Detecting quantum many-body states with imperfect measuring devices

TL;DR

This work models imperfect quantum measurements via a coarse-graining map $\mathcal{C}$ that aggregates particle information and reduces system size, enabling a rigorous analysis of which fine-grained states underlie a given coarse-grained observation. It derives exact preimage-volume formulas for the two-qubit case using geometry and confirms them with random-matrix theory extended to $N$ qubits, showing that observed coarse-grained states concentrate near the maximally mixed state as system size grows. The authors also compute the average preimage state, revealing a predominantly isotropic mixture with a singlet contribution for maximally mixed targets, and they examine separable versus entangled preimages, aided by a symmetry under unitary covariance. Numerically, Monte Carlo simulations validate the analytical predictions and demonstrate practical strategies to detect coarse-graining and estimate optimal neighborhood volumes, offering a principled framework for tomography and state-detection under imperfect devices.

Abstract

We study a coarse-graining map arising from incomplete and imperfect addressing of particles in a multipartite quantum system. In its simplest form, corresponding to a two-qubit state, the resulting channel produces a convex mixture of the two partial traces. We derive the probability density of obtaining a given coarse-grained state, using geometric arguments for two qubits coarse-grained to one, and random-matrix methods for larger systems. As the number of qubits increases, the probability density sharply concentrates around the maximally mixed state, making nearly pure coarse-grained states increasingly unlikely. For two qubits, we also compute the inverse state needed to characterize the effective dynamics under coarse-graining and find that the average preimage of the maximally mixed state contains a finite singlet component. Finally, we validate the analytical predictions by inferring the underlying probabilities from Monte-Carlo-generated coarse-grained statistics.

Figures (14)

• Figure 1: Visualization of a pure state formed by two qubits in the parameterization (\ref{['eq:app-parametrized-state']}). The state shown corresponds to the values $\eta={\frac{\pi}{6}}$,$\gamma=\frac{\pi}{3}$, $\theta_{1} ={\frac{3\pi}{4}}$, $\phi_{1}=\frac{\pi}{2}$, $\theta_{2}=\frac{\pi}{4}$ and $\phi_{2}= {\frac{7\pi}{4}}$. The orange and purple spheres correspond to the Bloch spheres of the reduced states of qubit 1 and 2, respectively. The Bloch vectors in both cases have a length $\cos\eta={\frac{\sqrt{3}}{2}}$. The gray hemisphere between the spheres is the entanglement hemisphere, it shows the concurrence angle $\eta$ and the relative phase $\gamma$. Note that the reduced states can only be pure when the entanglement arrow points to the north pole, i.e., when $\eta=0$.
• Figure 2: Green and red subspheres corresponding to the locus of states $\rho_{1}$ and $\rho_{2}$, respectively. Each state in the green subsphere has a corresponding state in the red subsphere, so that the sum of their Bloch vectors is equal to $\vb{r}_\mathrm{ts}$ (chosen here along the $z$ axis). The gray sphere is the Bloch sphere associated with the target state $\rho$, which can be seen at the blue dot. In both cases shown $h = 0.4$. In the left figure $r_\mathrm{ts}=0.3$ was used, so the subspheres are contained within the Bloch sphere; while in the right $r_\mathrm{ts}=0.5$. Separable states, represented by bold curves, are available only when $r_\mathrm{ts}\geq h$ as in the right figure. See the main text for details.
• Figure 3: Probability density $P_2(h;r_\mathrm{ts})$vs.$r_\mathrm{ts}$, for various values of $h$. In all cases, $P_2$ tends to zero for both $r_\mathrm{ts} \rightarrow 0$ (maximally mixed state) and $r_\mathrm{ts} \rightarrow 1$ (pure states). The cusp at $r_\mathrm{ts}=h$ corresponds to the two subspheres in Fig. \ref{['fig:esferas']} touching the outer Bloch sphere.
• Figure 4: The simplex $S_0$ and the image of 40,000 bipartite states in it.
• Figure 5: Histogram of the distribution of $a \equiv \rho_{00}$, for various values of $p_1$. Note that for $p_1=.5$ (orange/ochre histogram in the back) there is a cusp at $a=.5$, which is smoothed out for the other values of $p_1$.