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arXiv:1303.7154 · Physical Review A

Quantum state tomography from a sequential measurement of two variables in a single setup

Antonio Di Lorenzo

Published Oct 17, 2013• 13 citations
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Abstract

We demonstrate that the task of determining an unknown quantum state can be accomplished efficiently by making a sequential measurement of two observables, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}$, the eigenstates of which form bases connected by a discrete Fourier transform. The state can be pure or mixed, the dimension of the Hilbert space and the coupling strength are arbitrary, and the experimental setup is fixed. The concept of Moyal quasicharacteristic function is introduced for finite-dimensional Hilbert spaces.

DOI
10.1103/physreva.88.042114

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We demonstrate that the task of determining an unknown quantum state can be accomplished efficiently by making a sequential measurement of two observables, A\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}A\ifmmode^\elseˆ\fi​ and B\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}B\ifmmode^\elseˆ\fi​, the eigenstates of which form bases connected by a discrete Fourier transform. The state can be pure or mixed, the dimension of the Hilbert space and the coupling strength are arbitrary, and the experimental setup is fixed. The concept of Moyal quasicharacteristic function is introduced for finite-dimensional Hilbert spaces.
It is shown how it is possible to reconstruct the initial state of a one-dimensional system by sequentially measuring two conjugate variables. The procedure relies on the quasicharacteristic function, the Fourier transform of the Wigner quasiprobability. The proper characteristic function obtained by Fourier transforming the experimentally accessible joint probability of observing "position" then "momentum" (or vice versa) can be expressed as a product of the quasicharacteristic function of the two detectors and that unknown of the quantum system. This allows state reconstruction through the sequence (1) data collection, (2) Fourier transform, (3) algebraic operation, and (4) inverse Fourier transform. The strength of the measurement should be intermediate for the procedure to work.
For any finite-dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state. This means that such measurements can be used to distinguish an arbitrary pure state from any other state, pure or mixed, and the pure state can be reconstructed from the outcome distribution in a feasible way. The set of measurements we construct is independent of the unknown state, and therefore our results provide a fixed scheme for pure state tomography, as opposed to the adaptive (state-dependent) scheme proposed by Goyeneche et al. ( Phys. Rev. Lett. , 115 (2015) 090401). We show that our scheme is robust with respect to noise, in the sense that any measurement scheme which approximates these measurements well enough is equally suitable for pure state tomography. Finally, we present two convex programs which can be used to reconstruct the unknown pure state from the measurement outcome distributions.
We describe a quantum state tomography scheme which is applicable to a system described in a Hilbert space of arbitrary finite dimensionality and is constructed from sequences of two measurements. The scheme consists of measuring the various pairs of projectors onto two bases --which have no mutually orthogonal vectors--, the two members of each pair being measured in succession. We show that this scheme implies measuring the joint quasi-probability of any pair of non-degenerate observables having the two bases as their respective eigenbases. The model Hamiltonian underlying the scheme makes use of two meters initially prepared in an arbitrary given quantum state, following the ideas that were introduced by von Neumann in his theory of measurement.
We describe a quantum state tomography scheme which is applicable to a system described in a Hilbert space of arbitrary finite dimensionality and is constructed from sequences of two measurements. The scheme consists of measuring the various pairs of projectors onto two bases --which have no mutually orthogonal vectors--, the two members of each pair being measured in succession. We show that this scheme implies measuring the joint quasi-probability of any pair of non-degenerate observables having the two bases as their respective eigenbases. The model Hamiltonian underlying the scheme makes use of two meters initially prepared in an arbitrary given quantum state, following the ideas that were introduced by von Neumann in his theory of measurement.
We utilize a discrete (sequential) measurement protocol to investigate quantum process tomography of a single two-level quantum system, with an unknown initial state, undergoing Rabi oscillations. The ignorance of the dynamical parameters is encoded into a continuous-variable classical system which is coupled to the two-level quantum system via a generalized Hamiltonian. This combined estimate of the quantum state and dynamical parameters is updated by using the information obtained from sequential measurements on the quantum system and, after a sufficient waiting period, faithful state monitoring and parameter determination is obtained. Numerical evidence is used to demonstrate the convergence of the state estimate to the true state of the hybrid system.
Quantum state tomography is an important tool for quantum communication, computation, metrology, and simulation. Efficient quantum state tomography on a high dimensional quantum system is still a challenging problem. Here, we propose a novel quantum state tomography method, auxiliary Hilbert space tomography, to avoid pre-rotations before measurement in a quantum state tomography experiment. Our method requires projective measurements in a higher dimensional space that contains the subspace that includes the prepared states. We experimentally demonstrate this method with orbital angular momentum states of photons. In our experiment, the quantum state tomography measurements are as simple as taking a photograph with a camera. We experimentally verify our method with near-pure- and mixed-states of orbital angular momentum with dimension up to d=13d=13d=13, and achieve greater than 95 % state fidelities for all states tested. This method dramatically reduces the complexity of measurements for full quantum state tomography, and shows potential application in various high dimensional quantum information tasks.