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arXiv:0803.1946

Maximal qubit tomography

Tamás Tasnádi

Published Mar 13, 2008• 0 citations
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Abstract

A new qubit tomography protocol is introduced, based on a continuous positive operator valued measure, which is supported by the set of pure states, and equivariant under the symmetry group SO(3,R) of the qubit state space. Thus the sample data for the tomography protocol is a discrete set of points on the Bloch sphere obtained by a series of independent measurements on identical replicas of the qubit. Although the maximum likelihood estimator uniquely exists, it cannot be explicitly calculated. Instead, we present another convenient, simple estimator, show that it is unbiased and calculate its variance. Furthermore, a detailed account is given about known properties of related discrete qubit state estimation protocols (like ``minimal qubit tomography''), and the results are compared, discussed.

DOI
10.48550/arxiv.0803.1946

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A new qubit tomography protocol is introduced, based on a continuous positive operator valued measure, which is supported by the set of pure states, and equivariant under the symmetry group SO(3,R) of the qubit state space. Thus the sample data for the tomography protocol is a discrete set of points on the Bloch sphere obtained by a series of independent measurements on identical replicas of the qubit. Although the maximum likelihood estimator uniquely exists, it cannot be explicitly calculated. Instead, we present another convenient, simple estimator, show that it is unbiased and calculate its variance. Furthermore, a detailed account is given about known properties of related discrete qubit state estimation protocols (like ``minimal qubit tomography''), and the results are compared, discussed.
A statistically feasible data post-processing method for the conventional qubit state tomography is studied from an information geometrical point of view. It is shown that the space (−1,1)3(-1,1)^3(−1,1)3 of the Stokes parameters (ξ1,ξ2,ξ3)(ξ_1, ξ_2,ξ_3)(ξ1​,ξ2​,ξ3​) that specify qubit states should be regarded as a Riemannian manifold endowed with a metric gij:=δij/(1−(ξi)2)g_{ij}:=δ_{ij}/(1-(ξ_i)^2)gij​:=δij​/(1−(ξi​)2), and that the data processing based on the maximum likelihood method is realized by the orthogonal projection from the empirical distribution onto the Bloch sphere with respect to the metric gijg_{ij}gij​. An efficient algorithm for computing the maximum likelihood estimate is also proposed.
Tomography of a quantum state is usually based on a positive-operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows us to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside from the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multidimensional Laplace integrals appearing in Bayesian mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum nondemolition measurements relying on Rydberg atoms, and heterodyne fluorescence measurements of a superconducting qubit.
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.
In experiments it is often required that we determine an unknown quantum state produced by a source. Quantum Tomography accomplishes this through repeated measurements on the state. This process requires 4^n measurements for an n-qubit system. However, if it is known that the state is pure, fewer measurements are required. Many such results already exist, but they either use non-local measurements or they have some chance of failing. Our goal is to obtain the smallest set of Pauli operators which can uniquely determine any pure state among all states. This involves ensuring the complement of the span of the measurements must have two positive and two negative eigenvalues. This is nontrivial since there are few relationships between the eigenvalues of a set of operators and the eigenvalues their real linear combinations. We obtain the lower bound of 30 Paulis necessary for 3-qubit tomography which improves on any known result.
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. In this article, we review the concept of minimax state estimation, and 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.