We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this we impose two very natural assumptions: the continuity and reversibility of dynamics, and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories interacting dynamics is impossible. This result reveals that "existence of entanglement" is the requirement with minimal logical content which singles out quantum theory from our family of theories.
Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more represe…
Quantum theory and relativity are the pillar theories on which our understanding of physics is based. Poincaré invariance is a fundamental physical principle stating that the experimental results must be the same in all inertial reference frames in Minkowski spacetime. It is a basic condition imposed on quantum theory in order to construct quantum field theories, hence, it plays a fundamental role in the standard model of particle physics too. As is well known, Minkowski spacetime follows from clear physical principles, like the relativity principle and the invariance of the speed of light. He…
Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base …
For both unitary and open qubit dynamics, we compare asymmetry monotone-based bounds on the minimal time required for an initial qubit state to evolve to a final qubit state from which it is probabilistically distinguishable with fixed minimal error probability (i.e., the minimal error distinguishability time). For the case of unitary dynamics generated by a time-independent Hamiltonian, we derive a necessary and sufficient condition on two asymmetry monotones that guarantees that an arbitrary state of a two-level quantum system or a separable state of $N$ two-level quantum systems will unitar…
The computational efficiency of quantum mechanics can be defined in terms of the qubit circuit model, which is characterized by a few simple properties: each computational gate is a reversible transformation in a connected matrix group; single wires carry quantum bits, i.e. states of a three-dimensional Bloch ball; states on two or more wires are uniquely determined by local measurement statistics and their correlations. In this paper, we ask whether other types of computation are possible if we relax one of those characteristics (and keep all others), namely, if we allow wires to be described…
The article considers a two-level open quantum system, whose evolution is governed by the Gorini--Kossakowski--Lindblad--Sudarshan master equation with Hamiltonian and dissipation superoperator depending, correspondingly, on piecewise constant coherent and incoherent controls with constrained magnitudes. Additional constraints on controls' variations are also considered. The system is analyzed using Bloch parametrization of the system's density matrix. We adapt the section method for obtaining outer parallelepipedal and pointwise estimations of reachable and controllability sets in the Bloch b…
Quantum walks are processes that model dynamics in coherent systems. Their experimental implementations proved key to unveil novel phenomena in Floquet topological insulators. Here we realize a photonic quantum walk in the presence of a synthetic gauge field, which mimics the action of an electric field on a charged particle. By tuning the energy gaps between the two quasi-energy bands, we investigate intriguing system dynamics characterized by the interplay between Bloch oscillations and Landau-Zener transitions. When both gaps at quasi-energy values 0 and $π$ are vanishingly small, the Floqu…
Entanglement Dimensionality of Continuous Variable States From Phase-Space Quasi-Probabilities
Peking University（北京大学）；TU Wien（维也纳工业大学）；Shanxi University（山西大学）；Hefei National Laboratory（合肥国家实验室）；Austrian Academy of Sciences（奥地利科学院）；Universitat de Val\'encia - Consejo Superior de Investigaciones Cient\'ificas（瓦伦西亚大学-西班牙国家科研委员会）
Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band struc…
We classify the Markov chains that can be generated on the set of quantum states by a unitarily evolving 3-dim quantum system (qutrit) that is repeatedly measured with a projective measurement (PVM) consisting of one rank-2 projection and one rank-1 projection. The dynamics of such a system can be visualized as taking place on the union of a Bloch ball and a single point, which correspond to the respective projections. The classification is given in terms of the eigenvalues of the 2x2 matrix that describes the dynamics arising on the Bloch ball, i.e., on the 2-dim subsystem. We also express th…
Fundamental limits on state preparation for an open qubit
Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；National Research University Higher School of Economics（国立高等经济大学）
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem whether a complete set of five iso-entangled 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 …
An efficient simulator for quantum systems is one of the original goals for the efforts to develop a quantum computer [1]. In recent years, synthetic dimension in photonics [2] have emerged as a potentially powerful approach for simulation that is free from the constraint of geometric dimensionality. Here we demonstrate a quantum-correlated synthetic crystal, based upon a coherently-controlled broadband quantum frequency comb produced in a chip-scale dynamically modulated lithium niobate microresonator. The time-frequency entanglement inherent with the comb modes significantly extends the dime…
Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions
Technische Universität Wien（维也纳工业大学）；BCAM - Basque Center for Applied Mathematics（巴斯克应用数学中心）；University of Gdańsk（格但斯克大学）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）；University of the Basque Country UPV/EHU（巴斯克大学）；IKERBASQUE Basque Foundation for Science（伊克尔巴斯克科学基金会）
We numerically study the real-time dynamics of a single hole created in the $t-J$ model on a square lattice. Initially, the hole spreads ballistically with a velocity proportional to the hopping matrix element. At intermediate to long times, the dimensionality as well as the spin background determine the hole dynamics. A hole created in the ground state of a two dimensional quantum antiferromagnet propagates again ballistically at long times but with a velocity proportional to the spin exchange coupling, showing the formation of a magnetic polaron. We provide an intuitive explanation of this d…
