scholarly journals Distillation of secret key and entanglement from quantum states

2005 ◽  
Vol 461 (2053) ◽  
pp. 207-235 ◽  
Author(s):  
Igor Devetak ◽  
Andreas Winter
Keyword(s):  
Quantum Correlations ◽  
Quantum States ◽  
Von Neumann Entropy ◽  
Secret Key ◽  
Classical Communication ◽  
Von Neumann ◽  
Information Theoretic ◽  
Einstein Podolsky Rosen ◽  
Classical Quantum ◽  
The Difference

We study and solve the problem of distilling a secret key from quantum states representing correlation between two parties (Alice and Bob) and an eavesdropper (Eve) via one–way public discussion: we prove a coding theorem to achieve the ‘wire–tapper’ bound, the difference of the mutual information Alice–Bob and that of Alice–Eve, for so–called classical–quantum–quantum–correlations, via one–way public communication. This result yields information–theoretic formulae for the distillable secret key, giving ‘ultimate’ key rate bounds if Eve is assumed to possess a purification of Alice and Bob's joint state. Specializing our protocol somewhat and making it coherent leads us to a protocol of entanglement distillation via one–way LOCC (local operations and classical communication) which is asymptotically optimal: in fact we prove the so–called ‘hashing inequality’, which says that the coherent information (i.e. the negative conditional von Neumann entropy) is an achievable Einstein–Podolsky–Rosen rate. This result is known to imply a whole set of distillation and capacity formulae, which we briefly review.

scholarly journals A field theory study of entanglement wedge cross section: odd entropy

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Ali Mollabashi ◽  
Kotaro Tamaoka
Keyword(s):  
Cross Section ◽  
Entanglement Entropy ◽  
Quantum Correlations ◽  
Von Neumann Entropy ◽  
Mixed States ◽  
Scale Invariant ◽  
Gaussian States ◽  
Von Neumann ◽  
Free Scalar ◽  
The Difference

Abstract We study odd entanglement entropy (odd entropy in short), a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories. Our study is restricted to Gaussian states of scale-invariant theories as well as their finite temperature generalizations, for which we show that odd entropy is a well-defined measure for mixed states. Motivated from holographic results, the difference between odd and von Neumann entropy is also studied. In particular, we show that large amounts of quantum correlations ensure the odd entropy to be larger than von Neumann entropy, which is qualitatively consistent with the holographic CFT. In general cases, we also find that this difference is not even a monotonic function with respect to size of (and distance between) subsystems.

scholarly journals Purity-Based Continuity Bounds for von Neumann Entropy

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Junaid ur Rehman ◽  
Hyundong Shin
Keyword(s):  
Quantum States ◽  
Von Neumann Entropy ◽  
Trace Distance ◽  
Von Neumann ◽  
Information Theoretic

Abstract We propose continuity bounds for the von Neumann entropy of qubits whose difference in purity is bounded. Considering the purity difference of two qubits to capture the notion of distance between them results into bounds which are demonstrably tighter than the trace distance-based existing continuity bounds of quantum states. Continuity bounds can be utilized in bounding the information-theoretic quantities which are generally difficult to compute.

THE UNREASONABLE EFFECTIVENESS OF EXPONENTIALLY SUPPRESSED CORRECTIONS IN PRESERVING INFORMATION

2013 ◽  
Vol 22 (12) ◽  
pp. 1342030 ◽  
Author(s):  
KYRIAKOS PAPADODIMAS ◽  
SUVRAT RAJU
Keyword(s):  
Hilbert Space ◽  
Black Hole ◽  
Quantum Mechanics ◽  
Nonperturbative Effects ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Information Theoretic ◽  
Black Hole Evaporation ◽  
Strong Subadditivity ◽  
Unreasonable Effectiveness

We point out that nonperturbative effects in quantum gravity are sufficient to reconcile the process of black hole evaporation with quantum mechanics. In ordinary processes, these corrections are unimportant because they are suppressed by e-S. However, they gain relevance in information-theoretic considerations because their small size is offset by the corresponding largeness of the Hilbert space. In particular, we show how such corrections can cause the von Neumann entropy of the emitted Hawking quanta to decrease after the Page time, without modifying the thermal nature of each emitted quantum. Second, we show that exponentially suppressed commutators between operators inside and outside the black hole are sufficient to resolve paradoxes associated with the strong subadditivity of entropy without any dramatic modifications of the geometry near the horizon.

