scholarly journals Dually flat structures induced from monotone metrics on a two-level quantum state space

2020 ◽  
Vol 135 (10) ◽  
Author(s):  
Akio Fujiwara
Keyword(s):  
State Space ◽  
Quantum State ◽  
Information Geometry ◽  
Central Importance ◽  
Flat Structure ◽  
Statistical Manifolds ◽  
Flat Structures ◽  
Quantum Statistical

AbstractThe notion of dually flatness is of central importance in information geometry. Nevertheless, little is known about dually flat structures on quantum statistical manifolds except that the Bogoliubov metric admits a global dually flat structure on a quantum state space $${{\mathcal {S}}}({{\mathbb {C}}}^d)$$ S ( C d ) for any $$d\ge 2$$ d ≥ 2 . In this paper, we show that every monotone metric on a two-level quantum state space $${{\mathcal {S}}}({{\mathbb {C}}}^2)$$ S ( C 2 ) admits a local dually flat structure.

scholarly journals Some Illustrations of Information Geometry in Biology and Physics

2012 ◽  
pp. 287-315 ◽  
Author(s):  
C. T. J. Dodson
Keyword(s):  
Amino Acid ◽  
State Space ◽  
Information Geometry ◽  
Probability Density Functions ◽  
Density Functions ◽  
Binary Variables ◽  
Information Theoretic ◽  
Precise Meaning ◽  
Rate Processes ◽  
Intuitive Sense

Many real processes have stochastic features which seem to be representable in some intuitive sense as `close to Poisson’, `nearly random’, `nearly uniform’ or with binary variables `nearly independent’. Each of those particular reference states, defined by an equation, is unstable in the formal sense, but it is passed through or hovered about by the observed process. Information geometry gives precise meaning for nearness and neighbourhood in a state space of processes, naturally quantifying proximity of a process to a particular state via an information theoretic metric structure on smoothly parametrized families of probability density functions. We illustrate some aspects of the methodology through case studies: inhomogeneous statistical evolutionary rate processes for epidemics, amino acid spacings along protein chains, constrained disordering of crystals, distinguishing nearby signal distributions and testing pseudorandom number generators.

Charting the Shape of Quantum-State Space

2011 ◽  
Author(s):  
Christopher A. Fuchs ◽  
Timothy Ralph ◽  
Ping Koy Lam
Keyword(s):  
State Space ◽  
Quantum State

scholarly journals Quantum-inspired learning vector quantizers for prototype-based classification

2020 ◽  
Author(s):  
Thomas Villmann ◽  
Alexander Engelsberger ◽  
Jensun Ravichandran ◽  
Andrea Villmann ◽  
Marika Kaden
Keyword(s):  
State Space ◽  
Quantum State ◽  
Inner Product ◽  
Mathematical Framework ◽  
Feature Mapping ◽  
Quantum Bit ◽  
Vector Quantizers ◽  
New Ideas ◽  
Learning Scenarios ◽  
Bit Vector

AbstractPrototype-based models like the Generalized Learning Vector Quantization (GLVQ) belong to the class of interpretable classifiers. Moreover, quantum-inspired methods get more and more into focus in machine learning due to its potential efficient computing. Further, its interesting mathematical perspectives offer new ideas for alternative learning scenarios. This paper proposes a quantum computing-inspired variant of the prototype-based GLVQ for classification learning. We start considering kernelized GLVQ with real- and complex-valued kernels and their respective feature mapping. Thereafter, we explain how quantum space ideas could be integrated into a GLVQ using quantum bit vector space in the quantum state space $${\mathcal {H}}^{n}$$ H n and show the relations to kernelized GLVQ. In particular, we explain the related feature mapping of data into the quantum state space $${\mathcal {H}}^{n}$$ H n . A key feature for this approach is that $${\mathcal {H}}^{n}$$ H n is an Hilbert space with particular inner product properties, which finally restrict the prototype adaptations to be unitary transformations. The resulting approach is denoted as Qu-GLVQ. We provide the mathematical framework and give exemplary numerical results.

scholarly journals Quantum state space dimension as a quantum resource

2015 ◽  
Vol 13 (06) ◽  
pp. 1550039 ◽  
Author(s):  
A. Plastino ◽  
G. Bellomo ◽  
A. R. Plastino
Keyword(s):  
Quantum Information ◽  
State Space ◽  
Quantum State ◽  
Space Dimension ◽  
Quantum Systems ◽  
Quantum States ◽  
Information Tasks

We argue that the dimensionality of the space of quantum systems’ states should be considered as a legitimate resource for quantum information tasks. The assertion is supported by the fact that quantum states with discord-like capacities can be obtained from classically-correlated states in spaces of dimension large enough. We illustrate things with some simple examples that justify our claim.

scholarly journals Quantum Statistical Manifolds

Entropy ◽  
2018 ◽  
Vol 20 (6) ◽  
pp. 472 ◽  
Author(s):  
Jan Naudts
Keyword(s):  
Statistical Manifolds ◽  
Quantum Statistical

scholarly journals The D-CTC Condition is Generically Fulfilled in Classical (Non-quantum) Statistical Systems

2021 ◽  
Vol 51 (5) ◽  
Author(s):  
Jürgen Tolksdorf ◽  
Rainer Verch
Keyword(s):  
State Space ◽  
General Model ◽  
Quantum Systems ◽  
The State ◽  
Closed Timelike Curves ◽  
Original Proof ◽  
Quantum Nature ◽  
Model Independent ◽  
Quantum Statistical ◽  
Statistical Systems

AbstractThe D-CTC condition, introduced by David Deutsch as a condition to be fulfilled by analogues for processes of quantum systems in the presence of closed timelike curves, is investigated for classical statistical (non-quantum) bi-partite systems. It is shown that the D-CTC condition can generically be fulfilled in classical statistical systems, under very general, model-independent conditions. The central property used is the convexity and completeness of the state space that allows it to generalize Deutsch’s original proof for q-bit systems to more general classes of statistically described systems. The results demonstrate that the D-CTC condition, or the conditions under which it can be fulfilled, is not characteristic of, or dependent on, the quantum nature of a bi-partite system.

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