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 the 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 nonassociative manyvalued (MV) logic.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}
Quantum logic has some properties that clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:^{[6]}

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 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]"
(using some system of units where the reduced Planck's constant is 1) then we might observe that:

p and (q or r) = true
in other words, that the particle's momentum is between 0 and +1/6, and its position is between −1 and +3. On the other hand, the propositions "p and q" and "p and r" are both false, since they assert tighter restrictions on simultaneous values of position and momentum than is allowed by the uncertainty principle (they have combined uncertainty 1/3 < 1/2). So,

(p and q) or (p and r) = false
Thus the distributive law fails.
Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher 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. A more modern approach to the structure of quantum logic is to assume that it is a diagram – in the sense of category theory – of classical logics (see David Edwards).
Introduction
In his classic treatise Mathematical Foundations of Quantum Mechanics, John von Neumann noted that projections on a Hilbert space can be viewed as propositions about physical observables. The set of principles for manipulating these quantum propositions was called quantum logic by von Neumann and Birkhoff. In his book (also called Mathematical Foundations of Quantum Mechanics) G. Mackey attempted to provide a set of axioms for this propositional system as an orthocomplemented lattice. Mackey viewed elements of this set as potential yes or no questions an observer might ask about the state of a physical system, questions that would be settled by some measurement. Moreover Mackey defined a physical observable in terms of these basic questions. Mackey's axiom system is somewhat unsatisfactory though, since it assumes that the partially ordered set is actually given as the orthocomplemented closed subspace lattice of a separable Hilbert space. Piron, Ludwig and others have attempted to give axiomatizations that do not require such explicit relations to the lattice of subspaces.
The axioms are most commonly stated as algebraic equations concerning the poset and its operations; one set of axioms (taken from ^{[7]}) is as follows:

a = a^{\perp \perp}

\cup is commutative and associative.

There is a maximal element 1, and 1 = b \cup b^\perp for any b.

a \cup (a^\perp \cup b)^\perp = a .

The orthomodular law: If 1 = (a^\perp \cup b^\perp)^\perp \cup (a \cup b)^\perp then a = b .
Alternative formulations include Hilbertstyle propositional axioms,^{[8]} sequent calculi,^{[9]}^{[10]} and tableaux systems.^{[11]}
The remainder of this article assumes the reader is familiar with the spectral theory of selfadjoint operators on a Hilbert space. However, the main ideas can be understood using the finitedimensional spectral theorem.
Projections as propositions
The socalled Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest case of a single particle moving in R^{3}, the state space is the positionmomentum space R^{6}. We will merely note here that an observable is some realvalued 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.
It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to that of some Borel sets. The set of propositions is ordered by the natural ordering of sets and has a complementation operation. In terms of observables, the complement of the proposition {f ≥ a} is {f < a}.
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 settheoretic union:


\operatorname{LUB}(\{E_i\}) = \bigcup_{i=1}^\infty E_i.
In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined selfadjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projectionvalued measure E defined on the Borel subsets of R. In particular, for any bounded Borel function f, the following equation holds:

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 selfadjoint projection, and can be interpreted as the quantum analogue of the classical proposition

Measurement of A yields a value in the interval [a, b].
The propositional lattice of a quantum mechanical system
This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics. This is essentially Mackey's Axiom VII:

The orthocomplemented lattice Q of propositions of a quantum mechanical system is the lattice of closed subspaces of a complex Hilbert space H where orthocomplementation of V is the orthogonal complement V^{⊥}.
Q is also sequentially complete: any pairwise disjoint sequence{V_{i}}_{i} of elements of Q has a least upper bound. Here disjointness of W_{1} and W_{2} means W_{2} is a subspace of W_{1}^{⊥}. The least upper bound of {V_{i}}_{i} is the closed internal direct sum.
Henceforth we identify elements of Q with selfadjoint projections on the Hilbert space H.
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

