not settle whether the exponential scaling in \(M\) is necessary.
Transferable methods
Sampling a phase-space integral through homodyne slices
The estimator turns a \(2M\)-dimensional integral into random phases measured physically and radial variables integrated analytically, so that no sampling is wasted on the part that can be done by hand.
Write each mode's phase-space point in signed polar form, \((u_{q,\ell},u_{p,\ell})=s_\ell(\cos\theta_\ell,\sin\theta_\ell)\). The Jacobian is \(\prod_\ell|s_\ell|\). So drawing \(\theta_\ell\) uniformly on \([0,\pi)\) and \(s_\ell\) with density \(q_R(s)=|s|/R^2\) on \([-R,R]\) makes \(\mathbf u\) uniform on the product of disks.
By the Fourier-slice relation, conditional on the phases the outcomes satisfy \(\mathbb E[e^{i\mathbf s^{\mathsf T}(\mathbf x-\mathbf y)}]=\chi_\rho(\mathbf u)\chi_\sigma(-\mathbf u)\). Averaging over \(\mathbf u\), multiplying by the volume factor \(c_R=(R^2/2)^M\) and keeping the real part gives \(f_R\). So the statistic is unbiased.
The radial average can be done exactly, which can only reduce variance, by the law of total variance. For one mode, \[\psi_R(t)=\frac2{R^2}\int_0^Rs\cos(st)\,ds=\frac{2[Rt\sin(Rt)+\cos(Rt)-1]}{(Rt)^2}.\] So each setting contributes \[W_i=\frac{c_R}{n^2}\sum_{a,b}\prod_\ell\psi_R\big(x^{(a)}_{i,\ell}-y^{(b)}_{i,\ell}\big).\]
Editorial check: for two coherent states of one mode with exact overlap \(0.67032\) and \(R=6\), direct quadrature of \(f_R\) gave \(0.67032\), confirming the normalisation of the Plancherel identity and Jacobian. Simulating the estimator with \(n=1\) and \(2\times10^5\) settings gave \(0.687\pm0.018\), consistent with the target, and every sample respected \(|W|\le c_R=18\). The formula for \(\psi_R\) agreed with numerical integration to seven digits. The estimator's large standard error, with this many settings, shows why these counts are high.
How to reuse it
If a quantity is a phase-space integral of a product of two local functions, shared randomness plus independent slices estimates it without joint measurement. Do by hand any part of the average that needs no data. The method needs a shared phase reference; without one, the slices do not line up.
Bounding truncation bias by energy
Truncation needs a tail bound on \(\chi_\tau\), and the only assumption is finite energy. The argument converts mean energy into a quadratic moment of the characteristic function.
Define \(I_\tau=(2\pi)^{-M}\int\|\mathbf u\|^2|\chi_\tau(\mathbf u)|^2d\mathbf u\). Since commuting \(\hat R_j\) with \(W(\mathbf u)\) brings down a component of \(\Omega\mathbf u\), Plancherel gives \(I_\tau=\sum_j\|[\hat R_j,\tau]\|_2^2\).
Each term expands to \(2\operatorname{Tr}(\hat R_j^2\tau^2)-2\operatorname{Tr}(\hat R_j\tau\hat R_j\tau)\), and the second trace is nonnegative. With \(\sum_j\hat R_j^2=2\hat N+M\) and \(\tau^2\preceq\tau\), this gives \[I_\tau\le4\operatorname{Tr}(\tau\hat N)+2M\le4ME+2M.\]
Outside the truncation domain some mode has radius above \(R\), so \(\|\mathbf u\|^2/R^2\ge1\). Inserting this factor and applying Cauchy–Schwarz gives \[|f-f_R|\le\frac{\sqrt{I_\rho I_\sigma}}{R^2}\le\frac{4ME+2M}{R^2}.\] This inverse-square decay is the source of the extra cost: \(R^2\sim1/\epsilon\) makes \(c_R^2\sim\epsilon^{-2M}\). For Gaussian states, with exponential tails, \(R^2\sim\log(1/\epsilon)\).
Editorial check: for two vacuum states the exact bias is \(e^{-R^2/2}\), and it lies below both stated bounds at \(R=1,2,3,5\): for example \(0.0111\) against \(0.222\) and \(0.105\) at \(R=3\). We also recomputed the two copy-count formulas at \(M=1\), \(E=1\), \(\epsilon=0.1\), \(\delta=0.05\): about \(1.06\times10^7\) and \(9.5\times10^5\) settings. The Gaussian advantage is thus modest at this accuracy and grows only as \(\epsilon\) falls.
How to reuse it
For bosonic estimation tasks, a physical energy bound can stand in for compactness: turn it into a moment bound through commutators, then use Markov-style tail control. Expect only algebraic decay from one moment. Stronger tails need stronger assumptions, such as Gaussianity or bounds on higher moments of \(\hat N\).
Context & limitations
Why it matters. Homodyne detection is the standard measurement of continuous-variable optics, so the protocol needs no new hardware and no state reconstruction. The bounds also identify what costs what: the factor \(\epsilon^{-2M}\) comes from truncating phase space, not from statistics, and it is the only part that Gaussian structure improves.
Limitations. Copy counts are exponential in the mode number and are sufficient conditions, which the authors call conservative; the sections we read prove no matching lower bound. For instance, a single mode with \(E=1\), \(\epsilon=0.1\) and \(\delta=0.05\) gives about \(10^7\) phase settings by Equation (28). The platforms must share a calibrated phase reference and a quadrature normalisation, and the guarantee assumes a known energy bound. A separate paper on the same topic, 2610.39912, appeared the same day.