Physics:Quantum logic

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In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manip­ulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting point an obs­ervation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in classical mechanics forms a Boolean algebra, but the structure of experimental tests in quantum mechanics forms a much more complicated structure.

A number of other logics have also been proposed to analyze quantum-mechanical phenomena, unfortunately also under the name of "quantum logic(s)." They are not the subject of this article. For discussion of the similarities and differences between quantum logic and some of these competitors, see .

Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher Hilary Putnam, at least at one point in his career. This thesis was an important ingredient in Putnam's 1968 paper "Is Logic Empirical?" in which he analysed the epistemological status of the rules of propositional logic. Modern philosophers reject quantum logic as a basis for reasoning, because it lacks a material conditional; a common alternative is the system of linear logic, of which quantum logic is a fragment.

Mathematically, quantum logic is formulated by weakening the distributive law for a Boolean algebra, resulting in an ortho­complemented lattice. Quantum-mechanical observables and states can be defined in terms of functions on or to the lattice, giving an alternate formalism for quantum computations.

logic in the Quantum Collection.

Introduction

The most notable difference between quantum logic and classical logic is the failure of the propositional distributive law:

p and (q or r) = (p and q) or (p and r),

where the symbols p, q and r are propositional variables.

To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let

p = "the particle has momentum in the interval [0, +1/6]"

q = "the particle is in the interval "

r = "the particle is in the interval "

We might observe that:

p and (q or r) = true

in other words, that the state of the particle is a weighted superposition of momenta between 0 and +1/6 and positions between −1 and +3.

On the other hand, the propositions "p and q" and "p and r" each assert tighter restrictions on simultaneous values of position and momentum than are allowed by the uncertainty principle (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So there are no states that can support either proposition, and

(p and q) or (p and r) = false

History and modern criticism

In his classic 1932 treatise Mathematical Foundations of Quantum Mechanics, John von Neumann noted that projections on a Hilbert space can be viewed as propositions about physical observables; that is, as potential yes-or-no questions an observer might ask about the state of a physical system, questions that could be settled by some measurement. Principles for manipulating these quantum propositions were then called quantum logic by von Neumann and Birkhoff in a 1936 paper.

George Mackey, in his 1963 book (also called Mathematical Foundations of Quantum Mechanics), attempted to axiomatize quantum logic as the structure of an ortho­complemented lattice, and recognized that a physical observable could be defined in terms of quantum propositions. Although Mackey's presentation still assumed that the ortho­complemented lattice is the lattice of closed linear subspaces of a separable Hilbert space, Constantin Piron, Günther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space.

Inspired by Hans Reichenbach's recent defence of general relativity, the philosopher Hilary Putnam popularized Mackey's work in two papers in 1968 and 1975, in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist David Finkelstein. Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal. Later, Putnam retracted his views, albeit with much less fanfare, but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one. Their work proved fruitless, and now lies in poor repute.

Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true) that quantum logic is a logic, in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses. In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems. Tim Maudlin writes that quantum "logic 'solves' the [measurement] problem by making the problem impossible to state."

|Hilary Putnam|pp. 184-185}}

Quantum logic remains in limited use among logicians as an extremely pathological counterexample (Dalla Chiara and Giuntini: "Why quantum logics? Simply because 'quantum logics are there!'"). Although the central insight to quantum logic remains mathematical folklore as an intuition pump for categorification, discussions rarely mention quantum logic.

Quantum logic's best chance at revival is through the recent development of quantum computing, which has engendered a proliferation of new logics for formal analysis of quantum protocols and algorithms (see also ). The logic may also find application in (computational) linguistics.

Algebraic structure

Quantum logic can be axiomatized as the theory of propositions modulo the following identities:

• a=¬¬a
• ∨ is commutative and associative.
• There is a maximal element ⊤, and ⊤=b∨¬b for any b.
• a∨¬(¬a∨b)=a.

("¬" is the traditional notation for "not", "∨" the notation for "or", and "∧" the notation for "and".)

Some authors restrict to orthomodular lattices, which additionally satisfy the orthomodular law:

• If ⊤=¬(¬a∨¬b)∨¬(a∨b) then a=b.

("⊤" is the traditional notation for truth and ""⊥" the traditional notation for falsity.)

Alternative formulations include propositions derivable via a natural deduction, sequent calculus or tableaux system. Despite the relatively developed proof theory, quantum logic is not known to be decidable.

