|
|
| Line 1: |
Line 1: |
| 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 logic]] with the facts concerning the measurement of [[complementarity (physics)|complementary variable]]s in quantum mechanics, such as position and momentum.
| | It begins by consuming in a pair from the lovely island where your company peaceful village is part way through beaches and woods up until the enemies known when your BlackGuard led by Lieutenant Hammerman invades your snowdonia. After managing to guard against a smallish invasion force, he intentions to avenge his loss all through battle.<br><br>An individual are are a parent or simply gaurdian of any tad one who appreciates stealing part in video games, be familiar with multi-player and on-line video game titles options. These personality give your kid so that you interact with many other gamers throughout the complete world. As good as it's is for your tyke in order to talk about with others, you do not know who a single on the other quit is.<br><br>Many people which play clash of clans crave ways of getting free gems. If you liked this report and you would like to get a lot more facts concerning clash of clans cheat; [http://prometeu.net enquiry], kindly check out our own web site. The gems are very important because they give the player the features and the power increase their gaming experience. As opposed to various other equivalent games in cell phone websites, especially those even individuals use various crevices in buy to take advantage of these practical information relating to free, the nature of farmville and its represent does not enable your varieties of hacks that anyone can put to the activity. Everyone is always looking for ways for you to get free gems with regard to clash of clans nonetheless most important thing to do is to employ the best approach to earn these anyone for free, save them [http://Data.Gov.uk/data/search?q=suitably suitably] and use they will where necessary.<br><br>Be charged attention to how so much money your teenager is [http://www.wired.com/search?query=considered+spending considered spending] on video games. These products are usually cheap and there 's often the option together with buying more add-ons in just the game itself. Set monthly and yearly limits on the wide variety of money that is going to be spent on games. Also, carry conversations with your little children about budgeting.<br><br>Rugby season is here and furthermore going strong, and like for example many fans we incredibly long for Sunday afternoon when the games begin. If you have trialled and liked Soul Caliber, you will love this excellent game. The third best is the Processor Cell which will with little thought fill in some sections. Defeating players as if that by any now means necessary can be that this reason that pushes these folks to use Words with the Friends Cheat. Currently the app requires you on to answer 40 questions with varying degrees of difficulty.<br><br>Your war abject is agnate in your approved village, except that your gua abject will not gain resources. Barrio while in your warfare abject find it difficult to be anon improved or maybe a rearranged, as it on it's own mimics this adjustment and so accomplished completed advancement stages of your apple while having alertness day. Combat bases additionally never charges to take their of add-ons rearmed, defenses reloaded in addition to characters healed, as some people are consistently ready. The association alcazar in the your war abject expense be abounding alone about the one in your incredible whole village.<br><br>Video games are some of typically the finest kinds of excitement around. They are unquestionably also probably the for the most part pricey types of entertainment, with console games which range from $50 that will $60, and consoles on their own inside a person's 100s. It is often possible to spend lesser on clash of clans hack and console purchases, and you can track down out about them over the following paragraphs. |
| | |
| Quantum logic can be formulated either as a modified version of [[propositional logic]] or as a [[Noncommutative logic|noncommutative]] and non-associative [[Multi-valued logic|many-valued (MV) logic]].<ref>http://arxiv.org/abs/quant-ph/0101028v2 Maria Luisa Dalla Chiara and Roberto Giuntini. 2008. ''Quantum Logic''., 102 pages PDF</ref><ref>Dalla Chiara, M. L. and Giuntini, R.: 1994, Unsharp quantum logics, ''Foundations of Physics,'', '''24''', 1161–1177.</ref><ref>http://planetphysics.org/encyclopedia/QuantumLMAlgebraicLogic.html I. C. Baianu. 2009. Quantum LMn Algebraic Logic.</ref><ref>Georgescu, G. and C. Vraciu. 1970, On the characterization of centered Łukasiewicz algebras.,'' J. Algebra'', '''16''': 486-495.</ref><ref>Georgescu, G. 2006, N-valued Logics and Łukasiewicz-Moisil Algebras, ''Axiomathes'', '''16''' (1-2): 123-</ref>
| |
| | |
| 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 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 [[Hilary Putnam]], at least at one point in his career. This thesis was an important ingredient in Putnam's paper ''[[Is logic empirical?|Is Logic Empirical?]]'' in which he analysed the [[epistemology|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 (mathematics)|formalism]] for relating [[observable]]s, system preparation filters and states. In this view, the quantum logic approach resembles more closely the [[C*-algebra]]ic 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 lattice|orthocomplemented]] [[closed set|closed]] [[Linear subspace|subspace]] [[Lattice (order)|lattice]] of a separable Hilbert space. [[Constantin Piron|Piron]], Ludwig and others have attempted to give axiomatizations which 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 <ref>{{cite web|last=Megill|first=Norman|title=Quantum Logic Explorer|url=http://us.metamath.org/qlegif/mmql.html|work=Metamath|accessdate=2013-03-27}}</ref>) is as follows:
| |
| * <math> a = a^{\perp \perp} </math>
| |
| * <math> \cup </math> is commutative and associative.
