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In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. This research area and its name originated in the 1936 paper by G. Birkhoff and J. von Neumann, who attempted to reconcile some of the apparent inconsistencies of classical boolean logic with the facts related to measurement and observation in quantum mechanics.
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. Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself, to the physicist David Finkelstein. It should be noted, however, that 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. This logic has some unusual properties: For instance, the distributive law of propositional logic, that is the equivalence
fails in this logic.
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; in fact with some minor technical assumptions it can be subsumed by it. The similarities of the quantum logic formalism to a system of deductive logic is regarded more as a curiosity than as a fact of fundamental philosophical importance.
In his classic treatise Mathematical Foundations of Quantum Mechanics, 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) Mackey attempted to provide a set of axioms for this propositional system as an orthocomplemented partially ordered set. 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 which do not require such explicit relations to the lattice of subspaces.
The remainder of the following 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 using the finite-dimensional spectral theorem.
The so-called 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. We will merely note here that 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. A proposition concerning a classical system is a statement of the form
It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to that of some boolean algebra of subsets of the state space. By logic in this context we mean the rules that relate set operations and ordering relations, such as de Morgan's laws. These are analogous to the rules relating boolean conjunctives and material implication in classical propositional logic. For technical reasons, we will also assume that the algebra of subsets of the state space is that of all 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:
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 spectral measures.
Imagine a forensics lab which has some apparatus to measure the speed of a 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 {Ei}i is a sequence of pairwise orthogonal elements of Q then
The following highly non-trivial theorem is due to A. 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
for any self-adjoint projection E.
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.
Physicists commonly regard a density operator as being representated by a (possibly infinite) density matrix relative to some orthonormal basis.
For more information on statistics of quantum systems, see quantum statistical mechanics.
An automorphism of Q is a bijective mapping α:Q → Q which preserves the orthocomplemented structure of Q, that is:
for any sequence {Ei}i of pairwise orthogonal self-adjoint 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:
The mapping α* is bijective and preserves convex combinations of density operators. This means
whenever 1 = r1 + r2 and r1, r2 are non-negative real numbers. Now we use a theorem of deterministic way: if the system is in a state S at time t then at time s> t, the system is in a state Fs,t(S). Moreover, we assume
By Kadison's theorem, there is a 1-parameter family of unitary or anti-unitary operators {Ut}t such that
In fact,
Theorem. Under the above assumptions, there is a strongly continuous 1-parameter group of unitary operators {Ut}t such that the above equation holds.
Note that it easily from uniqueness from Kadison's theorem that
where &sigma(t,s) has modulus 1. Now the square of an ant-unitary is a unitary, so that all the Ut are unitary. The remainder of the argument shows that &sigma(t,s) can be chosen to be 1 (by modifying each Ut by a scalar of modulus 1.)
A convex combinations of statistical states S1 and S2 is a state of the form S = p2 S1 +p2 S2 where p1, p2 are non-negative and p1 + p2 =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 p1, p2 and depending on outcome choose system prepared to S1 or S2
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 one-dimensional 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 non-negative all the entries are non-negative 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 non-zero 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 1-dimensional space generated by a vector ψ of norm 1 then
for any E in Q. In physics jargon, if
where ψ has norm 1, then
Thus pure states can be identified with rays in the Hilbert space H.
Consider a quantum mechanical system with lattice Q which 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:
(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:
Operators of this kind are special cases of quantum operations.
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