Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeometric Measures of Information for Quantum State Characterization
182
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannons entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski. We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.
rate research
Read More
We study measures of quantum information when the space spanned by the set of accessible observables is not closed under products, i.e., we consider systems where an observer may be able to measure the expectation values of two operators, $langle O_1 rangle$ and $langle O_2 rangle$, but may not have access to $langle O_1 O_2 rangle$. This problem is relevant for the study of localized quantum information in gravity since the set of approximately-local operators in a region may not be closed under arbitrary products. While we cannot naturally associate a density matrix with a state in this setting, it is still possible to define a modular operator for a state, and distinguish between two states using a relative modular operator. These operators are defined on a little Hilbert space, which parameterizes small deformations of the system away from its original state, and they do not depend on the structure of the full Hilbert space of the theory. We extract a class of relative-entropy-like quantities from the spectrum of these operators that measure the distance between states, are monotonic under contractions of the set of available observables, and vanish only when the states are equal. Consequently, these distance-measures can be used to define measures of bipartite and multipartite entanglement. We describe applications of our measures to coarse-grained and fine-grained subregion dualities in AdS/CFT and provide a few sample calculations to illustrate our formalism.
Quantum mechanics can produce correlations that are stronger than classically allowed. This stronger-than-classical correlation is the fuel for quantum computing. In 1991 Schumacher forwarded a beautiful geometric approach, analogous to the well-known result of Bell, to capture non-classicality of this correlation for a singlet state. He used well-established information distance defined on an ensemble of identically-prepared states. He calculated that for certain detector settings used to measure the entangled state, the resulting geometry violated a triangle inequality -- a violation that is not possible classically. This provided a novel information-based geometric Bell inequality in terms of a covariance distance. Here we experimentally-reproduce his construction and demonstrate a definitive violation for a Bell state of two photons based on the usual spontaneous parametric down-conversion in a paired BBO crystal. The state we produced had a visibility of $V_{ad}=0.970$. We discuss generalizations to higher dimensional multipartite quantum states.
We use a novel form of quantum conditional probability to define new measures of quantum information in a dynamical context. We explore relationships between our new quantities and standard measures of quantum information such as von Neumann entropy. These quantities allow us to find new proofs of some standard results in quantum information theory, such as the concavity of von Neumann entropy and Holevos theorem. The existence of an underlying probability distribution helps to shed some light on the conceptual underpinnings of these results.
Complementary idempotent paravectors and their ordered compositions, are used to represent multivector basis elements of geometric Clifford algebra for 3D Euclidean space as the states of a geometric byte in a given frame of reference. Two layers of information, available in real numbers, are distinguished. The first layer is a continuous one. It is used to identify spatial orientations of similar geometric objects in the same computational basis. The second layer is a binary one. It is used to manipulate with 8D structure elements inside the computational basis itself. An oriented unit cube representation, rather than a matrix one, is used to visualize an inner structure of basis multivectors. Both layers of information are used to describe unitary operations -- reflections and rotations -- in Euclidian and Hilbert spaces. The results are compared with ones for quantum gates. Some consequences for quantum and classical information technologies are discussed.
An entanglement measure for a bipartite quantum system is a state functional that vanishes on separable states and that does not increase under separable (local) operations. It is well-known that for pure states, essentially all entanglement measures are equal to the v. Neumann entropy of the reduced state, but for mixed states, this uniqueness is lost. In quantum field theory, bipartite systems are associated with causally disjoint regions. There are no separable (normal) states to begin with when the regions touch each other, so one must leave a finite safety-corridor. Due to this corridor, the normal states of bipartite systems are necessarily mixed, and the v. Neumann entropy is not a good entanglement measure in the above sense. In this paper, we study various entanglement measures which vanish on separable states, do not increase under separable (local) operations, and have other desirable properties. In particular, we study the relative entanglement entropy, defined as the minimum relative entropy between the given state and an arbitrary separable state. We establish rigorous upper and lower bounds in various quantum field theoretic (QFT) models, as well as also model-independent ones. The former include free fields on static spacetime manifolds in general dimensions, or integrable models with factorizing $S$-matrix in 1+1 dimensions. The latter include bounds on ground states in general conformal QFTs, charged states (including charges with braid-group statistics) or thermal states in theories satisfying a nuclearity condition. Typically, the bounds show a divergent behavior when the systems get close to each other--sometimes of the form of a generalized area law--and decay when the systems are far apart. Our main technical tools are of operator algebraic nature.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
