Entirety of Quantum Uncertainty and Its Experimental Verification

As a foundation of quantum physics,uncertainty relations describe ultimate limit for the measurement uncertainty of incompatible observables.Traditionally,uncertainty relations are formulated by mathematical bounds for a specific state.Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables,ranging over all possible input states.We find that for the pair of position and momentum operators,Heisenberg's uncertainty principle points exactly to the attainable area of the variances of position and momentum.Moreover,for finite-dimensional systems,we prove that the corresponding area is necessarily semialgebraic;in other words,this set can be represented via finite polynomial equations and inequalities,or any finite union of such sets.In particular,we give the analytical characterization of the areas of variances of(a)a pair of one-qubit observables and(b)a pair of projective observables for arbitrary dimension,and give the first experimental observation of such areas in a photonic system.

Quantum state and process tomography via adaptive measurements

We investigate quantum state tomography(QST) for pure states and quantum process tomography(QPT) for unitary channels via adaptive measurements. For a quantum system with a d-dimensional Hilbert space, we first propose an adaptive protocol where only 2d. 1 measurement outcomes are used to accomplish the QST for all pure states. This idea is then extended to study QPT for unitary channels, where an adaptive unitary process tomography(AUPT) protocol of d2+d.1measurement outcomes is constructed for any unitary channel. We experimentally implement the AUPT protocol in a 2-qubit nuclear magnetic resonance system. We examine the performance of the AUPT protocol when applied to Hadamard gate, T gate(/8 phase gate), and controlled-NOT gate,respectively, as these gates form the universal gate set for quantum information processing purpose. As a comparison, standard QPT is also implemented for each gate. Our experimental results show that the AUPT protocol that reconstructing unitary channels via adaptive measurements significantly reduce the number of experiments required by standard QPT without considerable loss of fidelity.

Joint product numerical range and geometry of reduced density matrices

The reduced density matrices of a many-body quantum system form a convex set, whose three-dimensional projection is convex in R3. The boundary of may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces： symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti＇s theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range II of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that a ruled surface emerges naturally when taking a convex hull of ∏. We show that, a ruled surface on sitting in ∏ has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of , with two boundary pieces of symmetry breaking origin separated by two gapless lines.

Physical origins of ruled surfaces on the reduced density matrices geometry

The reduced density matrices （RDMs） of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces. In this paper, we study the physical origins of these ruled surfaces for bosonic systems. The emergence of ruled surfaces was recently proposed as signatures of symmetry- breaking phase. We show that, apart from being signatures of symmetry-brealdng, ruled surfaces can also be the consequence of gapless quantum systems by demonstrating an explicit example in terms of a two-mode Ising model. Our analysis was largely simplified by the quantum de Finetti＇s theorem--in the limit of large system size, these RDMs are the convex set of all the symmetric separable states. To distinguish ruled surfaces originated from gapless systems from those caused by symmetry- breaking, we propose to use the finite size scaling method for the corresponding geometry. This method is then applied to the two-mode XY model, successfully identifying a ruled surface as the consequence of gapless systems.

Experimental cryptographic verification for near-term quantum cloud computing

An important task for quantum cloud computing is to make sure that there is a real quantum computer running,instead of classical simulation.Here we explore the applicability of a cryptographic verification scheme for verifying quantum cloud computing.We provided a theoretical extension and implemented the scheme on a 5-qubit NMR quantum processor in the laboratory and a 5-qubit and 16-qubit processors of the IBM quantum cloud.We found that the experimental results of the NMR processor can be verified by the scheme with about 1.4%error,after noise compensation by standard techniques.However,the fidelity of the IBM quantum cloud is currently too low to pass the test(about 42%error).This verification scheme shall become practical when servers claim to offer quantum-computing resources that can achieve quantum supremacy.