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 On Entangled Information and Quantum Capacity

2001 ◽  
Vol 08 (01) ◽  
pp. 1-18 ◽  
Author(s):  
Viacheslav P. Belavkin
Keyword(s):  
Quantum Channel ◽  
Simple Algebra ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Input State ◽  
Von Neumann ◽  
Quantum Capacity ◽  
Different Types ◽  
Mixed Compound ◽  
True Quantum

The pure quantum entanglement is generalized to the case of mixed compound states to include the classical and quantum encodings as particular cases. The true quantum entanglements are characterized as transpose-CP but not CP maps. The entangled information is introduced as the relative entropy of the mutual and the input state and total information of the entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the supremum of the information over all c-entanglements, and the true quantum entropy, which is achieved at the standard entanglement. The q-capacity, defined as the supremum over all entanglements, doubles the c-capacity in the case of the simple algebra. The conditional q-entropy is positive, and q-information of a quantum channel is additive.

scholarly journals Measurement-induced randomness and state-merging

2018 ◽  
Vol 16 (02) ◽  
pp. 1850018
Author(s):  
Indranil Chakrabarty ◽  
Abhishek Deshpande ◽  
Sourav Chatterjee
Keyword(s):  
Quantum Information Processing ◽  
Classical System ◽  
Conditional Entropy ◽  
Von Neumann Entropy ◽  
Quantum Mechanical ◽  
Total System ◽  
Joint Entropy ◽  
Von Neumann ◽  
Classical Counterpart ◽  
The Cost

In this work we introduce the randomness which is truly quantum mechanical in nature arising as an act of measurement. For a composite classical system, we have the joint entropy to quantify the randomness present in the total system and that happens to be equal to the sum of the entropy of one subsystem and the conditional entropy of the other subsystem, given we know the first system. The same analogy carries over to the quantum setting by replacing the Shannon entropy by the von Neumann entropy. However, if we replace the conditional von Neumann entropy by the average conditional entropy due to measurement, we find that it is different from the joint entropy of the system. We call this difference Measurement Induced Randomness (MIR) and argue that this is unique of quantum mechanical systems and there is no classical counterpart to this. In other words, the joint von Neumann entropy gives only the total randomness that arises because of the heterogeneity of the mixture and we show that it is not the total randomness that can be generated in the composite system. We generalize this quantity for N-qubit systems and show that it reduces to quantum discord for two-qubit systems. Further, we show that it is exactly equal to the change in the cost quantum state merging that arises because of the measurement. We argue that for quantum information processing tasks like state merging, the change in the cost as a result of discarding prior information can also be viewed as a rise of randomness due to measurement.

Generalized Nonadditive Information Theory and Quantum Entanglement

2004 ◽  
Author(s):  
Sumiyoshi Abe
Keyword(s):  
Information Theory ◽  
Quantum Theory ◽  
Statistical Mechanics ◽  
Quantum Entanglement ◽  
Theory Of Measurement ◽  
Conditional Entropy ◽  
Tsallis Entropy ◽  
Von Neumann Entropy ◽  
Additive Theory ◽  
Von Neumann

Nonadditive classical information theory is developed in the axiomatic framework and then translated into quantum theory. The nonadditive conditional entropy associated with the Tsallis entropy indexed by q is given in accordance with the formalism of nonextensive statistical mechanics. The theory is applied to the problems of quantum entanglement and separability of the Werner-Popescu-type mixed state of a multipartite system, in order to examine if it has any points superior to the additive theory with the von Neumann entropy realized in the limit q → 1. It is shown that the nonadditive theory can lead to the necessary and sufficient condition for separability of the Werner-Popescu-type state, whereas the von Neumann theory can give only a much weaker condition…. Tsallis' nonextensive generalization of Boltzmann-Gibbs statistical mechanics [3, 15, 16] and its success in describing behaviors of a large class of complex systems naturally lead to the question of whether information theory can also admit an analogous generalization. If the answer is affirmative, then that will be of particular importance in connection with the problem of quantum entanglement and quantum theory of measurement [6, 8], in which necessities of a nonadditive information measure and an information content are suggested. One should also remember that there exists a conceptual similarity between a complex system and an entangled quantum system. In these systems, a "part" is indivisibly connected with the rest. An external operation on any part drastically influences the whole system, in general. Thus, the traditional reductionistic approach to an understanding of the nature of such a system may not work efficiently. In this chapter, we report a recent development in nonadditive quantum information theory based on the Tsallis entropy indexed by q [15] and its associated nonadditive conditional entropy [1]. This theory includes the ordinary additive theory with the von Neumann entropy in a special limiting case: q → To see if it has points superior to the additive theory, we apply it to the problems of separability and quantum entanglement.

