Quantum logic

In themathematical study of logicand thephysicalanalysis ofquantum foundations,quantum logicis a set of rules for manip­ulation ofpropositionsinspired by the structure ofquantum theory.The formal system takes as its starting point an obs­ervation ofGarrett BirkhoffandJohn 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 theepistemologicalstatus 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 lawfor a Boolean algebra, resulting in anortho­complemented lattice.Quantum-mechanicalobservablesandstatescan be defined in terms of functions on or to the lattice, giving an alternateformalismfor quantum computations.

Introduction

The most notable difference between quantum logic andclassical logicis the failure of thepropositionaldistributive law:

pand (qorr) = (pandq) or (pandr),

where the symbolsp,qandrare 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 constantis 1) let

p= "the particle hasmomentumin 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:

pand (qorr) =true

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

On the other hand, the propositions "pandq"and"pandr"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

(pandq) or (pandr) =false

History and modern criticism

In his classic 1932 treatiseMathematical Foundations of Quantum Mechanics,John von Neumannnoted thatprojectionson aHilbert spacecan be viewed as propositions about physical observables; that is, as potentialyes-or-no questionsan 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 calledquantum logicby von Neumann and Birkhoff in a 1936 paper.

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 bedefinedin terms of quantum propositions. Although Mackey's presentation still assumed that the ortho­complemented lattice is thelatticeofclosedlinear subspacesof aseparableHilbert space,Constantin Piron,Günther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space.

Inspired byHans Reichenbach's recent defence ofgeneral relativity,the philosopherHilary Putnampopularized 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, physicistDavid Finkelstein.Putnam hoped to develop a possible alternative tohidden variablesorwavefunction collapsein the problem ofquantum measurement,butGleason's theorempresents 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 theCopenhagen interpretationof 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 toclassical logic.It is far from evident (albeit true) 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.In particular, modernphilosophers of scienceargue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.Tim Maudlinwrites that quantum "logic 'solves' the[measurement] problemby making the problem impossible to state. "

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.... 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.
— Maudlin,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 remainsmathematical folkloreas an intuition pump forcategorification,discussions rarely mention quantum logic.

Quantum logic's best chance at revival is 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).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
• ∨ iscommutativeandassociative.
• There is a maximal element ⊤, and ⊤ =b∨¬bfor anyb.
• a∨¬(¬a∨b) =a.

( "¬" 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:

• If ⊤ = ¬(¬a∨¬b)∨¬(a∨b) thena=b.

( "⊤" is the traditional notation fortruthand "" ⊥ "the traditional notation forfalsity.)

Alternative formulations include propositions derivable via anatural deduction,sequent calculusortableauxsystem.Despite the relatively developedproof theory,quantum logic is not known to bedecidable.

Quantum logic as the logic of observables

The remainder of this article assumes the reader is familiar with thespectral theoryofself-adjoint operatorson a Hilbert space. However, the main ideas can be under­stood in thefinite-dimensionalcase.

Logic of classical mechanics

TheHamiltonianformulations ofclassical mechanicshave three ingredients:states,observablesanddynamics.In the simplest case of a single particle moving inR³,the state space is the position–momentum spaceR⁶.An observable is somereal-valued functionfon the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the valuef(x), that is the value offfor some particular system statex,is obtained by a process of measurement off.

Thepropositionsconcerning a classical system are generated from basic statements of the form

"Measurement offyields a value in the interval [a,b] for some real numbersa,b."

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 algebraofBorel subsetsof 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 conjunctivesand subset inclusion corresponding tomaterial implication.

In fact, a stronger claim is true: they must obey theinfinitary logicL_ω1,ω.

We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguishedorthocomplementationoperation: The lattice operations ofmeetandjoinare respectively set intersection and set union. The orthocomplementation operation is set complement. Moreover, this lattice issequentially complete,in the sense that any sequence {E_i}_i∈Nof elements of the lattice has aleast upper bound,specifically the set-theoretic union:$$\operatorname {LUB} (\{E_{i}\})=\bigcup _{i=1}^{\infty }E_{i}{\text{.}}$$

Propositional lattice of a quantum mechanical system

In theHilbert spaceformulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possiblyunbounded) densely definedself-adjoint operatorAon a Hilbert spaceH.Ahas aspectral decomposition,which is aprojection-valued measureE defined on the Borel subsets ofR.In particular, for any boundedBorel functionfonR,the following extension offto operators can be made:$$f(A)=\int _{\mathbb {R} }f(\lambda )\,d\operatorname {E} (\lambda ).$$

In casefis the indicator function of an interval [a,b], the operatorf(A) is a self-adjoint projection onto the subspace ofgeneralized eigenvectorsofAwith eigenvalue in[a,b].That subspace can be interpreted as the quantum analogue of the classical proposition

• Measurement ofAyields 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'sAxiom VII:

• The propositions of a quantum mechanical system correspond to the lattice of closed subspaces ofH;the negation of a propositionVis the orthogonal complementV^⊥.

The spaceQof quantum propositions is also sequentially complete: any pairwise-disjoint sequence {V_i}_iof elements ofQhas a least upper bound. Here disjointness ofW₁andW₂meansW₂is a subspace ofW₁^⊥.The least upper bound of {V_i}_iis the closed internaldirect sum.

Standard semantics

The standard semantics of quantum logic is that quantum logic is the logic ofprojection operatorsin aseparableHilbertorpre-Hilbert space,where an observablepis associated with theset of quantum statesfor whichp(when measured) haseigenvalue1. From there,

• ¬pis theorthogonal complementofp(since for those states, the probability of observingp,P(p) = 0),
• p∧qis the intersection ofpandq,and
• p∨q= ¬(¬p∧¬q) refers to states thatsuperposepandq.