I = p \vee q

0 = p\wedge q
have exactly one solution, namely the settheoretic complement of p. In these equations I refers to the atomic proposition that is identically true and 0 the atomic proposition that is identically false. In the case of the lattice of projections there are infinitely many solutions to the above equations.
Having made these preliminary remarks, we turn everything around and attempt to define observables within the projection lattice framework and using this definition establish the correspondence between selfadjoint operators and observables: A Mackey observable is a countably additive homomorphism from the orthocomplemented lattice of the Borel subsets of R to Q. To say the mapping φ is a countably additive homomorphism means that for any sequence {S_{i}}_{i} of pairwise disjoint Borel subsets of R, {φ(S_{i})}_{i} are pairwise orthogonal projections and

\varphi\left(\bigcup_{i=1}^\infty S_i\right) = \sum_{i=1}^\infty \varphi(S_i).
Theorem. There is a bijective correspondence between Mackey observables and densely defined selfadjoint operators on H.
This is the content of the spectral theorem as stated in terms of spectral measures.
Statistical structure
Imagine a forensics lab that has some apparatus to measure the speed of a bullet fired from a gun. Under carefully controlled conditions of temperature, humidity, pressure and so on the same gun is fired repeatedly and speed measurements taken. This produces some distribution of speeds. Though we will not get exactly the same value for each individual measurement, for each cluster of measurements, we would expect the experiment to lead to the same distribution of speeds. In particular, we can expect to assign probability distributions to propositions such as {a ≤ speed ≤ b}. This leads naturally to propose that under controlled conditions of preparation, the measurement of a classical system can be described by a probability measure on the state space. This same statistical structure is also present in quantum mechanics.
A quantum probability measure is a function P defined on Q with values in [0,1] such that P(0)=0, P(I)=1 and if {E_{i}}_{i} is a sequence of pairwise orthogonal elements of Q then

\operatorname{P}\!\left(\sum_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i).
The following highly nontrivial theorem is due to Andrew Gleason:
Theorem. Suppose H is a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure on Q there exists a unique trace class operator S such that

\operatorname{P}(E) = \operatorname{Tr}(S E)
for any selfadjoint projection E.
The operator S is necessarily nonnegative (that is all eigenvalues are nonnegative) and of trace 1. Such an operator is often called a density operator.
Physicists commonly regard a density operator as being represented by a (possibly infinite) density matrix relative to some orthonormal basis.
For more information on statistics of quantum systems, see quantum statistical mechanics.
Automorphisms
An automorphism of Q is a bijective mapping α:Q → Q that preserves the orthocomplemented structure of Q, that is

\alpha\!\left(\sum_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \alpha(E_i)
for any sequence {E_{i}}_{i} of pairwise orthogonal selfadjoint projections. Note that this property implies monotonicity of α. If P is a quantum probability measure on Q, then E → α(E) is also a quantum probability measure on Q. By the Gleason theorem characterizing quantum probability measures quoted above, any automorphism α induces a mapping α* on the density operators by the following formula:

\operatorname{Tr}(\alpha^*(S) E) = \operatorname{Tr}(S \alpha(E)).
The mapping α* is bijective and preserves convex combinations of density operators. This means

\alpha^*(r_1 S_1 + r_2 S_2) = r_1\alpha^*(S_1) + r_2 \alpha^*(S_2) \quad
whenever 1 = r_{1} + r_{2} and r_{1}, r_{2} are nonnegative real numbers. Now we use a theorem of Richard V. Kadison:
Theorem. Suppose β is a bijective map from density operators to density operators that is convexity preserving. Then there is an operator U on the Hilbert space that is either linear or conjugatelinear, preserves the inner product and is such that

\beta(S) = U S U^* \,
for every density operator S. In the first case we say U is unitary, in the second case U is antiunitary.
Remark. This note is included for technical accuracy only, and should not concern most readers. The result quoted above is not directly stated in Kadison's paper, but can be reduced to it by noting first that β extends to a positive trace preserving map on the trace class operators, then applying duality and finally applying a result of Kadison's paper.
The operator U is not quite unique; if r is a complex scalar of modulus 1, then r U will be unitary or antiunitary if U is and will implement the same automorphism. In fact, this is the only ambiguity possible.
It follows that automorphisms of Q are in bijective correspondence to unitary or antiunitary operators modulo multiplication by scalars of modulus 1. Moreover, we can regard automorphisms in two equivalent ways: as operating on states (represented as density operators) or as operating on Q.
Nonrelativistic dynamics
In nonrelativistic physical systems, there is no ambiguity in referring to time evolution since there is a global time parameter. Moreover an isolated quantum system evolves in a deterministic way: if the system is in a state S at time t then at time s > t, the system is in a state F_{s,t}(S). Moreover, we assume

The dependence is reversible: The operators F_{s,t} are bijective.