| |
| * There is a maximal element '''1''', and <math> 1 = b \cup b^\perp </math> for any ''b''.
| |
| * <math> a \cup (a^\perp \cup b)^\perp = a </math>.
| |
| * The orthomodular law: If <math> 1 = (a^\perp \cup b^\perp)^\perp \cup (a \cup b)^\perp </math> then <math> a = b </math>.
| |
| | |
| Alternative formulations include Hilbert-style propositional axioms,<ref>{{cite book|last=Kalmbach|first=G|title=Orthomodular Lattices|year=1983|page=236}}</ref> sequent calculi,<ref>{{cite journal|last=Cutland|first=N.J.|coauthors=Gibbins, P.F.|title=A regular sequent calculus for Quantum Logic in which V and & are dual|journal=Logique et Analyse - Nouvelle Serie|year=1982|volume=25|pages=221–248}}</ref><ref>{{cite journal|last=Nishimura|first=H.|title=Proof theory for minimal quantum logic|journal=International Journal of Theoretical Physics|year=1994|volume=33|pages=103–113, 1427–1443|bibcode = 1994IJTP...33..103N |doi = 10.1007/BF00671616 }}</ref> and tableaux systems.<ref>{{cite book|last=Egly|first=Uwe|title=Tableaux'99|year=1999|id={{citeseerx|10.1.1.88.9045}}|coauthors=Tompits, Hans|chapter=Gentzen-like Methods in Quantum Logic}}</ref>
| |
| | |
| 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.
| |
| | |
| == Projections as propositions ==
| |
| | |
| The so-called ''[[Hamiltonian mechanics|Hamiltonian]]'' formulations of [[classical mechanics]] have three ingredients: ''[[Classical mechanics|states]]'', ''[[observable]]s'' and ''[[Dynamics (mechanics)|dynamics]]''. In the simplest case of a single particle moving in '''R'''<sup>3</sup>, the state space is the position-momentum space '''R'''<sup>6</sup>. 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
| |
| | |
| * 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 (structure)|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 algebra|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:
| |
| | |
| * 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''<sub>''i''</sub>}<sub>''i''</sub> of elements of the lattice has a least upper bound, specifically the set-theoretic union:
| |
| | |
| :: <math> \operatorname{LUB}(\{E_i\}) = \bigcup_{i=1}^\infty E_i. </math>
| |
| | |
| 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:
| |
| | |
| :<math> f(A) = \int_{\mathbb{R}} f(\lambda) \, d \operatorname{E}(\lambda).</math>
| |
| | |
| 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''<sup>⊥</sup>.
| |
| | |
| ''Q'' is also sequentially complete: any pairwise disjoint sequence{''V''<sub>''i''</sub>}<sub>''i''</sub> of elements of ''Q'' has a least upper bound. Here disjointness of ''W''<sub>1</sub> and ''W''<sub>2</sub> means ''W''<sub>2</sub> is a subspace of ''W''<sub>1</sub><sup>⊥</sup>. The least upper bound of {''V''<sub>''i''</sub>}<sub>''i''</sub> 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
| |
| :<math> I = p \vee q</math>
| |
| :<math> 0 = p\wedge q </math>
| |
| 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 measure|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''<sub>''i''</sub>}<sub>''i''</sub> of pairwise disjoint Borel subsets of '''R''', {φ(''S''<sub>''i''</sub>)}<sub>''i''</sub> are pairwise orthogonal projections and
| |
| | |
| :<math> \varphi\left(\bigcup_{i=1}^\infty S_i\right) = \sum_{i=1}^\infty \varphi(S_i). </math>
| |
| | |
| '''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 measure]]s.
