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arXiv:2107.00121 · Physical Review A

Self-consistent state and measurement tomography with fewer measurements

A. Stephens, Jacob Cutshall, T. McPhee, M. Beck

Published Jul 19, 2021• 5 citations
arXivHTMLPDF

Abstract

We describe a technique for self-consistently characterizing both the quantum state of a single-qubit system, and the positive-operator-valued measure (POVM) that describes measurements on the system. The method works with only 10 measurements. We assume that a series of unitary transformations performed on the quantum state are fully known, while making minimal assumptions about both the density operator of the state and the POVM. The technique returns maximum-likely estimates of both the density operator and the POVM. To experimentally demonstrate the method, we perform reconstructions of over 300 state-measurement pairs, and we compare them to their expected density operators and POVMs. We find that 95% of the reconstructed POVMs have fidelities of 0.98 or greater, and 92% of the density operators have fidelities that are 0.98 or greater.

DOI
10.1103/physreva.104.012416

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We describe a technique for self-consistently characterizing both the quantum state of a single-qubit system, and the positive-operator-valued measure (POVM) that describes measurements on the system. The method works with only 10 measurements. We assume that a series of unitary transformations performed on the quantum state are fully known, while making minimal assumptions about both the density operator of the state and the POVM. The technique returns maximum-likely estimates of both the density operator and the POVM. To experimentally demonstrate the method, we perform reconstructions of over 300 state-measurement pairs, and we compare them to their expected density operators and POVMs. We find that 95% of the reconstructed POVMs have fidelities of 0.98 or greater, and 92% of the density operators have fidelities that are 0.98 or greater.
We propose an estimation method for quantum measurement tomography (QMT) based on semidefinite programming (SDP) and discuss how it may be employed to detect experimental imperfections, such as shot noise and/or faulty preparation of the input states on near-term quantum computers. Moreover, if the positive operator-valued measure (POVM) we aim to characterize is informationally complete, we put forward a method for self-consistent tomography, i.e., for recovering a set of input states and POVM effects that is consistent with the experimental outcomes and does not assume any a priori knowledge about the input states of the tomography. Contrary to many methods that have been discussed in the literature, our approach does not rely on additional assumptions such as low noise or the existence of a reliable subset of input states.
Quantum computers solve ever more complex tasks using steadily growing system sizes.Characterizing these quantum systems is vital, yet becoming increasingly challenging.The gold-standard method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge.The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)-a daunting bottleneck given the scale of existing and nearterm quantum devices.Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs).We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion-without requiring ancillary ions to assist in measurements.More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot.Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination.SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods.The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state.Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running).We demonstrate this on an eight-qubit entangled state (which has 2 8 • 2 8 -1 = 65 535 degrees of freedom), as well as for fast state identification.All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.
Abstract In this paper, we introduce an analytical framework for the reconstruction of quantum states. The reconstruction of an unknown quantum state requires the information of a complete set of observables, obtained through experimental measurements of Hermitian operators usually defined as positive-operator-valued measures (POVMs). The scheme involves a single-qubit unambiguous state discrimination POVM, which can be generalized to perform n-qubit measurements. We also use maximum likelihood estimation as a method in the reconstruction of the density matrix from experimental data and show that the expected value of the cleaner is independent of the parameter of the density operator.
Generalized quantum measurements, described by positive operator-valued measures (POVMs), are essential for modeling realistic processes in open quantum systems. While quantum process tomography can fully characterize a POVM, it is resource-intensive and impractical when only specific POVM elements or matrix elements of a particular POVM element are of interest. Direct quantum measurement tomography offers a more efficient alternative but typically relies on weak interactions and complex structures of the system, environment, and probe as the dimension of the system increases, limiting its precision and scalability. Furthermore, characterizing a POVM element alone is insufficient to determine the underlying physical mechanism, as multiple Kraus operators can yield the same measurement statistics. In this work, we present a unified framework for the direct characterization of individual matrix elements of Kraus operators associated with specific POVM elements and arbitrary input states without requiring weak interaction, complex structures of the system-environment-probe or full process and state tomography. This framework naturally extends to projective measurements, enabling direct observable tomography, and to the characterization of unitary operations. Our method also captures modular and weak values of observables and Kraus operators, without invoking weak interaction approximations. We demonstrate potential implementations in optical systems, highlighting the experimental feasibility of our approach.
We present a method for measuring quantum states encoded in the temporal modes of photons. The basis for the multilevel quantum states is defined by the use of modes propagating in a dispersive medium, which is a fiber in this case. The propagation and time-resolved single-photon detection allow us to define a positive-operator valued measure (POVM). The POVM depends on the amount of dispersion and the characteristics of a detector. This framework is numerically tested by performing quantum state tomography on a large number of states for a set of realistic experimental settings. Finally, the average fidelity between the expected and reconstructed states is computed for qubits, qutrits, and entangled qubits.
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.