The dependence is homogeneous: F_{s,t} = F_{s − t,0}.

The dependence is convexity preserving: That is, each F_{s,t}(S) is convexity preserving.

The dependence is weakly continuous: The mapping R→ R given by t → Tr(F_{s,t}(S) E) is continuous for every E in Q.
By Kadison's theorem, there is a 1parameter family of unitary or antiunitary operators {U_{t}}_{t} such that

\operatorname{F}_{s,t}(S) = U_{st} S U_{st}^*
In fact,
Theorem. Under the above assumptions, there is a strongly continuous 1parameter group of unitary operators {U_{t}}_{t} such that the above equation holds.
Note that it follows easily from uniqueness from Kadison's theorem that

U_{t+s} = \sigma(t,s) U_t U_s
where σ(t,s) has modulus 1. Now the square of an antiunitary is a unitary, so that all the U_{t} are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by modifying each U_{t} by a scalar of modulus 1.)
Pure states
A convex combination of statistical states S_{1} and S_{2} is a state of the form S = p_{1} S_{1} +p_{2} S_{2} where p_{1}, p_{2} are nonnegative and p_{1} + p_{2} =1. Considering the statistical state of system as specified by lab conditions used for its preparation, the convex combination S can be regarded as the state formed in the following way: toss a biased coin with outcome probabilities p_{1}, p_{2} and depending on outcome choose system prepared to S_{1} or S_{2}
Density operators form a convex set. The convex set of density operators has extreme points; these are the density operators given by a projection onto a onedimensional space. To see that any extreme point is such a projection, note that by the spectral theorem S can be represented by a diagonal matrix; since S is nonnegative all the entries are nonnegative and since S has trace 1, the diagonal entries must add up to 1. Now if it happens that the diagonal matrix has more than one nonzero entry it is clear that we can express it as a convex combination of other density operators.
The extreme points of the set of density operators are called pure states. If S is the projection on the 1dimensional space generated by a vector ψ of norm 1 then

\operatorname{Tr}(S E) = \langle E \psi  \psi \rangle
for any E in Q. In physics jargon, if

S =  \psi \rangle \langle \psi  ,
where ψ has norm 1, then

\operatorname{Tr}(S E) = \langle \psi  E  \psi \rangle .
Thus pure states can be identified with rays in the Hilbert space H.
The measurement process
Consider a quantum mechanical system with lattice Q that is in some statistical state given by a density operator S. This essentially means an ensemble of systems specified by a repeatable lab preparation process. The result of a cluster of measurements intended to determine the truth value of proposition E, is just as in the classical case, a probability distribution of truth values T and F. Say the probabilities are p for T and q = 1 − p for F. By the previous section p = Tr(S E) and q = Tr(S (I − E)).
Perhaps the most fundamental difference between classical and quantum systems is the following: regardless of what process is used to determine E immediately after the measurement the system will be in one of two statistical states:

If the result of the measurement is T


\frac{1}{\operatorname{Tr}(E S)} E S E. \,

If the result of the measurement is F


\frac{1}{\operatorname{Tr}((IE) S)}(I E) S (I E). \,
(We leave to the reader the handling of the degenerate cases in which the denominators may be 0.) We now form the convex combination of these two ensembles using the relative frequencies p and q. We thus obtain the result that the measurement process applied to a statistical ensemble in state S yields another ensemble in statistical state:

