Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Bloch vectors formalism for a finite-dimensional quantum system
346
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In the present article, we consistently develop the main issues of the Bloch vectors formalism for an arbitrary finite-dimensional quantum system. In the frame of this formalism, qudit states and their evolution in time, qudit observables and their expectations, entanglement and nonlocality, etc. are expressed in terms of the Bloch vectors -- the vectors in the Euclidean space $mathbb{R}^{d^{2}-1}$ arising under decompositions of observables and states in different operator bases. Within this formalism, we specify for all $dgeq2$ the set of Bloch vectors of traceless qudit observables and describe its properties; also, find for the sets of the Bloch vectors of qudit states, pure and mixed, the new compact expressions in terms of the operator norms that explicitly reveal the general properties of these sets and have the unified form for all $dgeq2$. For the sets of the Bloch vectors of qudit states under the generalized Gell-Mann representation, these general properties cannot be analytically extracted from the known equivalent specifications of these sets via the system of algebraic equations. We derive the general equations describing the time evolution of the Bloch vector of a qudit state if a qudit system is isolated and if it is open and find for both cases the main properties of the Bloch vector evolution in time. For a pure bipartite state of a dimension $d_{1}times d_{2}$, we quantify its entanglement in terms of the Bloch vectors for its reduced states. The introduced general formalism is important both for the theoretical analysis of quantum system properties and for quantum applications, in particular, for optimal quantum control, since, for systems where states are described by vectors in the Euclidean space, the methods of optimal control, analytical and numerical, are well developed.
rate research
Read More
Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result $x$ rather than y, for any pair (x,y). This induces a notion of optimal transport cost for a pair of probability distributions, and we include an appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a true value is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples.
In the present report we discuss measures of classicality/quantumness of states of finite-dimensional quantum systems, which are based on a deviation of quasiprobability distributions from true statistical distributions. Particularly, the dependence of the global indicator of classicality on the assigned geometry of a quantum state space is analysed for a whole family of Wigner quasiprobability representations. General considerations are exemplified by constructing the global indicator of classicality/quantumness for the Hilbert-Schmidt, Bures and Bogoliubov-Kubo-Mori ensembles of qubits and qutrits.
We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.
We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter $r$ of a Bell state $|Phi^{+} rangle$ and the completely incoherent state, $hat{rho}_r = r |Phi^{+} rangle langle Phi^{+}| + (1-r)frac{hat{mathbb{I}}}{N^2}$: we find that for $r_0 < r < 1$, where $r_0$ is obtained as a function of the dimensionality $N$, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.
One of the fundamental physical limits on the speed of time evolution of a quantum state is known in the form of the celebrated Mandelshtam-Tamm inequality. This inequality gives an answer to the question on how fast an isolated quantum system can evolve from its initial state to an orthogonal one. In its turn, the Fleming bound is an extension of the Mandelshtam-Tamm inequality that gives an optimal speed bound for the evolution between non-orthogonal initial and final states. In the present work, we are concerned not with a single state but with a whole (possibly infinite-dimensional) subspace of the system states that are subject to the Schroedinger evolution. By using the concept of maximal angle between subspaces we derive an optimal estimate on the speed of such a subspace evolution that may be viewed as a natural generalization of the Fleming bound.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
