A quantum system may be purified, i.e., projected into a pure state, faster if one applies feedback operations during the measurement process. However existing results suggest that such an enhancement is only possible when the measurement efficiency exceeds 0.5, which is difficult to achieve experimentally. We address the task of finding the global optimal feedback control for purifying a single qubit in the presence of measurement inefficiency. We use the Bloch vector length, a more physical and practical quantity than purity, to assess the quality of the state, and employ a backward iteration algorithm to find the globally optimal strategy. Our results show that a speedup is available for quantum efficiencies well below 0.5, which opens the possibility of experimental implementation in existing systems.
Recently two papers [K. Jacobs, Phys. Rev. A {\bf 67}, 030301(R) (2003); H. M. Wiseman and J. F. Ralph, New J. Physics {\bf 8}, 90 (2006)] have derived control strategies for rapid purification of qubits, optimized with respect to various goals. In the former paper the proof of optimality was not mathematically rigorous, while the latter gave only heuristic arguments for optimality. In this paper we provide rigorous proofs of optimality in all cases, by applying simple concepts from optimal control theory, including Bellman equations and verification theorems.
Quantum control is an essential tool for the operation of quantum technologies such as quantum computers, simulators, and sensors. Although there are sophisticated theoretical tools for developing quantum control protocols, formulating optimal protocols while incorporating experimental conditions remains a challenge. In this paper, motivated by recent advances in realization of real-time feedback control in circuit quantum electrodynamics systems, we study the effect of experimental imperfections on the optimality of qubit purification protocols. Specifically, we find that the optimal control …
It was first shown by Jacobs, in 2003, that the process of qubit state purification by continuous measurement of one observable can be enhanced, on average, by unitary feedback control. Here, we quantify this by the reduction in any one of the family of Rényi entropies $S_α$, with $0< α< \infty$, at some terminal time, revealing the rich structure of stochastic quantum control even for this simple problem. We generalize Jacobs' original argument, which was for (the unique) impurity measure with a linear evolution map under his protocol, by replacing linearity with convexity, thereby making it …
A method for producing an upper bound for all multipartite purification protocols is devised, based on knowing the optimal protocol for purifying bipartite states. When applied to a range of noise models, both local and correlated, the optimality of certain protocols can be demonstrated for a variety of graph and valence bond states.
We consider the problem of improving noisy quantum measurements by suitable preprocessing strategies making many noisy detectors equivalent to a single ideal detector. For observables pertaining to finite-dimensional systems (e.g. qubits or spins) we consider preprocessing strategies that are reminiscent of quantum error correction procedures and allows one to perfectly measure an observable on a single quantum system for increasing number of inefficient detectors. For measurements of observables with unbounded spectrum (e.g. photon number, homodyne and heterodyne detection), the purification …
We consider qubit purification under simultaneous continuous measurement of the three non-commuting qubit operators σ_x, σ_y, σ_z. The purification dynamics is quantified by (i) the average purification rate, and (ii) the mean time of reaching given level of purity, (1-ε). Under ideal measurements (detector efficiency η=1), we show in the first case an asymptotic mean purification speed-up of 4 as compared to a standard (classical) single-detector measurement. However by the second measure --- the mean time of first passage T(ε) of the purity --- the corresponding speed-up is only 2. We explai…
We quantise the generalised Hawk-Dove Game. By restricting the strategy space available to the players, we show that every game of this type can be extended into the quantum realm to produce a Pareto optimal evolutionarily stable strategy. This equilibrium replaces the inefficient classical one when the entanglement prepared in the game exceeds a critical threshold value, which we derive analytically.
In this paper, we study the quantum-state estimation problem in the framework of optimal design of experiments. We first find the optimal designs about arbitrary qubit models for popular optimality criteria such as A-, D-, and E-optimal designs. We also give the one-parameter family of optimality criteria which includes these criteria. We then extend a classical result in the design problem, the Kiefer-Wolfowitz theorem, to a qubit system showing the D-optimal design is equivalent to a certain type of the A-optimal design. We next compare and analyze several optimal designs based on the effici…
The common use in literature of orthogonal measurements in obtaining quantum discord for two-qubit states is discussed and compared with more general measurements. We prove the optimality of orthogonal measurements for rank 2 states. While for rank 3 and 4 mixed states they are not optimal, we present strong numerical evidence showing that they give the correct quantum discord up to minimal corrections. Based on the connection, through purification with an ancilla, between discord and entanglement of formation (EoF), we give a tight upper bound for the EoF of a $2\otimes N$ mixed state of rank…
In classical control theory, tracking refers to the ability to perform measurements and feedback on a classical system in order to enforce some desired dynamics. In this paper we investigate a simple version of quantum tracking, namely, we look at how to optimally transform the state of a single qubit into a given target state, when the system can be prepared in two different ways, and the target state depends on the choice of preparation. We propose a tracking strategy that is proved to be optimal for any input and target states. Applications in the context of state discrimination, state puri…
Security of practical continuous-variable quantum key distribution is addressed and a security analysis framework, which does not rely on phase-space symmetries of signal states and correlations, is developed. In a general purification-based approach, following optimality of Gaussian collective attacks, it is suggested to find an equivalent generally mixed two-mode state shared between the trusted parties and then purifying it using Bloch-Messiah decomposition. This allows to assess security of the schemes with arbitrary parameters, which can be typically expected in experiments. It also allow…
Quantum feedback for rapid state preparation in the presence of control imperfections
Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University（昆士兰大学量子计算与通信技术中心（澳大利亚研究委员会），量子动力学中心，格里菲斯大学）
Information encoded in networks of stationary, interacting spin-1/2 particles is central for many applications ranging from quantum spintronics to quantum information processing. Without control, however, information transfer through such networks is generally inefficient. Currently available control methods to maximize the transfer fidelities and speeds mainly rely on dynamic control using time-varying fields and often assume instantaneous readout. We present an alternative approach to achieving efficient, high-fidelity transfer of excitations by shaping the energy landscape via the design of…
