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 § Relationship to other logics. 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.
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 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 and Philosophical Debate 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 then-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 would agree that quantum logic is not a 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, some 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." Quantum logic remains in use among logicians and interests are expanding 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 § 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 ∨ 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:
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