scholarly journals Convergence Conditions of Mixed States and their von Neumann Entropy in Continuous Quantum Measurements

2014 ◽  
Vol 21 (04) ◽  
pp. 1450010
Author(s):  
Toru Fuda
Keyword(s):  
Quantum Measurements ◽  
Von Neumann Entropy ◽  
Projection Operators ◽  
Mixed States ◽  
Convergence Conditions ◽  
Von Neumann ◽  
Zeno Effect ◽  
The Difference ◽  
Arbitrary Precision ◽  
Continuous Quantum Measurements

By carrying out appropriate continuous quantum measurements with a family of projection operators, a unitary channel can be approximated in an arbitrary precision in the trace norm sense. In particular, the quantum Zeno effect is described as an application. In the case of an infinite dimension, although the von Neumann entropy is not necessarily continuous, the difference of the entropies between the states, as mentioned above, can be made arbitrarily small under some conditions.

Dynamics of Interacting Classical and Quantum Systems

2011 ◽  
Vol 18 (04) ◽  
pp. 339-351 ◽  
Author(s):  
Dariusz Chruściński ◽  
Andrzej Kossakowski ◽  
Giuseppe Marmo ◽  
E. C. G. Sudarshan
Keyword(s):  
Characteristic Feature ◽  
Quantum Dynamics ◽  
Quantum Discord ◽  
Main Idea ◽  
Quantum Correlations ◽  
Quantum Systems ◽  
Quantum States ◽  
Algebra Of Observables ◽  
Classical Quantum ◽  
True Quantum

We analyze the dynamics of coupled classical and quantum systems. The main idea is to treat both systems as true quantum ones and impose a family of superselection rules which imply that the corresponding algebra of observables of one subsystem is commutative and hence may be treated as a classical one. Equivalently, one may impose a special symmetry which restricts the algebra of observables to the 'classical' subalgebra. The characteristic feature of classical-quantum dynamics is that it leaves invariant a subspace of classical-quantum states, that is, it does not create quantum correlations as measured by the quantum discord.

QUANTUM ENTROPY AND INFORMATION IN DISCRETE ENTANGLED STATES

2001 ◽  
Vol 04 (02) ◽  
pp. 137-160 ◽  
Author(s):  
VIACHESLAV P. BELAVKIN ◽  
MASANORI OHYA
Keyword(s):  
Conditional Entropy ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Commutative Case ◽  
Maximal Abelian Subalgebra ◽  
Von Neumann ◽  
Abelian Subalgebra ◽  
Different Types ◽  
Classical Quantum ◽  
Entropic Measure

Quantum entanglements, describing truly quantum couplings, are studied and classified for discrete compound states. We show that classical-quantum correspondences such as quantum encodings can be treated as d-entanglements leading to a special class of separable compound states. The mutual information for the d-compound and for q-compound (entangled) states leads to two different types of entropies for a given quantum state. The first one is the von Neumann entropy, which is achieved as the supremum of the information over all d-entanglements, and the second one is the dimensional entropy, which is achieved at the standard entanglement, the true quantum entanglement, coinciding with a d-entanglement only in the commutative case. The q-conditional entropy and q-capacity of a quantum noiseless channel, defined as the supremum over all entanglements, is given as the logarithm of the dimensionality of the input von Neumann algebra. It can double the classical capacity, achieved as the supremum over all semiquantum couplings (d-entanglements, or encodings), which is bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra. The entropic measure for essential entanglement is introduced.

scholarly journals WIGNER ROTATIONS, BELL STATES, AND LORENTZ INVARIANCE OF ENTANGLEMENT AND VON NEUMANN ENTROPY

2004 ◽  
Vol 02 (02) ◽  
pp. 183-200 ◽  
Author(s):  
CHOPIN SOO ◽  
CYRUS C. Y. LIN
Keyword(s):  
Lorentz Invariance ◽  
Zeta Functions ◽  
Bell States ◽  
Von Neumann Entropy ◽  
Lorentz Transformations ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Einstein Podolsky Rosen ◽  
Definition Of ◽  
Reduced Density