Classical capacity of a noiseless quantum channel assisted by noisy entanglement

2001 ◽  
Vol 1 (3) ◽  
pp. 70-78
Author(s):  
M Horodecki ◽  
P Horodecki ◽  
R l Horodecki ◽  
D Leung ◽  
B Terha
Keyword(s):  
Mutual Information ◽  
General Formula ◽  
Quantum Channel ◽  
Classical Case ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Von Neumann ◽  
Classical Capacity ◽  
Quantum Mutual Information

We derive the general formula for the capacity of a noiseless quantum channel assisted by an arbitrary amount of noisy entanglement. In this capacity formula, the ratio of the quantum mutual information and the von Neumann entropy of the sender's share of the noisy entanglement plays the role of mutual information in the completely classical case. A consequence of our results is that bound entangled states cannot increase the capacity of a noiseless quantum channel.

scholarly journals MAXIMAL QUANTUM VIOLATION OF THE CGLMP INEQUALITY ON ITS BOTH SIDES

2008 ◽  
Vol 06 (05) ◽  
pp. 1067-1076 ◽  
Author(s):  
MING-GUANG HU ◽  
DONG-LING DENG ◽  
JING-LING CHEN
Keyword(s):  
Operator Method ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Von Neumann ◽  
Maximally Entangled States

We investigate the maximal violations for both sides of the d-dimensional CGLMP inequality by using the Bell operator method. It turns out that the maximal violations have a decelerating increase as the dimension increases and tend to a finite value at infinity. The numerical values are given out up to d = 106 for positively maximal violations and d = 2 × 105 for negatively maximal violations. Counterintuitively, the negatively maximal violations tend to be a little stronger than the positively maximal violations. Further we show the states corresponding to these maximal violations and compare them with the maximally entangled states by utilizing entangled degree defined by von Neumann entropy. It shows that their entangled degree tends to some nonmaximal value as the dimension increases.

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 SOME ENTANGLEMENT FEATURES OF HIGHLY ENTANGLED MULTIQUBIT STATES

2008 ◽  
Vol 06 (supp01) ◽  
pp. 605-611 ◽  
Author(s):  
A. BORRAS ◽  
M. CASAS ◽  
A. R. PLASTINO ◽  
A. PLASTINO
Keyword(s):  
Linear Entropy ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Marginal Density ◽  
Density Matrices ◽  
Von Neumann ◽  
Entanglement Measures

We explore some basic entanglement features of multiqubit systems that are relevant for the development of algorithms for searching highly entangled states. In particular, we compare the behaviours of multiqubit entanglement measures based (i) on the von Neumann entropy of marginal density matrices and (ii) on the linear entropy of those matrices.

Information mechanics

2017 ◽  
Author(s):  
Sandip Tiwari
Keyword(s):  
Information Flow ◽  
Irreversible Processes ◽  
State Machines ◽  
Science And Engineering ◽  
Von Neumann ◽  
Principle Of Maximum Entropy ◽  
Information Manipulation ◽  
Classical Quantum ◽  
Physical Laws ◽  
Principle Of Maximum

Information is physical, so its manipulation through devices is subject to its own mechanics: the science and engineering of behavioral description, which is intermingled with classical, quantum and statistical mechanics principles. This chapter is a unification of these principles and physical laws with their implications for nanoscale. Ideas of state machines, Church-Turing thesis and its embodiment in various state machines, probabilities, Bayesian principles and entropy in its various forms (Shannon, Boltzmann, von Neumann, algorithmic) with an eye on the principle of maximum entropy as an information manipulation tool. Notions of conservation and non-conservation are applied to example circuit forms folding in adiabatic, isothermal, reversible and irreversible processes. This brings out implications of fluctuation and transitions, the interplay of errors and stability and the energy cost of determinism. It concludes discussing networks as tools to understand information flow and decision making and with an introduction to entanglement in quantum computing.

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