Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Springer proceedings in mathematics & statistics

Generalized Bases with a Resolution of the Identity: A Cooperative Game Theory Approach

Apostol Vourdas

Published Jan 1, 2018• 0 citations
Publisher
DOI
10.1007/978-981-13-2179-5_10

Related papers

On the mathematical foundations of mutually unbiased bases

Koen Thas

Quantum Information Processing•Dec 21, 2017•3 citations

Renormalization of total sets of states into generalized bases with a resolution of the identity

Apostol Vourdas

Abstract A total set of states for which we have no resolution of the identity (a ‘pre-basis’), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a ‘generalized basis’, which is practically useful. The dresssing mechanism is inspired by Shapley’s methodology in cooperative game theory, and it uses Möbius transforms. There is non-independence and redundancy in these generalized bases, which is quantified with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coefficients. Also in a physical system with a ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.

arXiv:1708.05937•Journal of Physics A Mathematical and Theoretical•Jun 23, 2017•1 citation

A Cryptographic Solution to a Game Theoretic Problem

Yevgeniy Dodis, Shai Halevi, Tal Rabin

Lecture notes in computer science•Jan 1, 2000•85 citations

Random projectors with continuous resolutions of the identity in a finite-dimensional Hilbert space

Apostol Vourdas

Abstract Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley’s methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the ‘probabilistic quadrant’ , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables.

arXiv:1912.06421•Journal of Physics A Mathematical and Theoretical•Sep 27, 2019•0 citations

Classical Game Theory

Ramón Alonso‐Sanz

Emergence, complexity and computation•Jan 1, 2019•0 citations

Secret-Key Generation in Many-to-One Networks: An Integrated Game-Theoretic and Information-Theoretic Approach

Rémi A. Chou, Aylin Yener

This paper considers secret-key generation between several agents and a base station that observe independent and identically distributed realizations of correlated random variables. Each agent wishes to generate the longest possible individual key with the base station by means of public communication. All keys must be jointly kept secret from all external entities. In this many-to-one secret-key generation setting, it can be shown that the agents can take advantage of a collective protocol to increase the sum rate of their generated keys. However, when each agent is only interested in maximizing its own secret-key rate, agents may be unwilling to participate in a collective protocol. Furthermore, when such a collective protocol is employed, how to fairly allocate individual key rates arises as a valid issue. This paper studies the tension between cooperation and self-interest with a game-theoretic treatment. This paper establishes that cooperation is in the best interest of all individualistic agents and that there exist individual secret-key rate allocations that incentivize the agents to follow the protocol. In addition, an explicit coding scheme that achieves such allocations is proposed.

arXiv:1710.04643•IEEE Transactions on Information Theory•Feb 27, 2019•13 citations
Abstract A total set of states for which we have no resolution of the identity (a ‘pre-basis’), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a ‘generalized basis’, which is practically useful. The dresssing mechanism is inspired by Shapley’s methodology in cooperative game theory, and it uses Möbius transforms. There is non-independence and redundancy in these generalized bases, which is quantified with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coefficients. Also in a physical system with a ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.
Abstract Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley’s methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the ‘probabilistic quadrant’ , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables.
This paper considers secret-key generation between several agents and a base station that observe independent and identically distributed realizations of correlated random variables. Each agent wishes to generate the longest possible individual key with the base station by means of public communication. All keys must be jointly kept secret from all external entities. In this many-to-one secret-key generation setting, it can be shown that the agents can take advantage of a collective protocol to increase the sum rate of their generated keys. However, when each agent is only interested in maximizing its own secret-key rate, agents may be unwilling to participate in a collective protocol. Furthermore, when such a collective protocol is employed, how to fairly allocate individual key rates arises as a valid issue. This paper studies the tension between cooperation and self-interest with a game-theoretic treatment. This paper establishes that cooperation is in the best interest of all individualistic agents and that there exist individual secret-key rate allocations that incentivize the agents to follow the protocol. In addition, an explicit coding scheme that achieves such allocations is proposed.