| |
| | |
| == 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 [[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''<sub>''i''</sub>}<sub>''i''</sub> is a sequence of pairwise orthogonal elements of ''Q'' then
| |
| :<math> \operatorname{P}\!\left(\sum_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i). </math>
| |
| | |
| 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
| |
| :<math> \operatorname{P}(E) = \operatorname{Tr}(S E) </math>
| |
| 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]].
| |
| | |
| == Automorphisms ==
| |
| An ''[[automorphism]]'' of ''Q'' is a bijective mapping α:''Q'' → ''Q'' which preserves the orthocomplemented structure of ''Q'', that is
| |
| | |
| :<math> \alpha\!\left(\sum_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \alpha(E_i) </math>
| |
| | |
| for any sequence {''E''<sub>''i''</sub>}<sub>''i''</sub> 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:
| |
| | |
| :<math> \operatorname{Tr}(\alpha^*(S) E) = \operatorname{Tr}(S \alpha(E)).</math>
| |
| | |
| The mapping α* is bijective and preserves convex combinations of density operators. This means
| |
| :<math> \alpha^*(r_1 S_1 + r_2 S_2) = r_1\alpha^*(S_1) + r_2 \alpha^*(S_2) \quad </math>
| |
| whenever 1 = ''r''<sub>1</sub> + ''r''<sub>2</sub> and ''r''<sub>1</sub>, ''r''<sub>2</sub> 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
| |
| | |
| :<math> \beta(S) = U S U^* \, </math>
| |
| | |
| for every density operator ''S''. In the first case we say ''U'' is unitary, in the second case ''U'' is anti-unitary.
| |
| | |
| <blockquote style="background: white; border: 1px solid black; padding: 1em;">
| |
| '''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.
| |
| </blockquote>
| |
| | |
| 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 F<sub>''s'',''t''</sub>(''S''). Moreover, we assume
| |
| | |
| * The dependence is reversible: The operators F<sub>''s'',''t''</sub> are bijective.
| |
| | |
| * The dependence is homogeneous: F<sub>''s'',''t''</sub> = F<sub>''s'' − ''t'',0</sub>.
| |
| | |
| * The dependence is convexity preserving: That is, each F<sub>''s'',''t''</sub>(''S'') is convexity preserving.
| |
| | |
| * The dependence is weakly continuous: The mapping '''R'''→ '''R''' given by ''t'' → Tr(F<sub>''s'',''t''</sub>(''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 {''U''<sub>''t''</sub>}<sub>''t''</sub> such that
| |
| :<math> \operatorname{F}_{s,t}(S) = U_{s-t} S U_{s-t}^* </math>
| |
| | |
| In fact,
| |
| | |
| '''Theorem'''. Under the above assumptions, there is a strongly continuous 1-parameter group of unitary operators {''U''<sub>''t''</sub>}<sub>''t''</sub> such that the above equation holds.
| |
| | |
| Note that it follows easily from uniqueness from Kadison's theorem that
| |
| | |
| :<math> U_{t+s} = \sigma(t,s) U_t U_s </math>
| |
| | |
| where σ(t,s) has modulus 1. Now the square of an anti-unitary is a unitary, so that all the ''U''<sub>''t''</sub> are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by modifying each ''U''<sub>''t''</sub> by a scalar of modulus 1.)
