It has been recently shown that for quantum systems with dimension $d\ensuremath{\ge}4$ unknown pure states can be estimated via measurements on five bases only. Here, we study by means of numerical experiments the estimation accuracy of this method as a function of the dimension and the number of independently, identically prepared copies, that is, ensemble size. We show that the accuracy of this method can be greatly improved by modifying the estimation procedure, without increasing the number of measurement outcomes. The present estimation accuracy becomes approximately $d$ times smaller than the best accuracy achievable by tomographic methods for unknown mixed states. We also study the case of pure states affected by a low level of white noise. We show that the modified version of the method is very robust; that is, it remains to a large extent unaffected by the noise and achieves an accuracy similar to the case without noise.
It has been recently shown that for quantum systems with dimension d\ensuremath≥4 unknown pure states can be estimated via measurements on five bases only. Here, we study by means of numerical experiments the estimation accuracy of this method as a function of the dimension and the number of independently, identically prepared copies, that is, ensemble size. We show that the accuracy of this method can be greatly improved by modifying the estimation procedure, without increasing the number of measurement outcomes. The present estimation accuracy becomes approximately d times smaller than the best accuracy achievable by tomographic methods for unknown mixed states. We also study the case of pure states affected by a low level of white noise. We show that the modified version of the method is very robust; that is, it remains to a large extent unaffected by the noise and achieves an accuracy similar to the case without noise.
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.
Quantum tomography has become a key tool for the assessment of quantum states, processes, and devices. This drives the search for tomographic methods that achieve greater accuracy. In the case of mixed states of a single 2-dimensional quantum system adaptive methods have been recently introduced that achieve the theoretical accuracy limit deduced by Hayashi and Gill and Massar. However, accurate estimation of higher-dimensional quantum states remains poorly understood. This is mainly due to the existence of incompatible observables, which makes multiparameter estimation difficult. Here we present an adaptive tomographic method and show through numerical simulations that, after a few iterations, it is asymptotically approaching the fundamental Gill-Massar lower bound for the estimation accuracy of pure quantum states in high dimension. The method is based on a combination of stochastic optimization on the field of the complex numbers and statistical inference, exceeds the accuracy of any mixed-state tomographic method, and can be demonstrated with current experimental capabilities. The proposed method may lead to new developments in quantum metrology.
We present a tomographic method which requires only 4d-3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme, we have experimentally reconstructed a large number of pure states of dimension d=7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, which represents an improvement with respect to other similar methods and contributes to answering the question of how many observables are needed to uniquely determine any pure state.
We describe algorithms to obtain an approximate classical description of a d-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for ℓq-norm error up to logarithmic factors. As a special case, we show that it takes Θ(d/ε) applications of the unitaries to obtain an ε-ℓ2-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten q-norm estimates of a rank-r mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm (q=1) estimate can be obtained with O(dr/ε) queries. This improves (assuming our stronger input model) the ε-dependence over the algorithm of Haah et al.\ (2017) that uses a joint measurement on O(dr/ε2) copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to 1 in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an ℓ∞-norm estimate of a normalized vector induces a (slightly worse) ℓq-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al.\ (2021) when the measurement operators do not fully overlap.
It is widely accepted that the selection of measurement bases can affect the efficiency of quantum state estimation methods, precision of estimating an unknown state can be improved significantly by simply introduce a set of symmetrical measurement bases. Here we compare the efficiencies of estimations with different numbers of measurement bases by numerical simulation and experiment in optical system. The advantages of using a complete set of symmetrical measurement bases are illustrated more clearly.
It is widely accepted that the selection of measurement bases can affect the efficiency of quantum state estimation methods, precision of estimating an unknown state can be improved significantly by simply introduce a set of symmetrical measurement bases. Here we compare the efficiencies of estimations with different numbers of measurement bases by numerical simulation and experiment in optical system. The advantages of using a complete set of symmetrical measurement bases are illustrated more clearly.