Quantum state tomography and quantum detector tomography are two main problems in quantum system identification. In this paper, we study state tomography and detector tomography with prior knowledge about the true rank of the unknown state/detector. In this case, we propose a sufficient condition for static (non-adaptive) tomography algorithms to achieve the optimal infidelity scaling O(1/N) on the number of state copies N. Then we present two examples of such static quantum state tomography and quantum detector tomography algorithms. Numerical examples demonstrate the effectiveness of our algorithms.
Quantum state tomography and quantum detector tomography are two main problems in quantum system identification. In this paper, we study state tomography and detector tomography with prior knowledge about the true rank of the unknown state/detector. In this case, we propose a sufficient condition for static (non-adaptive) tomography algorithms to achieve the optimal infidelity scaling O(1/N) on the number of state copies N. Then we present two examples of such static quantum state tomography and quantum detector tomography algorithms. Numerical examples demonstrate the effectiveness of our algorithms.
Quantum tomography is a standard technique for characterizing, benchmarking and verifying quantum systems/devices and plays a vital role in advancing quantum technology and understanding the foundations of quantum mechanics. Achieving the highest possible tomography accuracy remains a central challenge. Here we unify the infidelity metrics for quantum state, detector and process tomography in a single index 1−F(S^,S), where S represents the true density matrix, POVM element, or process matrix, and S^ is its estimator. We establish a sufficient and necessary condition for any tomography protocol to attain the optimal scaling $1-F= O(1/N) $ where N is the number of state copies consumed, in contrast to the O(1/N) worst-case scaling of static methods. Guided by this result, we propose adaptive algorithms with provably optimal infidelity scalings for state, detector, and process tomography. Numerical simulations and quantum optical experiments validate the proposed methods, with our experiments reaching, for the first time, the optimal infidelity scaling in ancilla-assisted process tomography.
Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design optimal probe states for detector estimation based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the minimum UMSE and the minimum condition number for quantum detectors and provide concrete examples that can achieve optimal detector tomography. In order to enhance estimation precision, we also propose a two-step adaptive detector tomography algorithm and investigate how this adaptive strategy can be used to achieve efficient estimation of quantum detectors. Moreover, the superposition of coherent states are used as probe states for quantum detector tomography and the estimation error is analyzed. Numerical results demonstrate the effectiveness of both the proposed optimal and adaptive quantum detector tomography methods.
The estimation of all the parameters in an un-known quantum state or measurement device, known as quantum state tomography (QST) and quantum detector tomography (QDT), is fundamental to the characterization and control of quantum systems. This paper presents a unified state space framework for QST and QDT based on observable measurement results at different evolution moments, applicable to both closed and Markovian open quantum systems. We derive lower bounds on the number of sampling points required for a unique estimation and analyze the computational complexity and mean squared error (MSE) scaling. As the estimator may yield unphysical results, we introduce new correction techniques to enforce physicality, which may also improve the infidelity scaling. The effectiveness of the proposed methods is validated through numerical simulations.
Standard quantum tomography of a single qudit achieves an infidelity that scales in the worst case as O(1/N) for a sample of size N. Here, we propose a suitable generalization of the two-stage adaptive quantum tomography for a qubit to the case of a single qudit. This achieves an infidelity of the order of $O[1/\sqrt{{N}_{0}(N\ensuremath{-}{N}_{0})}]$ for all quantum states, where N0 and $N\ensuremath{-}{N}_{0}$ are the ensemble sizes employed in the two stages of the method. This result is based on a second-order Taylor series expansion of the infidelity that is obtained by means of the Fr\'echet derivative and measurement outcomes modeled by a multinomial distribution. Numerical simulations indicate that the choice N0=N/2 leads to an infidelity that scales approximately as O(1/N) for all quantum states in a wide range of dimensions, that is, a quadratic improvement of the infidelity when compared to standard quantum tomography in the case of low-rank states.
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small ε, estimating an unknown state on Cd of rank at most r to trace norm error ε with constant success probability requires, and is achievable with, Θ(ε2drmax{1,tr}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most t samples improve the complexity of algorithms making single-sample measurements by at most a factor t. Further, measuring order r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate.
For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on t samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
Abstract Quantum tomography is a process of quantum state recognition with multiple measurements. An essential goal for the quantum tomography algorithm is to find measurements that will maximize the useful information about an unknown quantum state obtained through measurements. One of the recently proposed methods of quantum tomography is the algorithm based on rank-preserving transformations. The main idea is to transform a basic measurement set in the way to provide a situation that is equivalent to measuring the maximally mixed state. As long as the tomography of the fully mixed state has the fastest convergence comparing to other states, this method is expected to be highly accurate. We present numerical and experimental comparisons of rank-preserving tomography with another adaptive method, which includes measurements in the estimator eigenbasis and with random-basis tomography. We also study ways to improve the efficiency of the rank-preserving transformations method using the transformation unitary freedom and a measurement set complementation.