We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein–Podolsky–Rosen–ell entangled states and their behaviour under the Lorentz group are analyzed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transformations can then be described succinctly. Reduced density matrices applicable to systems of identical particles are defined through Yang's prescription. The von Neumann entropy of each of the reduced density matrix is Lorentz invariant; and its relevance as a measure of entanglement is discussed, and illustrated with an explicit example. A regularization of the entropy in terms of generalized zeta functions is also suggested.

scholarly journals Quantum random state generation with predefined entanglement constraint

2014 ◽  
Vol 12 (05) ◽  
pp. 1450030 ◽  
Author(s):  
Anmer Daskin ◽  
Ananth Grama ◽  
Sabre Kais
Keyword(s):  
Error Correction ◽  
Weighted Graph ◽  
Quantum States ◽  
Von Neumann Entropy ◽  
Bipartite Entanglement ◽  
Von Neumann ◽  
Communication Algorithms ◽  
Equivalent Quantum ◽  
State Generation ◽  
Random Quantum States

Entanglement plays an important role in quantum communication, algorithms, and error correction. Schmidt coefficients are correlated to the eigenvalues of the reduced density matrix. These eigenvalues are used in von Neumann entropy to quantify the amount of the bipartite entanglement. In this paper, we map the Schmidt basis and the associated coefficients to quantum circuits to generate random quantum states. We also show that it is possible to adjust the entanglement between subsystems by changing the quantum gates corresponding to the Schmidt coefficients. In this manner, random quantum states with predefined bipartite entanglement amounts can be generated using random Schmidt basis. This provides a technique for generating equivalent quantum states for given weighted graph states, which are very useful in the study of entanglement, quantum computing, and quantum error correction.

scholarly journals Two Versions of the Projection Postulate: From EPR Argument to One-Way Quantum Computing and Teleportation

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Information Theory ◽  
Quantum Computing ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Von Neumann ◽  
Composite Systems ◽  
Einstein Podolsky Rosen ◽  
Epr Argument ◽  
Projection Postulate ◽  
The Difference

Nowadays it is practically forgotten that for observables with degenerate spectra the original von Neumann projection postulate differs crucially from the version of the projection postulate which was later formalized by Lüders. The latter (and not that due to von Neumann) plays the crucial role in the basic constructions of quantum information theory. We start this paper with the presentation of the notions related to the projection postulate. Then we remind that the argument of Einstein-Podolsky-Rosen against completeness of QM was based on the version of the projection postulate which is nowadays called Lüders postulate. Then we recall that all basic measurements on composite systems are represented by observables with degenerate spectra. This implies that the difference in the formulation of the projection postulate (due to von Neumann and Lüders) should be taken into account seriously in the analysis of the basic constructions of quantum information theory. This paper is a review devoted to such an analysis.

ENTANGLEMENT IN ONE-AND TWO-DIMENSIONAL ANDERSON MODELS WITH LONG-RANGE CORRELATED-DISORDER

2009 ◽  
Vol 07 (05) ◽  
pp. 959-968
Author(s):  
Z. Z. GUO ◽  
Z. G. XUAN ◽  
Y. S. ZHANG ◽  
XIAOWEI WU
Keyword(s):  
Long Range ◽  
Ground State ◽  
Two Dimensions ◽  
Von Neumann Entropy ◽  
Two Dimensional ◽  
Correlation Effects ◽  
Von Neumann ◽  
Range Correlation ◽  
The Difference ◽  
Anderson Models

The ground state entanglement in one- and two-dimensional Anderson models are studied with consideration of the long-range correlation effects and using the measures of concurrence and von Neumann entropy. We compare the effects of the long-range power-law correlation for the on-site energies on entanglement with the uncorrelated cases. We demonstrate the existence of the band structure of the entanglement. The intraband and interband jumping phenomena of the entanglement are also reported and explained to as the localization-delocalization transition of the system. We also demonstrated the difference between the results of one- and two-dimensions. Our results show that the correlation of the on-site energies increases the entanglement.

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