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.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 ofQimmediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a propositionp,the equations

⊤ =p∨qand

⊥ =p∧q

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 ofpsolves it; it need not be the orthocomplement).

More generally,propositional valuationhas unusual properties in quantum logic. An orthocomplemented lattice admitting atotallattice homomorphismto {⊥,⊤} must be Boolean. A standard workaround is to study maximal partial homomorphismsqwith a filtering property:

ifa≤bandq(a) = ⊤, thenq(b) = ⊤.

Failure of distributivity

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 byxand momentum byp,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 theFourier transformof asquare-integrablenonzero function with acompact supportisentireand hence does not have non-isolated zeroes. Therefore, there is no wave function that is bothnormalizablein momentum space and vanishes on preciselyx≥ 0. Thus,a∧band similarlya∧care 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, thenais true.

To understand more, letp₁andp₂be the momentum functions (Fourier transforms) for the projections of the particle wave function tox≤ 0 andx≥ 0 respectively. Let |p_i|↾_≥1be the restriction ofp_ito 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 definedpdifferently so as to make such states possible; also,a∧bcorresponds to |p₁|↾_≥1=0 andp₂=0). Meanwhile,acorresponds to states with |p|↾_≥1= 0. As an operator,p=p₁+p₂,and nonzero |p₁|↾_≥1and |p₂|↾_≥1might 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 aorthocomplemented latticeQ,a Mackey observable φ is acountably additive homomorphismfrom the orthocomplemented lattice of Borel subsets ofRtoQ.In symbols, this means that for any sequence {S_i}_iof pairwise-disjoint Borel subsets ofR,{φ(S_i)}_iare pairwise-orthogonal propositions (elements ofQ) and

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

Equivalently, a Mackey observable is aprojection-valued measureonR.

Theorem(Spectral theorem). IfQis the lattice of closed subspaces of HilbertH,then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators onH.

Quantum probability measures

Main articles:Gleason's theoremandQuantum statistical mechanics

Aquantum probability measureis a function P defined onQwith values in [0,1] such that P( "⊥)=0, P(⊤)=1 and if {E_i}_iis a sequence of pairwise-orthogonal elements ofQthen

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

Every quantum probability measure on the closed subspaces of a Hilbert space is induced by adensity matrix— anonnegative operatoroftrace1. Formally,

Theorem.SupposeQis the lattice of closed subspaces of a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measurePonQthere exists a uniquetrace classoperatorSsuch that$$\operatorname {P} (E)=\operatorname {Tr} (SE)$$for any self-adjoint projectionEinQ.

Relationship to other logics

Quantum logic embeds intolinear logicand themodal logicB.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 thededuction theorem.

Quantum logic admits no reasonablematerial conditional;anyconnectivethat ismonotonein a certain technical sense reduces the class of propositions to aBoolean algebra.Consequently, quantum logic struggles to represent the passage of time.One possible workaround is the theory ofquantum filtrationsdeveloped in the late 1970s and 1980s byBelavkin.It is known, however, that SystemBV,adeep inferencefragment oflinear logicthat is very close to quantum logic, can handle arbitrarydiscrete spacetimes.

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

Sources

Historical works

Arranged 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.JSTORj.ctt1wq8zhp.1955 editionavailable at theInternet Archive.
• G. BirkhoffandJ. von Neumann,"The Logic of Quantum Mechanics,"Annals of Mathematics,series II, vol. 37, issue 4, pp. 823–843, 1936. JSTOR1968621.DOI10.2307/1968621.
• G. Mackey,Mathematical Foundations of Quantum Mechanics,W. A. Benjamin, 1963. HathiTrust2027/mdp.39015001329567.
• H. Putnam,Is Logic Empirical?,Boston Studies in the Philosophy of Science V, ed. Robert S. Cohen and Marx W. Wartofsky, 1969.
• G. KalmbachOrthomodular Logic,Z. Logik und Grundl. Math., vol. 20, 1974, pp. 395-406.
• G. KalmbachOrthomodular 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. KalmbachOrthomodular Lattices,Academic Press, London, 1983

Modern philosophical perspectives

• Guido Bacciagaluppi, "Is Logic Empirical?",inHandbook of Quantum Logic and Quantum Structures: Quantum Logic,ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.
• Tim Maudlin,"The Tale of Quantum Logic" inHilary Putnam;Cambridge University Press"Contemporary Philosophy in Focus" series, 2005. DOI:10.1017/CBO9780511614187.006ISBN9780521012546.
• de Ronde, C.; Domenech, G.; Freytes, H."Quantum Logic in Historical and Philosophical Perspective".Internet Encyclopedia of Philosophy.
• Wilce, Alexander."Quantum Logic and Probability Theory".InZalta, Edward N.(ed.).Stanford Encyclopedia of Philosophy.

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. DOI10.1017/S0960129506005299arXiv2110.01361
• 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.
• M. L. Dalla Chiaraand R. Giuntini, "Quantum Logics",inHandbook of Philosophical Logic,vol. 6, D. Gabbay and F. Guenthner (eds.), Kluwer, 2002. arXivquant-ph/0101028
• M. L. Dalla Chiara, R. Giuntini, and R. Leporini, "Quantum Computational Logics: A Survey",inTrends in Logic,vol. 21, V. F. Hendricks and J. Malinowski (eds.), Springer, 2003. arXivquant-ph/0305029
• Norman Megill,Quantum Logic ExploreratMetamath,2019.
• N. Papanikolaou, "Reasoning Formally About Quantum Systems: An Overview",ACM SIGACT News,36(3), 2005. pp. 51–66. arXivcs/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.ISBN978-3-642-70029-3.
• Quantum Logicat thenLab
• C. Piron,Foundations of Quantum Physics,W. A. Benjamin, 1976.