Quantum logic
Encyclopedia
In quantum mechanics
, quantum logic is a set of rules for reasoning about proposition
s which 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 with the facts concerning the measurement of complementary variable
s 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 non-associative many-valued (MV) logic
.
Quantum logic has some properties which clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:
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
then the proposition "q or r" is true, so
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
. So,
Thus the distributive law fails.
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 paper Is Logic Empirical?
in which he analysed the epistemological status of the rules of propositional logic. Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein
. However, 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.
The more common view regarding quantum logic, however, is that it provides a formalism
for relating observable
s, 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 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).
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 which do not require such explicit relations to the lattice of subspaces.
The remainder of this article assumes the reader is familiar with the spectral theory
of self-adjoint operator
s on a Hilbert space. However, the main ideas can be understood using the finite-dimensional spectral theorem.
formulations of classical mechanics
have three ingredients: states, observable
s 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. The proposition
s concerning a classical system are generated from basic statements 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 set
s. 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 projection-valued measure
E defined on the Borel subsets of R. In particular, for any bounded Borel function f, the following equation holds:
In case f is the indicator function of an interval [a, b], the operator f(A) is a self-adjoint projection, and can be interpreted as the quantum analogue of the classical proposition
Q is also sequentially complete: any pairwise disjoint sequence{Vi}i of elements of Q has a least upper bound. Here disjointness of W1 and W2 means W2 is a subspace of W1⊥. The least upper bound of {Vi}i is the closed internal direct sum.
Henceforth we identify elements of Q with self-adjoint 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
have exactly one solution, namely the set-theoretic complement of p. In these equations I refers to the atomic proposition which is identically true and 0 the atomic proposition which 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 self-adjoint 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 {Si}i of pairwise disjoint Borel subsets of R, {φ(Si)}i are pairwise orthogonal projections and
Theorem. There is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H.
This is the content of the spectral theorem as stated in terms of spectral measures.
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 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
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 represented by a (possibly infinite) density matrix
relative to some orthonormal basis.
For more information on statistics of quantum systems, see quantum statistical mechanics
.
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 Richard V. Kadison:
Theorem. Suppose β is a bijective map from density operators to density operators which is convexity preserving. Then there is an operator U on the Hilbert space which is either linear or conjugate-linear, preserves the inner product and is such that
for every density operator S. In the first case we say U is unitary, in the second case U is anti-unitary.
The operator U is not quite unique; if r is a complex scalar of modulus 1, then r U will be unitary or anti-unitary 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 anti-unitary 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.
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 σ(t,s) has modulus 1. Now the square of an anti-unitary is a unitary, so that all the Ut are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by modifying each Ut by a scalar of modulus 1.)
Density operators form a convex set. The convex set of density operators has extreme point
s; 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.
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:
We see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as described above, is a special case of quantum operation
s.
formalism. On the other hand, quantum logics derived from MV-logic 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 super-geometry (which is needed to handle Fermi-fields) and non-commutative 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 S-matrix 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
. Alessio Guglielmi, Lutz Straßburger, and Richard Blute have also done work in this area.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, quantum logic is a set of rules for reasoning about proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s which takes the principles of quantum theory into account. This research area and its name originated in the 1936 paper by Garrett Birkhoff
Garrett Birkhoff
Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
and John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, who were attempting to reconcile the apparent inconsistency of classical with the facts concerning the measurement of complementary variable
Complementarity (physics)
In physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...
s 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
Noncommutative logic
Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus...
and non-associative many-valued (MV) logic
Multi-valued logic
In logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...
.
Quantum logic has some properties which clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:
- 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 is moving to the right"
- q = "the particle is in the interval [-1,1]"
- r = "the particle is not in the interval [-1,1]"
then the proposition "q or r" is true, so
- p and (q or r) = p
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
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
. 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 Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
, at least at one point in his career. This thesis was an important ingredient in Putnam's paper Is Logic Empirical?
Is logic empirical?
"Is logic empirical?" is the title of two articles that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds for revising classical logic...
in which he analysed the epistemological status of the rules of propositional logic. Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein
David Finkelstein
David Ritz Finkelstein is currently an emeritus professor of physics at the Georgia Institute of Technology. Finkelstein obtained his Ph.D. in physics at the Massachusetts Institute of Technology in 1953. From 1964 to 1976, he was professor of physics at Yeshiva University.In 1958 Charles W...
. However, this idea had been around for some time and had been revived several years earlier by George Mackey
George Mackey
George Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
's work on group representations and symmetry.
The more common view regarding quantum logic, however, is that it provides a formalism
Formalism
The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy. A practitioner of formalism is called a formalist. A formalist, with respect to some discipline, holds that there is no transcendent meaning to that discipline other than the literal...
for relating observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s, 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 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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
– of classical logics (see David Edwards).