Quantum logic as the logic of observables

The remainder of this article assumes the reader is familiar with the spectral theory of self-adjoint operators on a Hilbert space. However, the main ideas can be under­stood in the finite-dimensional case.

The logic of classical mechanics

The Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest case of a single particle moving in R³, the state space is the position–momentum space R⁶. An observable is some real-valued function f on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the value f(x), that is the value of f for some particular system state x, is obtained by a process of measurement of f.

The propositions concerning a classical system are generated from basic statements of the form

"Measurement of f yields a value in the interval [a, b] for some real numbers a, b."

through the conventional arithmetic operations and pointwise limits. It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to the Boolean algebra of Borel subsets of the state space. They thus obey the laws of classical propositional logic (such as de Morgan's laws) with the set operations of union and intersection corresponding to the Boolean conjunctives and subset inclusion corresponding to material implication.

In fact, a stronger claim is true: they must obey the infinitary logic L_ω1,ω.

We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguished orthocomplementation operation: The lattice operations of meet and join are respectively set intersection and set union. The orthocomplementation operation is set complement. Moreover, this lattice is sequentially complete, in the sense that any sequence {E_i}_i∈N of elements of the lattice has a least upper bound, specifically the set-theoretic union: $$\mathrm{LUB}(\{E_i\})=\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\text{.}$$

The propositional lattice of a quantum mechanical system

In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R. In particular, for any bounded Borel function f on R, the following extension of f to operators can be made: $$f(A)=\underset{\mathbb{ℝ}}{\int }f(\lambda )d\mathrm{E}(\lambda ).$$

In case f is the indicator function of an interval [a, b], the operator f(A) is a self-adjoint projection onto the subspace of generalized eigenvectors of A with eigenvalue in . That subspace can be interpreted as the quantum analogue of the classical proposition

• Measurement of A yields a value in the interval [a, b].

This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's Axiom VII:

• The propositions of a quantum mechanical system correspond to the lattice of closed subspaces of H; the negation of a proposition V is the orthogonal complement V^⊥.

The space Q of quantum propositions is also sequentially complete: any pairwise-disjoint sequence {V_i}_i of elements of Q has a least upper bound. Here disjointness of W₁ and W₂ means W₂ is a subspace of W₁^⊥. The least upper bound of {V_i}_i is the closed internal direct sum.

Standard semantics

The standard semantics of quantum logic is that quantum logic is the logic of projection operators in a separable Hilbert or pre-Hilbert space, where an observable p is associated with the set of quantum states for which p (when measured) has eigenvalue 1. From there,

• ¬p is the orthogonal complement of p (since for those states, the probability of observing p, P(p) = 0),
• p∧q is the intersection of p and q, and
• p∨q = ¬(¬p∧¬q) refers to states that superpose p and q.

This semantics has the nice property that the pre-Hilbert space is complete (i.e., Hilbert) if and only if the propositions satisfy the orthomodular law, a result known as the Solèr theorem. Although much of the development of quantum logic has been motivated by the standard semantics, it is not the characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic.

Differences with classical logic

The structure of Q immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a proposition p, the equations

⊤=p∨q and

⊥=p∧q

have exactly one solution, namely the set-theoretic complement of p. In the case of the lattice of projections there are infinitely many solutions to the above equations (any closed, algebraic complement of p solves it; it need not be the orthocomplement).

More generally, propositional valuation has unusual properties in quantum logic. An orthocomplemented lattice admitting a total lattice homomorphism to {⊥,⊤} must be Boolean. A standard workaround is to study maximal partial homomorphisms q with a filtering property:

if a≤b and q(a)=⊤, then q(b)=⊤.

Failure of distributivity

Expressions in quantum logic describe observables using a syntax that resembles classical logic. However, unlike classical logic, the distributive law a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) fails when dealing with noncommuting observables, such as position and momentum. This occurs because measurement affects the system, and measurement of whether a disjunction holds does not measure which of the disjuncts is true.

For example, consider a simple one-dimensional particle with position denoted by x and momentum by p, and define observables:

• a — |p| ≤ 1 (in some units)
• b — x < 0
• c — x ≥ 0

Now, position and momentum are Fourier transforms of each other, and the Fourier transform of a square-integrable nonzero function with a compact support is entire and hence does not have non-isolated zeroes. Therefore, there is no wave function that is both normalizable in momentum space and vanishes on precisely x ≥ 0. Thus, a ∧ b and similarly a ∧ c are false, so (a ∧ b) ∨ (a ∧ c) is false. However, a ∧ (b ∨ c) equals a, which is certainly not false (there are states for which it is a viable measurement outcome). Moreover: if the relevant Hilbert space for the particle's dynamics only admits momenta no greater than 1, then a is true.

