Generalized collective quantum tomography: algorithm design, optimization, and validation

TL;DR

The paper addresses the challenge of jointly characterizing quantum states, detectors, and processes by formulating a generalized collective tomography framework based on tensor-structured optimization. It introduces closed-form solutions with provable $O(1/N)$ mean-squared-error scaling and develops SOS optimization under semi-algebraic constraints to attain near-global optima, validated numerically and with two-copy experimental data that reveal purity information enhances estimation. The methods unify QST, QDT, and QPT into a single approach, provide explicit computational complexities, and demonstrate practicality through pure-state, unitary, and measurement tomography examples, as well as experimental data. The work significantly improves estimation accuracy by leveraging purity information from collective measurements and approaches the collective MSE bound, with potential impact on scalable quantum system identification and benchmarking.

Abstract

Quantum tomography is a fundamental technique for characterizing, benchmarking, and verifying quantum states and devices. It plays a crucial role in advancing quantum technologies and deepening our understanding of quantum mechanics. Collective quantum state tomography, which estimates an unknown state \r{ho} through joint measurements on multiple copies $ρ\otimes\cdots\otimesρ$ of the unknown state, offers superior information extraction efficiency. Here we extend this framework to a generalized setting where the target becomes $S_1\otimes\cdots\otimes S_n$, with each $S_i$ representing identical or distinct quantum states, detectors, or processes from the same category. We formulate these tasks as optimization problems and develop three algorithms for collective quantum state, detector and process tomography, respectively, each accompanied by an analytical characterization of the computational complexity and mean squared error (MSE) scaling. Furthermore, we develop optimal solutions of these optimization problems using sum of squares (SOS) techniques with semi-algebraic constraints. The effectiveness of our proposed methods is demonstrated through numerical examples. Additionally, we experimentally demonstrate the algorithms using two-copy collective measurements, where entangled measurements directly provide information about the state purity. Compared to existing methods, our algorithms achieve lower MSEs and approach the collective MSE bound by effectively leveraging purity information.

Figures (14)

• Figure 1: Schematic diagram for two-copy generalized collective QST: (a) The unknown input states are distinct (D-QST) as $\rho_1 \in \mathcal{S}_{d_1}$ and $\rho_2 \in \mathcal{S}_{d_2}$, which may have different dimensions. The measurement operators are $\{P_l\}_{l=1}^L \in \mathcal{D}_{d_1d_2}$. (b) The unknown input states are identical (I-QST) as $\rho_0\in \mathcal{S}_{d}$ and the measurement operators are $\{P_l\}_{l=1}^L\in \mathcal{D}_{d^2}$.
• Figure 2: Schematic diagram for two-copy generalized collective QDT: (a) the unknown detectors are distinct (D-QDT) as $\{P_l\}_{l=1}^{L} \in \mathcal{D}_{d_1}$ and $\{Q_k\}_{k=1}^{K} \in \mathcal{D}_{d_2}$, and the probe states are $\{\rho_{m}\}_{m=1}^{M}$ where $\rho_{m} \in \mathcal{S}_{d_1d_2}$. (b) The detectors are identical (I-QDT) as $\{P_l\}_{l=1}^{L} \in \mathcal{D}_{d}$, and the probe states are $\{\rho_{m}\}_{m=1}^{M}$ where $\rho_{m} \in \mathcal{S}_{d^2}$. This can be realized by, e.g., a fiber delay line Yokoyama2013hi2rohtua and a switch as in (c), where the switch toggled after the measurement of $\operatorname{Tr}_2(\rho_m)$ is completed.
• Figure 3: Schematic diagram for two-copy generalized collective QPT: (a) the unknown quantum processes are distinct (D-QPT) as $\mathcal{E}_1$ and $\mathcal{E}_2$ where the dimensions of the processes are $d_1$ and $d_2$, respectively. The input states are $\{\rho_{m}^{\operatorname{in}}\}_{m=1}^{M}$ where $\rho_{m}^{\operatorname{in}}\in \mathcal{S}_{d_1d_2}$ and the measurement operators are $\{P_l\}_{l=1}^{L} \in \mathcal{D}_{d_1d_2}$. (b) The unknown quantum processes are the same as $\mathcal{E}_0$ (I-QPT) with dimension $d$. The input states are $\{\rho_{m}^{\operatorname{in}}\}_{m=1}^{M}$ where $\rho_{m}^{\operatorname{in}}\in \mathcal{S}_{d^2}$ and the measurement operators are $\{P_l\}_{l=1}^{L} \in \mathcal{D}_{d^2}$. This can be realized by, e.g., two fiber delay lines and two switches as in (c), where the two switches both toggled after $\operatorname{Tr}_2(\rho_m^{\operatorname{in}})$ has passed through $\mathcal{E}_0$.
• Figure 4: Steps in our closed-form algorithm for the generalized collective QST.
• Figure 5: Steps in our closed-form algorithm for the generalized collective QDT.