Quantum set theory: Equality and order of internal reals defined through different quantum conditionals
- URL: http://arxiv.org/abs/2410.18347v1
- Date: Thu, 24 Oct 2024 00:56:43 GMT
- Title: Quantum set theory: Equality and order of internal reals defined through different quantum conditionals
- Authors: Masanao Ozawa,
- Abstract summary: We show how the form of the conditional follows from an analysis of experimental concepts in quantum theory.
For each of the above conditionals, we previously introduced an interpretation of quantum set theory.
We show that the externally defined set of the internal reals is identical for each choice among the above three conditionals.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: In quantum logic, there is a well-known arbitrariness in choosing a binary operation for conditionals. We currently have three candidates: the Sasaki conditional, the contrapositive Sasaki conditional, and the relevance conditional. A fundamental problem is to show how the form of the conditional follows from an analysis of experimental concepts in quantum theory. Here, we attempt such an analysis through quantum set theory. For each of the above conditionals, we previously introduced an interpretation of quantum set theory, a quantum logical truth value assignment to every set-theoretical statement, that satisfies both the full forms of De Morgan's laws and the transfer principle. In the present study, we explore the structure of the internal reals based on the new interpretations with different conditionals. We show that the externally defined set of the internal reals is identical for each choice among the above three conditionals, and that for the logic represented by a projection lattice on a Hilbert space, the internal reals are in one-to-one correspondence with the self-adjoint operators (or equivalently, quantum observables) affiliated with the von Neumann algebra generated by the logic. Moreover, we show that the truth values for their equality are also irrespective of the choice among the conditionals. Interestingly, however, the truth values for the order relation significantly depend on the underlying conditionals, whereas the order relation holds with the full truth value if and only if their corresponding self-adjoint operators satisfy Olson's spectral order relation irrespective of the choice of the conditional. We describe the difference of the interpretations of the order relation for two internal reals in terms of the order relation, well defined in quantum mechanics, of outcomes from the successive projective measurements of the corresponding two quantum observables.
Related papers
- The composition rule for quantum systems is not the only possible one [0.0]
We argue that the composition postulate deserves to be experimentally scrutinised independently of the other features of quantum theory.
We formulate a family of operational theories that are solely distinguished from quantum theory by their system-composition rule.
arXiv Detail & Related papers (2024-11-24T19:31:13Z) - A proof that no-signalling implies microcausality in quantum field theory [0.0]
We study some logical ins between fundamental properties in (relativistic) quantum theories.<n>An operational no-signalling condition is first introduced in the context of quantum mechanics.<n>We prove that it implies both microcausality and the spin-statistics theorem.
arXiv Detail & Related papers (2023-09-14T13:47:48Z) - Incompatibility of observables, channels and instruments in information
theories [68.8204255655161]
We study the notion of compatibility for tests of an operational probabilistic theory.
We show that a theory admits of incompatible tests if and only if some information cannot be extracted without disturbance.
arXiv Detail & Related papers (2022-04-17T08:44:29Z) - Stochastic approximate state conversion for entanglement and general quantum resource theories [41.94295877935867]
An important problem in any quantum resource theory is to determine how quantum states can be converted into each other.
Very few results have been presented on the intermediate regime between probabilistic and approximate transformations.
We show that these bounds imply an upper bound on the rates for various classes of states under probabilistic transformations.
We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels.
arXiv Detail & Related papers (2021-11-24T17:29:43Z) - Quantum conditional entropy from information-theoretic principles [10.674604700001966]
We show that any quantum conditional entropy must be negative on certain entangled states and must equal -log(d) on dxd maximally entangled states.
We also prove the non-negativity of conditional entropy on separable states, and we provide a generic definition for the dual of a quantum conditional entropy.
arXiv Detail & Related papers (2021-10-28T17:44:54Z) - Quantum realism: axiomatization and quantification [77.34726150561087]
We build an axiomatization for quantum realism -- a notion of realism compatible with quantum theory.
We explicitly construct some classes of entropic quantifiers that are shown to satisfy almost all of the proposed axioms.
arXiv Detail & Related papers (2021-10-10T18:08:42Z) - Quantum indistinguishability through exchangeable desirable gambles [69.62715388742298]
Two particles are identical if all their intrinsic properties, such as spin and charge, are the same.
Quantum mechanics is seen as a normative and algorithmic theory guiding an agent to assess her subjective beliefs represented as (coherent) sets of gambles.
We show how sets of exchangeable observables (gambles) may be updated after a measurement and discuss the issue of defining entanglement for indistinguishable particle systems.
arXiv Detail & Related papers (2021-05-10T13:11:59Z) - Experimental Validation of Fully Quantum Fluctuation Theorems Using
Dynamic Bayesian Networks [48.7576911714538]
Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small systems.
We experimentally verify detailed and integral fully quantum fluctuation theorems for heat exchange using two quantum-correlated thermal spins-1/2 in a nuclear magnetic resonance setup.
arXiv Detail & Related papers (2020-12-11T12:55:17Z) - Operational Resource Theory of Imaginarity [48.7576911714538]
We show that quantum states are easier to create and manipulate if they only have real elements.
As an application, we show that imaginarity plays a crucial role for state discrimination.
arXiv Detail & Related papers (2020-07-29T14:03:38Z) - Conceptual variables, quantum theory, and statistical inference theory [0.0]
A different approach towards quantum theory is proposed in this paper.
The basis is to be conceptual variables, physical variables that may be accessible or inaccessible, i.e., it may be possible or impossible to assign numerical values to them.
arXiv Detail & Related papers (2020-05-15T08:08:55Z) - Quantum Set Theory: Transfer Principle and De Morgan's Laws [0.0]
Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers.
We introduce a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws.
arXiv Detail & Related papers (2020-02-16T21:59:20Z) - Bohr meets Rovelli: a dispositionalist account of the quantum limits of
knowledge [0.0]
I argue that the no-go theorems reflect on a formal level those practical and experimental settings that are needed to come to know the properties of physical systems.
I show that, as a consequence of a relationist and perspectival approach to quantum mechanics, the quantum state of the universe regarded as an isolated system cannot be known in principle.
arXiv Detail & Related papers (2020-01-13T22:45:09Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.