To understand more, let p₁ and p₂ be the momenta for the restriction of the particle wave function to x < 0 and x ≥ 0 respectively (with the wave function zero outside of the restriction). Let |p|↾_>1 be the restriction of |p| to momenta that are (in absolute value) >1.

(a ∧ b) ∨ (a ∧ c) corresponds to states with |p₁|↾_>1 = |p₂|↾_>1 = 0 (this holds even if we defined p differently so as to make such states possible; also, a ∧ b corresponds to |p₁|↾_>1=0 and p₂=0). As an operator, p=p₁+p₂, and nonzero |p₁|↾_>1 and |p₂|↾_>1 might interfere to produce zero |p|↾_>1. Such interference is key to the richness of quantum logic and quantum mechanics.

Relationship to quantum measurement

Mackey observables

Given a orthocomplemented lattice Q, a Mackey observable φ is a countably additive homomorphism from the orthocomplemented lattice of Borel subsets of R to Q. In symbols, this means that for any sequence {S_i}_i of pairwise-disjoint Borel subsets of R, {φ(S_i)}_i are pairwise-orthogonal propositions (elements of Q) and

$\phi \left(\bigcup _{i=1}^{\mathrm{\infty }}{S}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\phi (S_i).$

Equivalently, a Mackey observable is a projection-valued measure on R.

Theorem (Spectral theorem). If Q is the lattice of closed subspaces of Hilbert H, then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H.

Quantum probability measures

A quantum probability measure is a function P defined on Q with values in [0,1] such that P("⊥)=0, P(⊤)=1 and if {E_i}_i is a sequence of pairwise-orthogonal elements of Q then

$\mathrm{P}\left(\underset{i=1}{\overset{\mathrm{\infty }}{\bigvee }}{E}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\mathrm{P}(E_i).$

Every quantum probability measure on the closed subspaces of a Hilbert space is induced by a density matrix — a nonnegative operator of trace 1. Formally,

Theorem. Suppose Q is the lattice of closed subspaces of a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure P on Q there exists a unique trace class operator S such that $$\mathrm{P}(E)=\mathrm{Tr}(SE)$$ for any self-adjoint projection E in Q.

Relationship to other logics

Quantum logic embeds into linear logic and the modal logic B. Indeed, modern logics for the analysis of quantum computation often begin with quantum logic, and attempt to graft desirable features of an extension of classical logic thereonto; the results then necessarily embed quantum logic.

The orthocomplemented lattice of any set of quantum propositions can be embedded into a Boolean algebra, which is then amenable to classical logic.

Limitations

Although many treatments of quantum logic assume that the underlying lattice must be orthomodular, such logics cannot handle multiple interacting quantum systems. In an example due to Foulis and Randall, there are orthomodular propositions with finite-dimensional Hilbert models whose pairing admits no orthomodular model. Likewise, quantum logic with the orthomodular law falsifies the deduction theorem.

Quantum logic admits no reasonable material conditional; any connective that is monotone in a certain technical sense reduces the class of propositions to a Boolean algebra. Consequently, quantum logic struggles to represent the passage of time. One possible workaround is the theory of quantum filtrations developed in the late 1970s and 1980s by Belavkin. It is known, however, that System BV, a deep inference fragment of linear logic that is very close to quantum logic, can handle arbitrary discrete spacetimes.

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

9. Physics:Quantum mechanics measurements

10. Physics:Quantum state

11. Physics:Quantum system

12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

19. Physics:Quantum Measurement problem

20. Physics:Quantum Bell's theorem

21. Physics:Quantum Hidden variable theory

22. Physics:Quantum nonlocality

23. Physics:Quantum contextuality

24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

37. Physics:Quantum Runge–Lenz vector

38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (21) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Aharonov-Casher effect

