Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. After reviewing the concept of minimax state estimation, we use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. After reviewing the concept of minimax state estimation, we use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.

International Journal of Quantum InformationJun 1, 201211 citations

Minimax Quantum Tomography: Estimators and Relative Entropy Bounds

Christopher Ferrie, Robin Blume-Kohout

A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O(1/sqrt[N])-in contrast to that of classical probability estimation, which is O(1/N)-where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O(1/sqrt[N])-in contrast to that of classical probability estimation, which is O(1/N)-where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.

arXiv:1503.03100Physical Review LettersMar 4, 201617 citations

Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators

Steven T. Flammia, David Gross, Yi-Kai Liu +1 more

Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.

arXiv:1205.2300New Journal of PhysicsSep 27, 2012356 citations

Optimal error regions for quantum state estimation

Jiangwei Shang, Hui Khoon Ng, Arun Sehrawat +2 more

An estimator is a state that represents one's best guess of the actual state of the quantum system for the given data. Such estimators are points in the state space. To be statistically meaningful, they have to be endowed with error regions, the generalization of error bars beyond one dimension. As opposed to standard ad hoc constructions of error regions, we introduce the maximum-likelihood region—the region of largest likelihood among all regions of the same size—as the natural counterpart of the popular maximum-likelihood estimator. Here, the size of a region is its prior probability. A related concept is the smallest credible region—the smallest region with pre-chosen posterior probability. In both cases, the optimal error region has constant likelihood on its boundary. This surprisingly simple characterization permits concise reporting of the error regions, even in high-dimensional problems. For illustration, we identify optimal error regions for single-qubit and two-qubit states from computer-generated data that simulate incomplete tomography with few measured copies.An estimator is a state that represents one's best guess of the actual state of the quantum system for the given data. Such estimators are points in the state space. To be statistically meaningful, they have to be endowed with error regions, the generalization of error bars beyond one dimension. As opposed to standard ad hoc constructions of error regions, we introduce the maximum-likelihood region—the region of largest likelihood among all regions of the same size—as the natural counterpart of the popular maximum-likelihood estimator. Here, the size of a region is its prior probability. A related concept is the smallest credible region—the smallest region with pre-chosen posterior probability. In both cases, the optimal error region has constant likelihood on its boundary. This surprisingly simple characterization permits concise reporting of the error regions, even in high-dimensional problems. For illustration, we identify optimal error regions for single-qubit and two-qubit states from computer-generated data that simulate incomplete tomography with few measured copies.

arXiv:1302.4081New Journal of PhysicsDec 13, 201358 citations

Informationally incomplete quantum tomography

Yong Siah Teo, Jaroslav Řeháček, Z. Hradil

Abstract In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/ process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantumstate and quantum-process tomography are arranged together in the form of a tutorial review article that is catered to students and researchers who are new to the field of quantum tomography, with focus on maximum-likelihood related techniques as instructive examples to illustrate these ideas.Abstract In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/ process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantumstate and quantum-process tomography are arranged together in the form of a tutorial review article that is catered to students and researchers who are new to the field of quantum tomography, with focus on maximum-likelihood related techniques as instructive examples to illustrate these ideas.

arXiv:1309.2906Quantum Measurements and Quantum MetrologyNov 12, 201313 citations

Efficient Learning of Arbitrary Single-Copy Quantum States

Shibdas Roy, Filippo Caruso, Srushti Patil +1 more

Quantum state tomography is the problem of estimating a given quantum state. Usually, it is required to run the quantum experiment - state preparation, state evolution, measurement - several times to be able to estimate the output quantum state of the experiment, because an exponentially high number of copies of the state is required. In this work, we present an efficient algorithm to estimate with a small but nonzero probability of error the output state of the experiment using a single copy of the state, without knowing the evolution dynamics of the state. It also does not destroy the original state, which can be recovered easily for any further quantum processing. As an example, it is usually required to repeat a quantum image processing experiment many times, since many copies of the state of the output image are needed to extract the information from all its pixels. The information from $\mathcal{N}$ pixels of the image can be inferred from a single run of the image processing experiment in our algorithm, to efficiently estimate the density matrix of the image state.Quantum state tomography is the problem of estimating a given quantum state. Usually, it is required to run the quantum experiment - state preparation, state evolution, measurement - several times to be able to estimate the output quantum state of the experiment, because an exponentially high number of copies of the state is required. In this work, we present an efficient algorithm to estimate with a small but nonzero probability of error the output state of the experiment using a single copy of the state, without knowing the evolution dynamics of the state. It also does not destroy the original state, which can be recovered easily for any further quantum processing. As an example, it is usually required to repeat a quantum image processing experiment many times, since many copies of the state of the output image are needed to extract the information from all its pixels. The information from$\mathcal{N}$pixels of the image can be inferred from a single run of the image processing experiment in our algorithm, to efficiently estimate the density matrix of the image state.

arXiv:2310.19748IEEE International Conference on Quantum Control, Computing and Learning (qCCL)Jun 25, 20250 citations

\mathcal{N}

pixels of the image can be inferred from a single run of the image processing experiment in our algorithm, to efficiently estimate the density matrix of the image state.