Priority Choice Experimental Two-Qubit Tomography: Measuring One by One All Elements of Density Matrices

Realistic Protocol to Measure Entanglement at Finite Temperatures

It is desirable to relate entanglement of many-body systems to measurable observables. In systems with a conserved charge, it was recently shown that the number entanglement entropy (NEE)-i.e., the entropy change due to an unselective subsystem charge measurement-is an entanglement monotone. Here we derive finite-temperature equilibrium relations between Rényi moments of the NEE, and multipoint charge correlations. These relations are exemplified in quantum dot systems where the desired charge correlations can be measured via a nearby quantum point contact. In quantum dots recently realizing the multichannel Kondo effect we show that the NEE has a nontrivial universal temperature dependence which is now accessible using the proposed methods.

Tomographic entanglement indicators from NMR experiments

In recent years, the performance of different entanglement indicators obtained directly from tomograms has been assessed in continuous-variable and hybrid quantum systems. In this paper, we carry out this task in the case of spin systems. We compute the entanglement indicators from actual experimental data obtained from three liquid-state NMR experiments, and compare them with standard entanglement measures calculated from the corresponding density matrices, both experimentally reconstructed and numerically computed. The gross features of entanglement dynamics and spin squeezing properties are found to be reproduced by these entanglement indicators. However, the extent to which these indicators and spin squeezing track the entanglement during time evolution of the multipartite systems in the NMR experiments is very sensitive to the precise nature and strength of interactions as well as the manner in which the full system is partitioned into subsystems. We also use the IBM quantum computer to implement equivalent circuits that capture the dynamics of the multipartite system in one of the NMR experiments and carry out a similar comparative assessment of the performance of tomographic indicators. This exercise shows that these indicators can estimate the degree of entanglement without necessitating detailed state reconstruction procedures, establishing the advantage of the tomographic approach.

Isoentangled Mutually Unbiased Bases, Symmetric Quantum Measurements, and Mixed-State Designs

Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.

Experimental Measurement of the Hilbert-Schmidt Distance between Two-Qubit States as a Means for Reducing the Complexity of Machine Learning

We report on the experimental measurement of the Hilbert-Schmidt distance between two two-qubit states by many-particle interference. We demonstrate that our three-step method for measuring distances in the Hilbert space is far less complex than reconstructing density matrices and that it can be applied in quantum-enhanced machine learning to reduce the complexity of calculating Euclidean distances between multidimensional points, which can be especially interesting for near term quantum technologies and quantum artificial intelligence research. Our results are also a novel example of applying mixed states in quantum information processing. Usually working with mixed states is undesired, but here it gives the possibility of encoding extra information as the degree of coherence between the given two dimensions of the density matrix.

A reconstruction algorithm for compressive quantum tomography using various measurement sets

Compressed sensing (CS) has been verified that it offers a significant performance improvement for large quantum systems comparing with the conventional quantum tomography approaches, because it reduces the number of measurements from O(d²) to O(rd log(d)) in particular for quantum states that are fairly pure. Yet few algorithms have been proposed for quantum state tomography using CS specifically, let alone basis analysis for various measurement sets in quantum CS. To fill this gap, in this paper an efficient and robust state reconstruction algorithm based on compressive sensing is developed. By leveraging the fixed point equation approach to avoid the matrix inverse operation, we propose a fixed-point alternating direction method algorithm for compressive quantum state estimation that can handle both normal errors and large outliers in the optimization process. In addition, properties of five practical measurement bases (including the Pauli basis) are analyzed in terms of their coherences and reconstruction performances, which provides theoretical instructions for the selection of measurement settings in the quantum state estimation. The numerical experiments show that the proposed algorithm has much less calculating time, higher reconstruction accuracy and is more robust to outlier noises than many existing state reconstruction algorithms.