Asymptotically optimal purification and dilution of mixed qubit and Gaussian states

Given an ensemble of mixed qubit states, it is possible to increase the purity of the constituent states using a procedure known as state purification. The reverse operation, which we refer to as dilution, reduces the level of purity present in the constituent states. In this paper we find asymptotically optimal procedures for purification and dilution of an ensemble of i.i.d. mixed qubit states, for some given input and output purities and an asymptotic output rate. Our solution involves using the statistical tool of local asymptotic normality, which recasts the qubit problem in terms of attenuation and amplification of a single displaced Gaussian state. Therefore, to obtain the qubit solutions, we must first solve the analogous problems in the Gaussian setup. We provide full solutions to all of the above, for the (global) trace norm figure of merit.Comment: 11 pages, 6 figure

Fisher information and asymptotic normality in system identification for quantum Markov chains

This paper deals with the problem of estimating the coupling constant $\theta$ of a mixing quantum Markov chain. For a repeated measurement on the chain's output we show that the outcomes' time average has an asymptotically normal (Gaussian) distribution, and we give the explicit expressions of its mean and variance. In particular we obtain a simple estimator of $\theta$ whose classical Fisher information can be optimized over different choices of measured observables. We then show that the quantum state of the output together with the system, is itself asymptotically Gaussian and compute its quantum Fisher information which sets an absolute bound to the estimation error. The classical and quantum Fisher informations are compared in a simple example. In the vicinity of $\theta=0$ we find that the quantum Fisher information has a quadratic rather than linear scaling in output size, and asymptotically the Fisher information is localised in the system, while the output is independent of the parameter.Comment: 10 pages, 2 figures. final versio

Asymptotically optimal quantum channel reversal for qudit ensembles and multimode Gaussian states

We investigate the problem of optimally reversing the action of an arbitrary quantum channel C which acts independently on each component of an ensemble of n identically prepared d-dimensional quantum systems. In the limit of large ensembles, we construct the optimal reversing channel R* which has to be applied at the output ensemble state, to retrieve a smaller ensemble of m systems prepared in the input state, with the highest possible rate m/n. The solution is found by mapping the problem into the optimal reversal of Gaussian channels on quantum-classical continuous variable systems, which is here solved as well. Our general results can be readily applied to improve the implementation of robust long-distance quantum communication. As an example, we investigate the optimal reversal rate of phase flip channels acting on a multi-qubit register.Comment: 17 pages, 3 figure

Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation

In a unified viewpoint in quantum channel estimation, we compare the Cramer-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramer-Rao bound in the phase estimation case while the both bounds coincide when the minimum mean square error decreases with the order O(1/n). We also derive a sufficient condition for that the minimum mean square error decreases with the order O(1/n).Comment: In this revision, some unlcear parts are clarifie

Quantum learning: optimal classification of qubit states

Pattern recognition is a central topic in Learning Theory with numerous applications such as voice and text recognition, image analysis, computer diagnosis. The statistical set-up in classification is the following: we are given an i.i.d. training set $(X_{1},Y_{1}),... (X_{n},Y_{n})$ where $X_{i}$ represents a feature and $Y_{i}\in \{0,1\}$ is a label attached to that feature. The underlying joint distribution of $(X,Y)$ is unknown, but we can learn about it from the training set and we aim at devising low error classifiers $f:X\to Y$ used to predict the label of new incoming features. Here we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown qubit states. Given a number of `training' copies from each of the states, we would like to `learn' about them by performing a measurement on the training set. The outcome is then used to design mesurements for the classification of future systems with unknown labels. We find the asymptotically optimal classification strategy and show that typically, it performs strictly better than a plug-in strategy based on state estimation. The figure of merit is the excess risk which is the difference between the probability of error and the probability of error of the optimal measurement when the states are known, that is the Helstrom measurement. We show that the excess risk has rate $n^{-1}$ and compute the exact constant of the rate.Comment: 24 pages, 4 figure