\operatorname{M}_E(S) = E S E + (I  E) S (I  E). \,
We see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as described above, is a special case of quantum operations.
Limitations
Quantum logic derived from propositional logic provides a satisfactory foundation for a theory of reversible quantum processes. Examples of such processes are the covariance transformations relating two frames of reference, such as change of time parameter or the transformations of special relativity. Quantum logic also provides a satisfactory understanding of density matrices. Quantum logic can be stretched to account for some kinds of measurement processes corresponding to answering yesno questions about the state of a quantum system. However, for more general kinds of measurement operations (that is quantum operations), a more complete theory of filtering processes is necessary. Such a theory of quantum filtering was developed in the late 1970s and 1980s by Belavkin ^{[12]}^{[13]} (see also Bouten et al.^{[14]}). A similar approach is provided by the consistent histories formalism. On the other hand, quantum logics derived from MVlogic extend its range of applicability to irreversible quantum processes and/or 'open' quantum systems.
In any case, these quantum logic formalisms must be generalized in order to deal with supergeometry (which is needed to handle Fermifields) and noncommutative geometry (which is needed in string theory and quantum gravity theory). Both of these theories use a partial algebra with an "integral" or "trace". The elements of the partial algebra are not observables; instead the "trace" yields "greens functions", which generate scattering amplitudes. One thus obtains a local Smatrix theory (see D. Edwards).
Since around 1978 the Flato school (see F. Bayen) has been developing an alternative to the quantum logics approach called deformation quantization (see Weyl quantization).
In 2004, Prakash Panangaden described how to capture the kinematics of quantum causal evolution using System BV, a deep inference logic originally developed for use in structural proof theory.^{[15]} Alessio Guglielmi, Lutz Straßburger, and Richard Blute have also done work in this area.^{[16]}
See also
References

^ http://arxiv.org/abs/quantph/0101028v2 Maria Luisa Dalla Chiara and Roberto Giuntini. 2008. Quantum Logic., 102 pages PDF

^ Dalla Chiara, M. L. and Giuntini, R.: 1994, Unsharp quantum logics, Foundations of Physics,, 24, 1161–1177.

^ http://planetphysics.org/encyclopedia/QuantumLMAlgebraicLogic.html I. C. Baianu. 2009. Quantum LMn Algebraic Logic.

^ Georgescu, G. and C. Vraciu. 1970, On the characterization of centered Łukasiewicz algebras., J. Algebra, 16: 486–495.

^ Georgescu, G. 2006, Nvalued Logics and ŁukasiewiczMoisil Algebras, Axiomathes, 16 (1–2): 123

^ "Quantum logic" entry by Peter Forrest in the Routledge Encyclopedia of Philosophy, Vol. 7 (1998), p. 882ff: "[Quantum logic] differs from the standard sentential calculus ... The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity."

^ Megill, Norman. "Quantum Logic Explorer". Metamath. Retrieved 20130327.

^ Kalmbach, G (1983). Orthomodular Lattices. p. 236.

^ Cutland, N.J.; Gibbins, P.F. (1982). "A regular sequent calculus for Quantum Logic in which V and & are dual". Logique et Analyse  Nouvelle Serie 25: 221–248.

^ Nishimura, H. (1994). "Proof theory for minimal quantum logic". International Journal of Theoretical Physics 33: 103–113, 1427–1443.

^ Egly, Uwe; Tompits, Hans (1999). "Gentzenlike Methods in Quantum Logic". Tableaux'99.

^ V. P. Belavkin (1978). "Optimal quantum filtration of Makovian signals [In Russian]". Problems of Control and Information Theory, 7 (5): 345–360.

^ V. P. Belavkin (1992). "Quantum stochastic calculus and quantum nonlinear filtering". Journal of Multivariate Analysis 42 (2): 171–201.

^ Luc Bouten, Ramon van Handel, Matthew R. James (2009). "A discrete invitation to quantum filtering and feedback control". SIAM Review 51: 239–316.

^ http://cs.bath.ac.uk/ag/p/BVQuantCausEvol.pdf

^ DI & CoS  Current Research Topics and Open Problems
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, Vol. 37, pp. 823–843, 1936.

D. Cohen, An Introduction to Hilbert Space and Quantum Logic, SpringerVerlag, 1989. This is a thorough but elementary and wellillustrated introduction, suitable for advanced undergraduates.

David Edwards, The Mathematical Foundations of Quantum Mechanics, Synthese, Volume 42, Number 1/September, 1979, pp. 1–70.

D. Edwards, The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Supersymmetry, Part I: Lattice Field Theories, International J. of Theor. Phys., Vol. 20, No. 7 (1981).

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

G. Ludwig, Foundations of Quantum Mechanics, SpringerVerlag, 1983.

G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by Dover 2004).

J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. Reprinted in paperback form.

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

H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.
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