77. Physics:Quantum Scattering theory

78. Physics:Quantum Scattering matrix

79. Physics:Quantum S-matrix

80. Physics:Quantum tunnelling

81. Physics:Quantum speed limit

82. Physics:Quantum revival

83. Physics:Quantum reflection

84. Physics:Quantum oscillations

85. Physics:Quantum jump

86. Physics:Quantum boomerang effect

87. Physics:Quantum chaos

7. Measurement and information (9) Back to index

88. Physics:Quantum Measurement theory

89. Physics:Quantum Measurement operators

90. Physics:Quantum Projective measurement

91. Physics:Quantum POVM

92. Physics:Quantum Weak measurement

93. Physics:Quantum Measurement collapse

94. Physics:Quantum entanglement

95. Physics:Quantum Zeno effect

96. Physics:Quantum limit

8. Quantum information and computing (15) Back to index

97. Physics:Quantum information theory

98. Physics:Quantum Qubit

99. Physics:Quantum Entanglement

100. Physics:Quantum Bell state

101. Physics:Quantum Gates and circuits

102. Physics:Quantum BB84

103. Physics:Quantum No-cloning theorem

104. Physics:Quantum Computing Algorithms in the NISQ Era

105. Physics:Quantum Noisy Qubits

106. Physics:Quantum error correction

107. Physics:Quantum Boson sampling

108. Physics:Quantum random access code

109. Physics:Quantum pseudo-telepathy

110. Physics:Quantum network

111. Physics:Quantum money

9. Quantum optics and experiments (10) Back to index

112. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

113. Physics:Quantum Stern-Gerlach experiment

114. Physics:Quantum Experimental quantum physics

115. Physics:Quantum optics

116. Physics:Quantum optics beam splitter experiments

117. Physics:Quantum Mach-Zehnder interferometer

118. Physics:Quantum Hong-Ou-Mandel effect

119. Physics:Quantum eraser experiment

120. Physics:Quantum delayed-choice quantum eraser

121. Physics:Quantum Ultra fast lasers

122. Template:Quantum optics operators

10. Open quantum systems (15) Back to index

123. Physics:Quantum Open systems

124. Physics:Quantum channel

125. Physics:Quantum Kraus operators

126. Physics:Quantum Amplitude damping

127. Physics:Quantum Phase damping

128. Physics:Quantum Depolarizing channel

129. Physics:Quantum Master equation

130. Physics:Quantum Lindblad equation

131. Physics:Quantum Decoherence

132. Physics:Quantum Dynamical decoupling

133. Physics:Quantum dissipation

134. Physics:Quantum Markov semigroup

135. Physics:Quantum Markovian dynamics

136. Physics:Quantum Non-Markovian dynamics

137. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

138. Physics:Quantum field theory (QFT) basics

139. Physics:Quantum field theory (QFT) core

140. Physics:Quantum Fields and Particles

141. Physics:Quantum Second quantization

142. Physics:Quantum Fock space

143. Physics:Quantum Harmonic Oscillator field modes

144. Physics:Quantum Creation and annihilation operators

145. Physics:Quantum vacuum fluctuations

146. Physics:Quantum Casimir effect

147. Physics:Quantum Propagators in quantum field theory

148. Physics:Quantum Feynman diagrams

149. Physics:Quantum Path integral formulation

150. Physics:Quantum Renormalization in field theory

151. Physics:Quantum Renormalization group

152. Physics:Quantum Field Theory Gauge symmetry

153. Physics:Quantum Spontaneous symmetry breaking

154. Physics:Quantum Non-Abelian gauge theory

155. Physics:Quantum Electrodynamics (QED)

156. Physics:Quantum chromodynamics (QCD)

157. Physics:Quantum Electroweak theory

158. Physics:Quantum Standard Model

159. Physics:Quantum triviality

160. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

161. Physics:Quantum Statistical mechanics

162. Physics:Quantum Partition function

163. Physics:Quantum Distribution functions

164. Physics:Quantum Liouville equation

165. Physics:Quantum Kinetic theory

166. Physics:Quantum Boltzmann equation

167. Physics:Quantum BBGKY hierarchy

168. Physics:Quantum Relaxation and thermalization

169. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (17) Back to index

170. Physics:Quantum Band structure

171. Physics:Quantum Fermi surfaces

172. Physics:Quantum Landau levels

173. Physics:Quantum fractional Hall effect

174. Physics:Quantum Semiconductor physics

175. Physics:Quantum Phonons

176. Physics:Quantum Electron-phonon interaction

177. Physics:Quantum Exchange interaction

178. Physics:Quantum Superconductivity

179. Physics:Quantum Topological phases of matter

