Qubit Tomography Research Articles

Quantum tomography of entangled qubits by time-resolved single-photon counting with time-continuous measurements

In this article, we introduce a framework for entanglement characterization by time-resolved single-photon counting with measurement operators defined in the time domain. For a quantum system with unitary dynamics, we generate time-continuous measurements by shifting from the Schrödinger picture to the Heisenberg representation. In particular, we discuss this approach in reference to photonic tomography. To make the measurement scheme realistic, we impose timing uncertainty on photon counts along with the Poisson noise. Then, the framework is tested numerically on quantum tomography of qubits. Next, we investigate the accuracy of the model for polarization-entangled photon pairs. Entanglement detection and precision of state reconstruction are quantified by figures of merit and presented on graphs versus the amount of time uncertainty.

Quantum Tomography of Pure States with Projective Measurements Distorted by Experimental Noise

The article undertakes the problem of pure state estimation from projective measurements based on photon counting. Two generic frames for qubit tomography are considered -- one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases (MUBs). Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements. The accuracy of each frame is quantified by the average fidelity and purity. The efficiency of the frames is compared and discussed. The method can be generalized to higher-dimensional states and transferred to other fields where the problem of complex vectors reconstruction appears.

Adaptive tomography of qubits: purity versus statistical fluctuations

The success of quantum information processing applications relies on accurate and efficient characterization of quantum states, especially nearly-pure states. In this work, we investigate a procedure for adaptive qubit state tomography which achieves O(1/N) scaling in accuracy for large N. We analyze the performance of the adaptive protocol on the characterization of pure, mixed and nearly-pure states, and clarify the interplay between the purity of the state and the statistical fluctuations of the measurement results. Our results highlight the subtle difference between the characterization of nearly-pure states and those of pure or highly mixed states.

Minimax estimation of qubit states with Bures risk

The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of n independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of 1/n. However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states.We propose qubit estimation strategies based on separate adaptive measurements, and collective measurements, that achieve 1/n scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource n to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us to derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the ‘standard’ rate n−1.

Single plane minimal tomography of double slit qubits

The determination of the density matrix of an ensemble of identically prepared quantum systems by performing a series of measurements, known as quantum tomography, is minimal when the number of outcomes is minimal. The most accurate minimal quantum tomography of qubits, sometimes called a tetrahedron measurement, corresponds to projections over four states which can be represented on the Bloch sphere as the vertices of a regular tetrahedron. We investigate whether it is possible to implement the tetrahedron measurement of double slit qubits of light, using measurements performed on a single plane. Assuming Gaussian slits and free propagation, we demonstrate that a judicious choice of the detection plane and the double slit geometry allows the implementation of a tetrahedron measurement. Finally, we consider possible sets of values which could be used in actual experiments.

Employing circuit QED to measure non-equilibrium work fluctuations

We study an interferometric method for measuring the statistics of work performed on a driven quantum system, which has been put forward recently (Dorner et al 2013 Phys. Rev. Lett. 110 230601, Mazzola et al 2013 Phys. Rev. Lett. 110 230602). This method allows the replacement of two projective measurements of the energy of the driven system with qubit tomography of an ancilla that is appropriately coupled to it. We highlight that this method could be employed to obtain the work statistics of closed as well as open driven system, even in the strongly dissipative regime. We then illustrate an implementation of the method in a circuit QED setup, which allows one to experimentally obtain the work statistics of a parametrically driven harmonic oscillator. Our implementation is an extension of the original method, in which two ancilla-qubits are employed and the work statistics are retrieved through two-qubit state tomography. Our simulations demonstrate the experimental feasibility.

Flux Noise Probed with Real Time Qubit Tomography in a Josephson Phase Qubit

We measure the dependence of qubit phase coherence and flux noise on inductor loop geometry. While wider inductor traces change neither the flux noise power spectrum nor the qubit dephasing time, increased inductance leads to a simultaneous increase in both. Using our new tomographic protocol for measuring low frequency flux noise, we make a direct comparison between the flux noise spectrum and qubit phase decay, finding agreement within 10% of theory.

Choice of measurement sets in qubit tomography

Optimal generalized measurements for state estimation are well understood. However, practical quantum state tomography is typically performed using a fixed set of projective measurements, and the question of how to choose these measurements has been largely unexplored in the literature. In this work, we develop theoretical asymptotic bounds for the average fidelity of pure qubit tomography using measurement sets whose axes correspond to faces of Platonic solids. We also present comprehensive simulations of maximum likelihood tomography for mixed qubit states using the Platonic solid measurements. We show that overcomplete measurement sets can be used to improve the accuracy of tomographic reconstructions.

Wigner tomography of two-qubit states and quantum cryptography

Tomography of the two-qubit density matrix shared by Alice and Bob is an essential ingredient for guaranteeing an acceptable margin of confidentiality during the establishment of a secure fresh key through the quantum key distribution scheme. We show how the Singapore protocol for key distribution is optimal from this point of view, due to the fact that it is based on so-called symmetric informationally complete positive-operator-valued measure (SIC POVM) qubit tomography which allows the most accurate full tomographic reconstruction of an unknown density matrix on the basis of a restricted set of experimental data. We illustrate with the help of experimental data the deep connections that exist between SIC POVM tomography and discrete Wigner representations. We also emphasize the special role played by Bell states in this approach and propose a protocol for quantum key distribution during which a third party is able to concede or to deny a posteriori to the authorized users the ability to build a fresh cryptographic key.