Michel R. P. Planat, Metod Saniga, Maurice R. Kibler
备注：
13 pages, 6 figures Fig. 3 improved, typos corrected; Version 4: Final Version Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
发表信息：
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2 (2006) Paper 066, 14 pages
The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15$\times$15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of $n$ copies of the Galois field GF(2), with $n$ = 2, 3 and 4.
We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class $S_1$ to the Euclidean space $\ell_2$, which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that pr…
Discrete quantum trajectories of systems under random unitary gates and projective measurements have been shown to feature transitions in the entanglement scaling that are not encoded in the density matrix. In this paper, we study the projective transverse field Ising model, a stochastic model with two noncommuting projective measurements and no unitary dynamics. We numerically demonstrate that their competition drives an entanglement transition between two distinct steady states that both exhibit area law entanglement, and introduce a classical but nonlocal model that captures the entanglemen…
Starting from a state of low quantum entanglement, local unitary time evolution increases the entanglement of a quantum many-body system. In contrast, local projective measurements disentangle degrees of freedom and decrease entanglement. We study the interplay of these competing tendencies by considering time evolution combining both unitary and projective dynamics. We begin by constructing a toy model of Bell pair dynamics which demonstrates that measurements can keep a system in a state of low (i.e. area law) entanglement, in contrast with the volume law entanglement produced by generic pur…
We investigate entanglement of purification in conformal field theory. By using Reeh-Schlieder theorem, we construct a set of the purification states for $ρ_{AB}$, where $ρ_{AB}$ is reduced density matrix for subregion $AB$ of a global state $ρ$. The set can be approximated by acting all the unitary observables,located in the complement of subregion $AB$, on the global state $ρ$, as long as the global state $ρ$ is \text{cyclic} for every local algebra, e.g., the vacuum state. Combining with the gravity explanation of unitary operations in the context of the so-called surface/state corresponden…
Non-equilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce an abrupt change in the scaling-law of the bipartite entanglement entropy, thus suggesting the existence of different non-equilibrium regimes. However, our understanding of how these regimes appear and whether they survive in the thermodynamic limit is much less established. Here we investigate these questions on a one-dimensional quadratic fermionic model: this allows us to reach system sizes…
For a projective measurement, the Born rule provides the probability for an outcome in terms of the inner product between a projector and a quantum state. If the projector represents a pure entangled state and the state for a composite system is separable, then we cannot get probability 1 for the outcome. This insight delivers a single condition for entanglement detection. By applying local unitary transformations from the Clifford group, we turn one condition into many. Furthermore, we present two equivalent schemes---one employs global and other requires local projective measurements---to te…
We calculate analytically the Rényi bipartite entanglement entropy $S_α$ of the ground state of $1+1$ dimensional conformal field theories (CFT) after performing a projective measurement in a part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region $A$ separates the two parts $B$ and $\bar{B}$, the entanglement entropy between $B$ and $\bar{B}$ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on t…
The present paper illustrates the realization of an atom-optomechanical system where an atomic ensemble is confined in a ring optomechanical cavity consisting of a fixed mirror and two movable ones. An analysis of the dynamics and the linearization of the equations allows to derive the multimode covariance matrix. Under realistic experimental conditions, we numerically simulate the steady-state bipartite and tripartite continuous variable entanglement using the logarithmic negativity, and analyze the shared entanglement in the multimode system. The introduction of the atomic medium allows to o…
We investigate a variant of the SSH model consisting of an SSH chain with an embedded Aharonov-Bohm quantum ring. The embedded ring gives rise to domain wall states whose energy levels are in the band gap. The dependence of some of the states on the ring's magnetic flux provides a mechanism to control the entanglement between the two SSH edge fermion modes. The concurrence between the edge modes depends periodically on the magnetic flux. In the deep topological zone, for appropriate values of the flux, the concurrence reaches the extreme values of $0$ and $1$, corresponding to vanishing entang…
We study the effect of projective measurements on the entanglement structure of quantum black holes. It is shown that the entanglement verification in monitored quantum circuits, recently discussed in condensed matter physics, is equivalent to the information recovery from a black hole with projective measurements. This correspondence provides useful predictions about non-perturbative effects on quantum gravity and some insights on the black hole interior as well as the final state proposal.
Dual-unitary circuits have emerged as a minimal model for chaotic quantum many-body dynamics in which the dynamics of correlations and entanglement remains tractable. Simultaneously, there has been intense interest in the effect of measurements on the dynamics of quantum information in many-body systems. In this work we introduce a class of models combining dual-unitary circuits with particular projective measurements that allow the exact computation of dynamical correlations of local observables, entanglement growth, and steady-state entanglement. We identify a symmetry preventing a measureme…
We study the entanglement behavior of a random unitary circuit punctuated by projective measurements at the measurement-driven phase transition in one spatial dimension. We numerically study the logarithmic entanglement negativity of two disjoint intervals and find that it scales as a power of the cross-ratio. We investigate two systems: (1) Clifford circuits with projective measurements, and (2) Haar random local unitary circuit with projective measurements. Remarkably, we identify a power-law behavior of entanglement negativity at the critical point. Previous results of entanglement entropy …
