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In themathematical study of logic and thephysical analysis ofquantum foundations,quantum logic is a set of rules for manipulation ofpropositions inspired by the structure ofquantum theory. The formal system takes as its starting point an observation ofGarrett Birkhoff andJohn von Neumann, that the structure of experimental tests in classical mechanics forms aBoolean 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§ Relationship to other logics.
Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopherHilary 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 theepistemological status of the rules of propositional logic. Modern philosophers reject quantum logic as a basis for reasoning, because it lacks amaterial conditional; a common alternative is the system oflinear logic, of which quantum logic is a fragment.
Mathematically, quantum logic is formulated by weakening thedistributive law for a Boolean algebra, resulting in anorthocomplemented lattice. Quantum-mechanicalobservables andstates can be defined in terms of functions on or to the lattice, giving an alternateformalism for quantum computations.
The most notable difference between quantum logic andclassical logic is the failure of thepropositionaldistributive law:[1]
where the symbolsp,q andr are propositional variables.
To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where thereduced Planck constant is 1) let[Note 1]
We might observe that:
in other words, that the state of the particle is a weightedsuperposition of momenta between 0 and +1/6 and positions between −1 and +3.
On the other hand, the propositions "p andq" and "p andr" each assert tighter restrictions on simultaneous values of position and momentum than are allowed by theuncertainty 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
In his classic 1932 treatiseMathematical Foundations of Quantum Mechanics,John von Neumann noted thatprojections on aHilbert space can be viewed as propositions about physical observables; that is, as potentialyes-or-no questions an observer might ask about the state of a physical system, questions that could be settled by some measurement.[2] Principles for manipulating these quantum propositions were then calledquantum logic by von Neumann and Birkhoff in a 1936 paper.[3]
George Mackey, in his 1963 book (also calledMathematical Foundations of Quantum Mechanics), attempted to axiomatize quantum logic as the structure of anorthocomplemented lattice, and recognized that a physical observable could bedefined in terms of quantum propositions. Although Mackey's presentation still assumed that the orthocomplemented lattice is thelattice ofclosedlinear subspaces of aseparable Hilbert space,[4]Constantin Piron, Günther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space.[5]
Inspired byHans Reichenbach's then-recent defence ofgeneral relativity, the philosopherHilary Putnam popularized Mackey's work in two papers in 1968 and 1975,[6] in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicistDavid Finkelstein.[7] Putnam hoped to develop a possible alternative tohidden variables orwavefunction collapse in the problem ofquantum measurement, butGleason's theorem presents severe difficulties for this goal.[6][8] Later, Putnam retracted his views, albeit with much less fanfare,[6] but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with theCopenhagen 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.[9] Their work proved fruitless, and now lies in poor repute.[10]
Most philosophers would agree that quantum logic is not a competitor toclassical logic. It is far from evident (albeit true[11]) that quantum logic is alogic, in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.[12][13] In particular, some modernphilosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.[14]Tim Maudlin writes that quantum "logic 'solves' the[measurement] problem by making the problem impossible to state."[15]
Quantum logic remains in use among logicians[16] and interests are expanding through the recent development ofquantum computing, which has engendered a proliferation of new logics for formal analysis of quantum protocols and algorithms (see also§ Relationship to other logics).[17] The logic may also find application in (computational) linguistics.
Quantum logic can be axiomatized as the theory of propositions modulo the following identities:[18]
("¬" is the traditional notation for "not", "∨" the notation for "or", and "∧" the notation for "and".)
Some authors restrict toorthomodular lattices, which additionally satisfy the orthomodular law:[19]
("⊤" is the traditional notation fortruth and ""⊥" the traditional notation forfalsity.)
Alternative formulations include propositions derivable via anatural deduction,[16]sequent calculus[20][21] ortableaux system.[22] Despite the relatively developedproof theory, quantum logic is not known to bedecidable.[18]
The remainder of this article assumes the reader is familiar with thespectral theory ofself-adjoint operators on a Hilbert space. However, the main ideas can be understood in thefinite-dimensional case.
TheHamiltonian formulations ofclassical mechanics have three ingredients:states,observables anddynamics. In the simplest case of a single particle moving inR3, the state space is the position–momentum spaceR6. An observable is somereal-valued functionf on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the valuef(x), that is the value off for some particular system statex, is obtained by a process of measurement off.
Thepropositions concerning a classical system are generated from basic statements of the form
through the conventional arithmetic operations andpointwise limits. It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to theBoolean algebra ofBorel subsets of the state space. They thus obey the laws ofclassicalpropositional logic (such asde Morgan's laws) with the set operations of union and intersection corresponding to theBoolean conjunctives and subset inclusion corresponding tomaterial implication.
In fact, a stronger claim is true: they must obey theinfinitary logicLω1,ω.