Spectral thresholding quantum tomography for low rank states

The estimation of high dimensional quantum states is an important statistical problem arising in current quantum technology applications. A key example is the tomography of multiple ions states, employed in the validation of state preparation in ion trap experiments (Häffner et al 2005 Nature 438 643). Since full tomography becomes unfeasible even for a small number of ions, there is a need to investigate lower dimensional statistical models which capture prior information about the state, and to devise estimation methods tailored to such models. In this paper we propose several new methods aimed at the efficient estimation of low rank states and analyse their performance for multiple ions tomography. All methods consist in first computing the least squares estimator, followed by its truncation to an appropriately chosen smaller rank. The latter is done by setting eigenvalues below a certain 'noise level' to zero, while keeping the rest unchanged, or normalizing them appropriately. We show that (up to logarithmic factors in the space dimension) the mean square error of the resulting estimators scales as where r is the rank, is the dimension of the Hilbert space, and N is the number of quantum samples. Furthermore we establish a lower bound for the asymptotic minimax risk which shows that the above scaling is optimal. The performance of the estimators is analysed in an extensive simulations study, with emphasis on the dependence on the state rank, and the number of measurement repetitions. We find that all estimators perform significantly better than the least squares, with the 'physical estimator' (which is a bona fide density matrix) slightly outperforming the other estimators

Local asymptotic normality for qubit states

We consider n identically prepared qubits and study the asymptotic properties of the joint state \rho^{\otimes n}. We show that for all individual states \rho situated in a local neighborhood of size 1/\sqrt{n} of a fixed state \rho^0, the joint state converges to a displaced thermal equilibrium state of a quantum harmonic oscillator. The precise meaning of the convergence is that there exist physical transformations T_{n} (trace preserving quantum channels) which map the qubits states asymptotically close to their corresponding oscillator state, uniformly over all states in the local neighborhood. A few consequences of the main result are derived. We show that the optimal joint measurement in the Bayesian set-up is also optimal within the pointwise approach. Moreover, this measurement converges to the heterodyne measurement which is the optimal joint measurement of position and momentum for the quantum oscillator. A problem of local state discrimination is solved using local asymptotic normality.Comment: 16 pages, 3 figures, published versio

Identification of single-input–single-output quantum linear systems

The purpose of this paper is to investigate system identification for single-input–single-output general (active or passive) quantum linear systems. For a given input we address the following questions: (1) Which parameters can be identified by measuring the output? (2) How can we construct a system realization from sufficient input-output data? We show that for time-dependent inputs, the systems which cannot be distinguished are related by symplectic transformations acting on the space of system modes. This complements a previous result of Guţă and Yamamoto [IEEE Trans. Autom. Control 61, 921 (2016)] for passive linear systems. In the regime of stationary quantum noise input, the output is completely determined by the power spectrum. We define the notion of global minimality for a given power spectrum, and characterize globally minimal systems as those with a fully mixed stationary state. We show that in the case of systems with a cascade realization, the power spectrum completely fixes the transfer function, so the system can be identified up to a symplectic transformation. We give a method for constructing a globally minimal subsystem direct from the power spectrum. Restricting to passive systems the analysis simplifies so that identifiability may be completely understood from the eigenvalues of a particular system matrix

Statistical analysis of compressive low rank tomography with random measurements

We consider the statistical problem of 'compressive' estimation of low rank states (r«d ) with random basis measurements, where r, d are the rank and dimension of the state respectively. We investigate whether for a fixed sample size N, the estimation error associated with a 'compressive' measurement setup is 'close' to that of the setting where a large number of bases are measured. We generalise and extend previous results, and show that the mean square error (MSE) associated with the Frobenius norm attains the optimal rate rd/N with only O(rlogd) random basis measurements for all states. An important tool in the analysis is the concentration of the Fisher information matrix (FIM). We demonstrate that although a concentration of the MSE follows from a concentration of the FIM for most states, the FIM fails to concentrate for states with eigenvalues close to zero. We analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the MSE about its optimal despite a lack of concentration of the FIM for states close to the boundary of the Bloch sphere. We also consider the estimation error in terms of a different metric–the quantum infidelity. We show that a concentration in the mean infidelity (MINF) does not exist uniformly over all states, highlighting the importance of loss function choice. Specifically, we show that for states that are nearly pure, the MINF scales as 1/√N but the constant converges to zero as the number of settings is increased. This demonstrates a lack of 'compressive' recovery for nearly pure states in this metric

Quantum computing for pattern classification

It is well known that for certain tasks, quantum computing outperforms classical computing. A growing number of contributions try to use this advantage in order to improve or extend classical machine learning algorithms by methods of quantum information theory. This paper gives a brief introduction into quantum machine learning using the example of pattern classification. We introduce a quantum pattern classification algorithm that draws on Trugenberger's proposal for measuring the Hamming distance on a quantum computer (CA Trugenberger, Phys Rev Let 87, 2001) and discuss its advantages using handwritten digit recognition as from the MNIST database.Comment: 14 pages, 3 figures, presented at the 13th Pacific Rim International Conference on Artificial Intelligenc