180. Physics:Quantum anyon

181. Physics:Quantum well

182. Physics:Quantum spin liquid

183. Physics:Quantum spin Hall effect

184. Physics:Quantum phase transition

185. Physics:Quantum critical point

186. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

187. Physics:Quantum Fusion reactions and Lawson criterion

188. Physics:Quantum Plasma (fusion context)

189. Physics:Quantum Magnetic confinement fusion

190. Physics:Quantum Inertial confinement fusion

191. Physics:Quantum Plasma instabilities and turbulence

192. Physics:Quantum Tokamak core plasma

193. Physics:Quantum Tokamak edge physics and recycling asymmetries

194. Physics:Quantum Stellarator

15. Timeline (8) Back to index

195. Physics:Quantum mechanics/Timeline

196. Physics:Quantum mechanics/Timeline/Pre-quantum era

197. Physics:Quantum mechanics/Timeline/Old quantum theory

198. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

199. Physics:Quantum mechanics/Timeline/Quantum field theory era

200. Physics:Quantum mechanics/Timeline/Quantum information era

201. Physics:Quantum mechanics/Timeline/Quantum technology era

202. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

203. Physics:Quantum topology

204. Physics:Quantum battery

205. Physics:Quantum Supersymmetry

206. Physics:Quantum Black hole thermodynamics

207. Physics:Quantum Holographic principle

208. Physics:Quantum gravity

209. Physics:Quantum De Sitter invariant special relativity

210. Physics:Quantum Doubly special relativity

211. Physics:Quantum arithmetic geometry

212. Physics:Quantum unsolved problems

213. Physics:Quantum Yang-Mills mass gap

214. Physics:Quantum gravity problem

215. Physics:Quantum black hole information paradox

216. Physics:Quantum dark matter problem

217. Physics:Quantum neutrino mass problem

218. Physics:Quantum matter-antimatter asymmetry problem

Sources

Historical works

Organized chronologically

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, trans. Robert T. Beyer, ed. Nicholas A. Wheeler; Princeton University Press, 2018 (original 1932). pp. 160-164. [[Organization:JSTOR
• [[Biography:Garrett Birkhoff
• [[Biography:George Mackey
• [[Biography:Hilary Putnam
• G. Kalmbach Orthomodular Logic, Z. Logik und Grundl. Math., vol. 20, 1974, pp. 395-406.
• G. Kalmbach Orthomodular Logic as a Hilbert Type Calculus, in Current Issues in Quantum Logic, Plenum Press, New York, ed. E. Beltrametti et al., 1981, pp. 333-340
• G. Kalmbach Orthomodular Lattices, Academic Press, London, 1983

Modern philosophical perspectives

• Guido Bacciagaluppi, "Is Logic Empirical?", in Handbook of Quantum Logic and Quantum Structures: Quantum Logic, ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.
• [[Biography:Tim Maudlin

Mathematical study and computational applications

• A. Baltag and S. Smets, "LQP: The Dynamic Logic of Quantum Information", Mathematical Structures in Computer Science, vol. 16, issue 3, pp. 491-525, 2006. DOI 10.1017/S0960129506005299 arXiv 2110.01361
• A. Baltag, J. Bergfeld, K. Kishida, J. Sack, S. Smets and S. Zhong, "PLQP & Company: Decidable Logics for Quantum Algorithms", International Journal of Theoretical Physics, vol. 53, issue 10, pp. 3628-3647, 2014.
• [[Biography:Maria Luisa Dalla Chiara
• M. L. Dalla Chiara, R. Giuntini, and R. Leporini, "Quantum Computational Logics: A Survey", in Trends in Logic, vol. 21, V. F. Hendricks and J. Malinowski (eds.), Springer, 2003. arXiv quant-ph/0305029
• Norman Megill, Quantum Logic Explorer at [[Software:Metamath
• N. Papanikolaou, "Reasoning Formally About Quantum Systems: An Overview", ACM SIGACT News, 36(3), 2005. pp. 51–66. arXiv cs/0508005.

Quantum foundations

• D. Cohen, An Introduction to Hilbert Space and Quantum Logic, Springer-Verlag, 1989. Elementary and well-illustrated; suitable for advanced undergraduates.
• Günther Ludwig, Der Grundlagen der Quantenmechanik (in German), Springer, 1954. The definitive work. Released in English as:
  • Günther Ludwig, Foundations of Quantum Mechanics, vol. 1, trans. Carl A. Hein; Springer-Verlag, 1983.
  • Günther Ludwig, An Axiomatic Basis for Quantum Mechanics, vol. 1: "Derivation of Hilbert Space Structure", trans. Leo F. Boron, ed. Karl Just; Springer, 1985. DOI: 10.1007/978-3-642-70029-3. ISBN 978-3-642-70029-3.
• Quantum Logic
• C. Piron, Foundations of Quantum Physics, W. A. Benjamin, 1976.

Source attribution: Quantum logic