We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguishedorthocomplementation operation: The lattice operations ofmeet andjoin are respectively set intersection and set union. The orthocomplementation operation is set complement. Moreover, this lattice issequentially complete, in the sense that any sequence {Ei}i∈N of elements of the lattice has aleast upper bound, specifically the set-theoretic union:
In theHilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possiblyunbounded) densely definedself-adjoint operatorA on a Hilbert spaceH.A has aspectral decomposition, which is aprojection-valued measure E defined on the Borel subsets ofR. In particular, for any boundedBorel functionf onR, the following extension off to operators can be made:
In casef is the indicator function of an interval [a,b], the operatorf(A) is a self-adjoint projection onto the subspace ofgeneralized eigenvectors ofA with eigenvalue in[a,b]. That subspace can be interpreted as the quantum analogue of the classical proposition
This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey'sAxiom VII:
The spaceQ of quantum propositions is also sequentially complete: any pairwise-disjoint sequence {Vi}i of elements ofQ has a least upper bound. Here disjointness ofW1 andW2 meansW2 is a subspace ofW1⊥. The least upper bound of {Vi}i is the closed internaldirect sum.
The standard semantics of quantum logic is that quantum logic is the logic ofprojection operators in aseparableHilbert orpre-Hilbert space, where an observablep is associated with theset of quantum states for whichp (when measured) haseigenvalue 1. From there,
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 theSolèr theorem.[23] Although much of the development of quantum logic has been motivated by the standard semantics, it is not characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic.[16]
The structure ofQ immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a propositionp, the equations
have exactly one solution, namely the set-theoretic complement ofp. In the case of the lattice of projections there are infinitely many solutions to the above equations (any closed, algebraic complement ofp solves it; it need not be the orthocomplement).
More generally,propositional valuation has unusual properties in quantum logic. An orthocomplemented lattice admitting atotallattice homomorphism to {⊥,⊤} must be Boolean. A standard workaround is to study maximal partial homomorphismsq with a filtering property:
Expressions in quantum logic describe observables using a syntax that resembles classical logic. However, unlike classical logic, the distributive lawa ∧ (b ∨c) = (a ∧b) ∨ (a ∧c) fails when dealing withnoncommuting 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 byx and momentum byp, and define observables:
Now, position and momentum are Fourier transforms of each other, and theFourier transform of asquare-integrable nonzero function with acompact support isentire and hence does not have non-isolated zeroes. Therefore, there is no wave function that is bothnormalizable in momentum space and vanishes on preciselyx ≥ 0. Thus,a ∧b and similarlya ∧c are false, so (a ∧b) ∨ (a ∧c) is false. However,a ∧ (b ∨c) equalsa, which is certainly not false (there are states for which it is a viablemeasurement outcome). Moreover: if the relevant Hilbert space for the particle's dynamics only admits momenta no greater than 1, thena is true.
To understand more, letp1 andp2 be the momentum functions (Fourier transforms) for the projections of the particle wave function tox ≤ 0 andx ≥ 0 respectively. Let |pi|↾≥1 be the restriction ofpi 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 definedp differently so as to make such states possible; also,a ∧b corresponds to |p1|↾≥1=0 andp2=0). Meanwhile,a corresponds to states with |p|↾≥1 = 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.
Given aorthocomplemented latticeQ, a Mackey observable φ is acountably additive homomorphism from the orthocomplemented lattice of Borel subsets ofR toQ. In symbols, this means that for any sequence {Si}i of pairwise-disjoint Borel subsets ofR, {φ(Si)}i are pairwise-orthogonal propositions (elements ofQ) and
Equivalently, a Mackey observable is aprojection-valued measure onR.
Theorem (Spectral theorem). IfQ is the lattice of closed subspaces of HilbertH, then there is a bijective correspondence between Mackey observables and densely defined self-adjoint operators onH.
Aquantum probability measure is a function P defined onQ with values in [0,1] such that P("⊥)=0, P(⊤)=1 and if {Ei}i is a sequence of pairwise-orthogonal elements ofQ then
Every quantum probability measure on the closed subspaces of a Hilbert space is induced by adensity matrix — anonnegative operator oftrace 1. Formally,
Quantum logic embeds intolinear logic[25] and themodal logicB.[16] 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.[26][27]
The orthocomplemented lattice of any set of quantum propositions can be embedded into a Boolean algebra, which is then amenable to classical logic.[28]
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.[8] Likewise, quantum logic with the orthomodular law falsifies thededuction theorem.[29]
Quantum logic admits no reasonablematerial conditional; anyconnective that ismonotone in a certain technical sense reduces the class of propositions to aBoolean algebra.[30] Consequently, quantum logic struggles to represent the passage of time.[25] One possible workaround is the theory ofquantum filtrations developed in the late 1970s and 1980s byBelavkin.[31][32] It is known, however, that SystemBV, adeep inference fragment oflinear logic that is very close to quantum logic, can handle arbitrarydiscrete spacetimes.[33]