| |
| | |
| == Pure states ==
| |
| A convex combination of statistical states ''S''<sub>1</sub> and ''S''<sub>2</sub> is a state of the form ''S'' = ''p''<sub>1</sub> ''S''<sub>1</sub> +''p''<sub>2</sub> ''S''<sub>2</sub> where ''p''<sub>1</sub>, ''p''<sub>2</sub> are non-negative and ''p''<sub>1</sub> + ''p''<sub>2</sub> =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''<sub>1</sub>, ''p''<sub>2</sub> and depending on outcome choose system prepared to ''S''<sub>1</sub> or ''S''<sub>2</sub>
| |
| | |
| 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 state]]s. If ''S'' is the projection on the 1-dimensional space generated by a vector ψ of norm 1 then
| |
| :<math> \operatorname{Tr}(S E) = \langle E \psi | \psi \rangle </math>
| |
| for any ''E'' in ''Q''. In physics jargon, if
| |
| :<math>S = | \psi \rangle \langle \psi | , </math>
| |
| where ψ has norm 1, then
| |
| :<math> \operatorname{Tr}(S E) = \langle \psi | E | \psi \rangle .</math>
| |
| | |
| 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'''
| |
| ::<math> \frac{1}{\operatorname{Tr}(E S)} E S E. \, </math>
| |
| * If the result of the measurement is '''F'''
| |
| ::<math> \frac{1}{\operatorname{Tr}((I-E) S)}(I- E) S (I- E). \, </math>
| |
| (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:
| |
| | |
| :<math> \operatorname{M}_E(S) = E S E + (I - E) S (I - E). \, </math>
| |
| | |
| We see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as described above, is a special case of [[quantum operation]]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 a theory of [[quantum filtering]] was developed in the late 1970s and 1980s by [[Viacheslav Belavkin|Belavkin]] <ref name=Belavkin78>{{cite journal
| |
| | author = V. P. Belavkin
| |
| | title = Optimal quantum filtration of Makovian signals [In Russian]
| |
| | journal = Problems of Control and Information Theory,
| |
| | volume = 7
| |
| | number = 5
| |
| | pages = 345–360
| |
| | year = 1978
| |
| }}</ref><ref name=Belavkin92>{{cite journal
| |
| | author = V. P. Belavkin
| |
| | title = Quantum stochastic calculus and quantum nonlinear filtering
| |
| | journal = Journal of Multivariate Analysis
| |
| | volume = 42
| |
| | number = 2
| |
| | year = 1992
| |
| | pages = 171–201
| |
| | doi = 10.1016/0047-259X(92)90042-E
| |
| | arxiv = math/0512362}}</ref> (see also Bouten et al.<ref name=Bouten2009>{{cite journal
| |
| | author = Luc Bouten, Ramon van Handel, Matthew R. James
| |
| | title = A discrete invitation to quantum filtering and feedback control
| |
| | journal = SIAM Review
| |
| | volume = 51
| |
| | pages = 239–316
| |
| | year = 2009
| |
| | doi = 10.1137/060671504
| |
| | arxiv = math/0606118v4
| |
| |bibcode = 2009SIAMR..51..239B }}</ref>). A similar approach is provided by the [[consistent histories]] 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]].<ref>http://cs.bath.ac.uk/ag/p/BVQuantCausEvol.pdf</ref> [[Alessio Guglielmi]], [[Lutz Straßburger]], and [[Richard Blute]] have also done work in this area.<ref>[http://alessio.guglielmi.name/res/cos/crt.html#CQE DI & CoS - Current Research Topics and Open Problems]</ref>
| |
| | |
| == See also ==
| |
| * [[Linear logic]]
| |
| * [[Mathematical formulation of quantum mechanics]]
| |
| * [[Multi-valued logic]]
| |
| * [[Vector logic]]
| |
| * [[Quasi-set theory]]
| |
| * [[HPO formalism]] (An approach to temporal quantum logic)
| |
| * [[Quantum field theory]]
| |
| | |
| == References ==
| |
| {{reflist|2}}
| |
| | |
| == 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).
| |
| * [[David Finkelstein|D. Finkelstein]], ''Matter, Space and Logic'', Boston Studies in the Philosophy of Science Vol. V, 1969
| |
| * [[Andrew Gleason|A. Gleason]], ''Measures on the Closed Subspaces of a Hilbert Space'', Journal of Mathematics and Mechanics, 1957.
| |
| * [[Richard Kadison|R. Kadison]], ''Isometries of Operator Algebras'', Annals of Mathematics, Vol. 54, pp. 325–338, 1951
| |
| * G. Ludwig, ''Foundations of Quantum Mechanics'', Springer-Verlag, 1983.
| |
| * [[George Mackey|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.
| |
| * [[Constantin Piron|C. Piron]], ''Foundations of Quantum Physics'', W. A. Benjamin, 1976.
| |
| * [[Hilary Putnam|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.
| |
| | |
| == External links ==
| |
| * [http://plato.stanford.edu/entries/qt-quantlog/ Stanford Encyclopedia of Philosophy entry on Quantum Logic and Probability Theory]
| |
| * [http://us.metamath.org/qlegif/mmql.html Quantum Logic Explorer] at Metamath
| |
| | |
| {{DEFAULTSORT:Quantum Logic}}
| |
| [[Category:Mathematical logic]]
| |
| [[Category:Non-classical logic]]
| |
| [[Category:Quantum mechanics]]
| |