Introduction
In his classic treatise Mathematical Foundations of Quantum Mechanics, John von NeumannJohn von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
noted that projections on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
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
Closed
Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
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 this article assumes the reader is familiar with the spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
of self-adjoint operator
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
s on a Hilbert space. However, the main ideas can be understood using the finite-dimensional spectral theorem.
Projections as propositions
The so-called HamiltonianHamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
formulations of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
have three ingredients: states, observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s and dynamics
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
. 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. The proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s 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 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 set
Borel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
s. 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 {Ei}i of elements of the lattice has a least upper bound, specifically the set-theoretic union:
In the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely-defined self-adjoint operator
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure
Projection-valued measure
In mathematics, particularly functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a Hilbert space...
E defined on the Borel subsets of R. In particular, for any bounded Borel function f, the following equation holds:
In case f is the indicator function of an interval [a, b], the operator f(A) is a self-adjoint 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{Vi}i of elements of Q has a least upper bound. Here disjointness of W1 and W2 means W2 is a subspace of W1⊥. The least upper bound of {Vi}i is the closed internal direct sum.
Henceforth we identify elements of Q with self-adjoint 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
have exactly one solution, namely the set-theoretic complement of p. In these equations I refers to the atomic proposition which is identically true and 0 the atomic proposition which 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 self-adjoint 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 {Si}i of pairwise disjoint Borel subsets of R, {φ(Si)}i are pairwise orthogonal projections and
Theorem. There is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H.
This is the content of the spectral theorem as stated in terms of spectral measures.
Statistical structure
Imagine a forensics lab which 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 probabilityProbability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
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 Andrew Gleason
Andrew Gleason
Andrew Mattei Gleason was an American mathematician and the eponym of Gleason's theorem and the Greenwood–Gleason graph. After briefly attending Berkeley High School he graduated from Roosevelt High School in Yonkers, then Yale University in 1942, where he became a Putnam Fellow...
:
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
Trace class
In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....
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 represented by a (possibly infinite) density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
relative to some orthonormal basis.
For more information on statistics of quantum systems, see quantum statistical mechanics
Quantum statistical mechanics
Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be...
.
Automorphisms
An automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the 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 Richard V. Kadison:
Theorem. Suppose β is a bijective map from density operators to density operators which is convexity preserving. Then there is an operator U on the Hilbert space which is either linear or conjugate-linear, preserves the inner product and is such that
for every density operator S. In the first case we say U is unitary, in the second case U is anti-unitary.
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 anti-unitary 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 anti-unitary 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.
Non-relativistic dynamics
In non-relativistic 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 Fs,t(S). Moreover, we assume- The dependence is reversible: The operators Fs,t are bijective.
- The dependence is homogeneous: Fs,t = Fs − t,0.
- The dependence is convexity preserving: That is, each Fs,t(S) is convexity preserving.
- The dependence is weakly continuous: The mapping R→ R given by t → Tr(Fs,t(S) E) is continuous for every E in Q.
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 σ(t,s) has modulus 1. Now the square of an anti-unitary is a unitary, so that all the Ut are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by modifying each Ut by a scalar of modulus 1.)
Pure states
A convex combination of statistical states S1 and S2 is a state of the form S = p1 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 S2Density operators form a convex set. The convex set of density operators has extreme point
Extreme point
In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S...
s; 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.
The measurement process
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:
- If the result of the measurement is T
-
- If the result of the measurement is F
(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:
We see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as described above, is a special case of quantum operation
Quantum operation
In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan...
s.
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 yes-no 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 an approach is provided by the consistent historiesConsistent histories
In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology...
formalism. On the other hand, quantum logics derived from MV-logic 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 super-geometry (which is needed to handle Fermi-fields) and non-commutative 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 S-matrix 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
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
).
In 2004, Prakash Panangaden described how to capture the kinematics of quantum causal evolution using System BV, a deep inference
Deep inference
Deep inference names a general idea in structural proof theory that breaks with the classical sequent calculus by generalising the notion of structure to permit inference to occur in contexts of high structural complexity...
logic originally developed for use in structural proof theory
Structural proof theory
In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.-Analytic proof:...
. Alessio Guglielmi, Lutz Straßburger, and Richard Blute have also done work in this area.
See also
- Mathematical formulation of quantum mechanicsMathematical formulation of quantum mechanicsThe mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
- Multi-valued logicMulti-valued logicIn logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...
- Quasi-set theoryQuasi-set theoryQuasi-set theory is a formal mathematical theory for dealing with collections of indistinguishable objects, mainly motivated by the assumption that certain objects treated in quantum physics are indistinguishable and don't have individuality.-Motivation:...
- HPO formalismHPO formalismThe History Projection Operator formalism is an approach to temporal quantum logic developed by Chris Isham. It deals with the logical structure of quantum mechanical propositions asserted at different points in time.- Introduction :...
(An approach to temporal quantum logic) - Quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
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, Springer-Verlag, 1989. This is a thorough but elementary and well-illustrated 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 Super-symmetry, 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, Springer-Verlag, 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.