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.

scholarly journals On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 539 ◽  
Author(s):  
Lu Wei
Keyword(s):  
Pure State ◽  
Entanglement Entropy ◽  
Tsallis Entropy ◽  
Von Neumann Entropy ◽  
Quadratic Entropy ◽  
Von Neumann ◽  
Special Cases ◽  
Variance Formula ◽  
Finite Sums ◽  
Random Pure State

The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.

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 THE HATSOPOULOS–GYFTOPOULOS RESOLUTION OF THE SCHRÖDINGER–PARK PARADOX ABOUT THE CONCEPT OF "STATE" IN QUANTUM STATISTICAL MECHANICS

2006 ◽  
Vol 21 (37) ◽  
pp. 2799-2811 ◽  
Author(s):  
GIAN PAOLO BERETTA
Keyword(s):  
Information Theory ◽  
Quantum Theory ◽  
Statistical Mechanics ◽  
Quantum Statistical Mechanics ◽  
Quantum Information Theory ◽  
Individual State ◽  
Fundamental Difficulty ◽  
Quantum Statistical ◽  
Theory Interpretation

A seldom recognized fundamental difficulty undermines the concept of individual "state" in the present formulations of quantum statistical mechanics (and in its quantum information theory interpretation as well). The difficulty is an unavoidable consequence of an almost forgotten corollary proved by Schrödinger in 1936 and perused by Park, Am. J. Phys.36, 211 (1968). To resolve it, we must either reject as unsound the concept of state, or else undertake a serious reformulation of quantum theory and the role of statistics. We restate the difficulty and discuss a possible resolution proposed in 1976 by Hatsopoulos and Gyftopoulos, Found. Phys.6, 15; 127; 439; 561 (1976).

scholarly journals Does spacetime inherently has von Neumann entropy?

2020 ◽  
Author(s):  
William Icefield
Keyword(s):  
Black Hole ◽  
Quantum Theory ◽  
Black Hole Entropy ◽  
Black Hole Thermodynamics ◽  
Von Neumann Entropy ◽  
Considerable Difficulty ◽  
Von Neumann ◽  
Thermodynamic Entropy

There has been considerable difficulty in equating thermodynamic entropy, suggested in classical and black hole thermodynamics, with von Neumann entropy. Successful derivations of black hole entropy from purely classical origins and recent doubts as to whether we can really equate von Neumann entropy with thermodynamic entropy open up the possibility that spacetime inherently encodes entropy. In this understanding, any quantum theory defined on some spacetime or worldsheet inherently calls for another quantum theory that explains entropy encoded by spacetime.

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 measurements

2009 ◽  
Author(s):  
Stephen Barnett
Keyword(s):  
Quantum Theory ◽  
Theory Of Measurement ◽  
Mathematical Formulation ◽  
Hermitian Operator ◽  
Physical Nature ◽  
Measurement Process ◽  
General Description ◽  
Von Neumann ◽  
Measured System ◽  
The Relationship

Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .

Entanglement for quantum walks on the line

2011 ◽  
Vol 11 (9&10) ◽  
pp. 855-866
Author(s):  
Yusuke Ide ◽  
Norio Konno ◽  
Takuya Machida
Keyword(s):  
Information Theory ◽  
Shannon Entropy ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Initial State ◽  
Von Neumann ◽  
Time Quantum ◽  
The One ◽  
Path Counting

The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.

scholarly journals Quantum thermodynamics and quantum entanglement entropies in an expanding universe

2017 ◽  
Vol 32 (15) ◽  
pp. 1750066 ◽  
Author(s):  
Mehrnoosh Farahmand ◽  
Hosein Mohammadzadeh ◽  
Hossein Mehri-Dehnavi
Keyword(s):  
Scalar Field ◽  
Quantum Entanglement ◽  
Particle Creation ◽  
Space Time ◽  
Von Neumann Entropy ◽  
Quantum Thermodynamics ◽  
Entanglement Generation ◽  
Von Neumann ◽  
Underlying Space ◽  
The Universe

We investigate an asymptotically spatially flat Robertson–Walker space–time from two different perspectives. First, using von Neumann entropy, we evaluate the entanglement generation due to the encoded information in space–time. Then, we work out the entropy of particle creation based on the quantum thermodynamics of the scalar field on the underlying space–time. We show that the general behavior of both entropies are the same. Therefore, the entanglement can be applied to the customary quantum thermodynamics of the universe. Also, using these entropies, we can recover some information about the parameters of space–time.

On the general quantum theory of measurement

1969 ◽  
Vol 65 (1) ◽  
pp. 111-122 ◽  
Author(s):  
Arthur I. Fine
Keyword(s):  
Quantum Theory ◽  
Theory Of Measurement ◽  
Quantum Measurements ◽  
Initial State ◽  
Von Neumann ◽  
General Quantum ◽  
Von Neumann Measurements ◽  
Quantum Theory Of Measurement

We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.

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