Quantum logic

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical logic with the facts concerning the measurement of complementary variables in quantum mechanics, such as position and momentum.

Quantum logic can be formulated either as a modified version of propositional logic or as a noncommutative and non-associative many-valued logic.

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. Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein. However, this idea had been around for some time and had been revived several years earlier by George Mackey's work on group representations and symmetry.

The more common view regarding quantum logic, however, is that it provides a formalism for relating observables, system preparation filters and states. In this view, the quantum logic approach resembles more closely the C*-algebraic approach to quantum mechanics. The similarities of the quantum logic formalism to a system of deductive logic may then be regarded more as a curiosity than as a fact of fundamental philosophical importance.

The need of quantum logic for describing quantum mechanics has been questioned. Alternative logical approaches to address the apparent paradoxical nature of quantum mechanics include the use of partial Boolean algebras as introduced in the formalism for the Kochen-Specker theorem and explored further in works on quantum contextuality.

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 [−1, 1]"

r = "the particle is in the interval [1, 3]"

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

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. 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. Putnam later retracted his views, albeit with much less fanfare.

Quantum logic remains in limited use among logicians as an extremely pathological counterexample (della 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, discussions rarely mention quantum logic.

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 of elements of the lattice has a least upper bound, specifically the set-theoretic union:

$$\operatorname {LUB} (\{E_{i}\})=\bigcup _{i=1}^{\infty }E_{i}{\text{.}}$$

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:

$f(A)=\int _{\mathbb {R} }f(\lambda )\,d\operatorname {E} (\lambda ).$

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 [a,b]. 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. But note that this property is not peculiar 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).

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=0 and |p₂|↾_>1 (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

$\varphi \left(\bigcup _{i=1}^{\infty }S_{i}\right)=\sum _{i=1}^{\infty }\varphi (S_{i}).$

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

Main articles: Gleason's theorem and Quantum statistical mechanics

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

$\operatorname {P} \!\left(\bigvee _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\operatorname {P} (E_{i}).$

The following highly non-trivial theorem is due to Andrew Gleason:

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

$\operatorname {P} (E)=\operatorname {Tr} (SE)$

for any self-adjoint projection E in Q.

The operator S is necessarily non-negative (that is all eigenvalues are non-negative) and of trace 1. Such an operator is often called a density operator or density matrix, after its coordinatization.

Relationship to other logics

Quantum logic embeds into linear logic and the modal logic B.

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.

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.

Criticism

Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident 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.

The philosopher of science Tim Maudlin notes that quantum logic "'solves' the [measurement] problem by making the problem impossible to state" and writes

the horse of quantum logic has been so thrashed, whipped and pummeled, and is so thoroughly deceased that...the question is not whether the horse will rise again, it is: how in the world did this horse get here in the first place? The tale of quantum logic is not the tale of a promising idea gone bad, it is rather the tale of the unrelenting pursuit of a bad idea. And this pursuit is by no means Putnam’s alone: many, many philosophers and physicists have become convinced that a change of logic (and most dramatically, the rejection of classical logic) will somehow help in understanding quantum theory, or is somehow suggested or forced on us by quantum theory. But quantum logic, even through its many incarnations and variations, both in technical form and in interpretation, has never yielded the goods.

See also

• Fuzzy logic
• HPO formalism (An approach to temporal quantum logic)
• Linear logic
• Mathematical formulation of quantum mechanics
• Multi-valued logic
• Quantum Bayesianism
• Quantum cognition
• Quantum contextuality
• Quantum field theory
• Quantum probability
• Quasi-set theory
• Solèr's theorem
• Vector logic

Further reading

• S. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, 1995.
• F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization I, II, Ann. Phys. (N.Y.), 111 (1978) pp. 61–110, 111–151.
• G. Birkhoff and J. von Neumann, "The Logic of Quantum Mechanics," Annals of Mathematics, series II, vol. 37, issue 4, pp. 823–843, 1936. JSTOR 1968621. DOI 10.2307/1968621.
• D. Cohen, An Introduction to Hilbert Space and Quantum Logic, Springer-Verlag, 1989. This is a thorough but elementary and well-illustrated introduction, suitable for advanced undergraduates.
• M.L. Dalla Chiara. R. Giuntini, G. Sergioli, "Probability in quantum computation and in quantum computational logics" . Mathematical Structures in Computer Sciences, ISSN 0960-1295, Vol.24, Issue 3, Cambridge University Press (2014).
• D. Finkelstein, Matter, Space and Logic, Boston Studies in the Philosophy of Science Vol. V, 1969
• A. Gleason, Measures on the Closed Subspaces of a Hilbert Space, Journal of Mathematics and Mechanics, 1957.
• R. Kadison, Isometries of Operator Algebras, Annals of Mathematics, Vol. 54, pp. 325–338, 1951. JSTOR 1969534.}
• G. Ludwig, Foundations of Quantum Mechanics, Springer-Verlag, 1983.
• G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (HathiTrust 2027/mdp.39015001329567; Dover paperback reprint 2004).
• Tim Maudlin, "The Tale of Quantum Logic" in Hilary Putnam; Cambridge University Press "Contemporary Philosophy in Focus" series, 2005. pp. 184-5. DOI: 10.1017/CBO9780511614187.006 ISBN 9780521012546.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955 (paperback reprint; original 1932).
• R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999. An extraordinarily lucid discussion of some logical and philosophical issues of quantum mechanics, with careful attention to the history of the subject. Also discusses consistent histories.
• N. Papanikolaou, Reasoning Formally About Quantum Systems: An Overview, ACM SIGACT News, 36(3), pp. 51–66, 2005.
• C. Piron, Foundations of Quantum Physics, W. A. Benjamin, 1976.
• H. Putnam, Is Logic Empirical?, Boston Studies in the Philosophy of Science Vol. V, 1969
• de Ronde, C.; Domenech, G.; Freytes, H. "Quantum Logic in Historical and Philosophical Perspective". Internet Encyclopedia of Philosophy.
• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.
• Wilce, Alexander. "Quantum Logic and Probability Theory". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.

External links

• Quantum Logic at the nLab
• Norman Megill, Quantum Logic Explorer at Metamath, 2019.