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Register a new userFive measurement bases determine pure quantum states on any dimension
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A long standing problem in quantum mechanics is the minimum number of observables required for the characterisation of unknown pure quantum states. The solution to this problem is specially important for the developing field of high-dimensional quantum information processing. In this work we demonstrate that any pure d-dimensional state is unambiguously reconstructed by measuring 5 observables, that is, via projective measurements onto the states of 5 orthonormal bases. Thus, in our method the total number of different measurement outcomes (5d) scales linearly with d. The state reconstruction is robust against experimental errors and requires simple post-processing, regardless of d. We experimentally demonstrate the feasibility of our scheme through the reconstruction of 8-dimensional quantum states, encoded in the momentum of single photons.
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We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation $hat{U}$ that transforms a given unknown state $|psi_taurangle$ to a known fiducial state $|frangle$. Then, after completion of the learning process, we can estimate and reproduce $|psi_taurangle$ based on the learned $hat{U}$ and $|frangle$. To realize this idea, we cast a random-based learning algorithm, called `single-shot measurement learning, in which the learning rule is based on an intuitive and reasonable criterion: the greater the number of success (or failure), the less (or more) changes are imposed. Remarkably, the learning process occurs by means of a single-shot measurement outcome. We demonstrate that our method works effectively, i.e., the learning is completed with a {em finite} number, say $N$, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large $N$, namely $simeq O(N^{-1})$ scales of average infidelity. This result is comparable to those yielded from the standard quantum tomographic method in the case where additional information is available. It highlights a non-trivial message, that is, a random-based adaptive strategy can potentially be as accurate as other standard statistical approaches.
We experimentally implement a machine-learning method for accurately identifying unknown pure quantum states. The method, called single-shot measurement learning, achieves the theoretical optimal accuracy for $epsilon = O(N^{-1})$ in state learning and reproduction, where $epsilon$ and $N$ denote the infidelity and number of state copies, without employing computationally demanding tomographic methods. This merit results from the inclusion of weighted randomness in the learning rule governing the exploration of diverse learning routes. We experimentally verify the advantages of our scheme by using a linear-optics setup to prepare and measure single-photon polarization qubits. The experimental results show highly accurate state learning and reproduction exhibiting infidelity of $O(N^{-0.983})$ down to $10^{-5}$, without estimation of the experimental parameters.
We consider a family of quantum channels characterized by the fact that certain (in general nonorthogonal) Pure states at the channel entrance are mapped to (tensor) Products of Pure states (PPP, hence pcubed) at the complementary outputs (the main output and the environment) of the channel. The pcubed construction, a reformulation of the twisted-diagonal procedure by M. M Wolf and D. Perez-Garcia, [Phys. Rev. A 75, 012303 (2007)], can be used to produce a large class of degradable quantum channels; degradable channels are of interest because their quantum capacities are easy to calculate. Several known types of degradable channels are either pcubed channels, or subchannels (employing a subspace of the channel entrance), or continuous limits of pcubed channels. The pcubed construction also yields channels which are neither degradable nor antidegradable (i.e., the complement of a degradable channel); a particular example of a qutrit channel of this type is studied in some detail. Determining whether a pcubed channel is degradable or antidegradable or neither is quite straightforward given the pure input and output states that characterize the channel. Conjugate degradable pcubed channels are always degradable.
The principle of superposition is an intriguing feature of Quantum Mechanics, which is regularly exploited at various instances. A recent work [PRL textbf{116}, 110403 (2016)] shows that the fundamentals of Quantum Mechanics restrict the superposition of two arbitrary pure states of a quantum system, even though it is possible to superpose two quantum states with partial prior knowledge. The prior knowledge imposes geometrical constraints on the choice of input pure states. We discuss an experimentally feasible protocol to superpose multiple pure states of a $d$ dimensional quantum system and carry out an explicit experimental realization to superpose two single-qubit pure states on a two-qubit NMR quantum information processor.
The project concerns the interplay among quantum mechanics, statistical mechanics and thermodynamics, in isolated quantum systems. The underlying goal is to improve our understanding of the concept of thermal equilibrium in quantum systems. First, I investigated the role played by observables and measurements in the emergence of thermal behaviour. This led to a new notion of thermal equilibrium which is specific for a given observable, rather than for the whole state of the system. The equilibrium picture that emerges is a generalization of statistical mechanics in which we are not interested in the state of the system but only in the outcome of the measurement process. I investigated how this picture relates to one of the most promising approaches for the emergence of thermal behaviour in isolated quantum systems: the Eigenstate Thermalization Hypothesis. Then, I applied the results to study some equilibrium properties of many-body localised systems. Despite the localization phenomenon, which prevents thermalization of subsystems, I was able to show that we can still use the predictions of statistical mechanics to describe the equilibrium of some observables. Moreover, the intuition developed in the process led me to propose an experimentally accessible way to unravel the interacting nature of many-body localised systems. Second, I exploited the Concentration of Measure phenomenon to study the macroscopic properties of the basis states of Loop Quantum Gravity. These techniques were previously used to explain why the thermal behaviour in quantum systems is such an ubiquitous phenomenon, at the macroscopic scale. I focused on the local properties, their thermodynamic behaviour and interplay with the semiclassical limit. This was motivated by the necessity to understand, from a quantum gravity perspective, how and why a classical horizon exhibits thermal properties.
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