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57-850: HPO may refer to: HPO formalism , an approach to temporal quantum logic Hamilton Philharmonic Orchestra , a symphony orchestra based in Ontario, Canada High performance organization , a conceptual framework for organizations Highway Post Office , of the United States Postal Service Homeowner Protection Office , in British Columbia, Canada Hpon language , spoken in Burma Human Phenotype Ontology Humanist Party of Ontario ,

114-513: A Hilbert space can be viewed as propositions about physical observables; that is, as potential yes-or-no questions an observer might ask about the state of a physical system, questions that could be settled by some measurement. Principles for manipulating these quantum propositions were then called quantum logic by von Neumann and Birkhoff in a 1936 paper. George Mackey , in his 1963 book (also called Mathematical Foundations of Quantum Mechanics ), attempted to axiomatize quantum logic as

171-477: A is true. To understand more, let p 1 and p 2 be the momentum functions (Fourier transforms) for the projections of the particle wave function to x ≤ 0 and x ≥ 0 respectively. Let | p i |↾ ≥1 be the restriction of p i to momenta that are (in absolute value) ≥1. ( a ∧ b ) ∨ ( a ∧ c ) corresponds to states with | p 1 |↾ ≥1 = | p 2 |↾ ≥1 = 0 (this holds even if we defined p differently so as to make such states possible; also,

228-511: A material conditional ; a common alternative is the system of linear logic , of which quantum logic is a fragment. Mathematically, quantum logic is formulated by weakening the distributive law for a Boolean algebra, resulting in an ortho­complemented lattice . Quantum-mechanical observables and states can be defined in terms of functions on or to the lattice, giving an alternate formalism for quantum computations. The most notable difference between quantum logic and classical logic

285-500: A separable Hilbert or pre-Hilbert space , where an observable p is associated with the set of quantum states for which p (when measured) has eigenvalue 1. From there, This semantics has the nice property that the pre-Hilbert space is complete (i.e., Hilbert) if and only if the propositions satisfy the orthomodular law, a result known as the Solèr theorem . Although much of the development of quantum logic has been motivated by

342-408: A ∧ b corresponds to | p 1 |↾ ≥1 =0 and p 2 =0). Meanwhile, a corresponds to states with | p |↾ ≥1 = 0. As an operator, p = p 1 + p 2 , and nonzero | p 1 |↾ ≥1 and | p 2 |↾ ≥1 might interfere to produce zero | p |↾ ≥1 . Such interference is key to the richness of quantum logic and quantum mechanics. Given a orthocomplemented lattice Q , a Mackey observable φ

399-419: A Boolean algebra, which is then amenable to classical logic. Although many treatments of quantum logic assume that the underlying lattice must be orthomodular, such logics cannot handle multiple interacting quantum systems. In an example due to Foulis and Randall, there are orthomodular propositions with finite-dimensional Hilbert models whose pairing admits no orthomodular model. Likewise, quantum logic with

456-629: A Hilbert space (See quantum logic ). The HPO formalism is a natural extension of these ideas to propositions about the system that are concerned with more than one time. A homogeneous history proposition α {\displaystyle \,\alpha } is a sequence of single-time propositions α t i {\displaystyle \alpha _{t_{i}}} specified at different times t 1 < t 2 < … < t n {\displaystyle t_{1}<t_{2}<\ldots <t_{n}} . These times are called

513-483: A classical system are generated from basic statements of the form through the conventional arithmetic operations and pointwise limits . It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to the Boolean algebra of Borel subsets of the state space. They thus obey the laws of classical propositional logic (such as de Morgan's laws ) with

570-430: A filtering property: Expressions in quantum logic describe observables using a syntax that resembles classical logic. However, unlike classical logic, the distributive law a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c ) fails when dealing with noncommuting observables , such as position and momentum. This occurs because measurement affects the system, and measurement of whether a disjunction holds does not measure which of

627-566: A political party in Canada Hydrogen peroxide , an oxidizer Premnaspirodiene oxygenase , an enzyme Hippo, a protein kinase involved in the Hippo signaling pathway Hyperparameter optimization , a technique used in automated machine learning Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title HPO . If an internal link led you here, you may wish to change

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684-413: Is a countably additive homomorphism from the orthocomplemented lattice of Borel subsets of R to Q . In symbols, this means that for any sequence { S i } i of pairwise-disjoint Borel subsets of R , {φ( S i )} i are pairwise-orthogonal propositions (elements of Q ) and Equivalently, a Mackey observable is a projection-valued measure on R . Theorem ( Spectral theorem ). If Q

741-400: 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 . The standard semantics of quantum logic is that quantum logic is the logic of projection operators in

798-412: Is associated with a Hilbert space H {\displaystyle {\mathcal {H}}} . States of the system at a fixed time are represented by normalised vectors in the space and physical observables are represented by Hermitian operators on H {\displaystyle {\mathcal {H}}} . A physical proposition P {\displaystyle \,P} about

855-415: Is both normalizable in momentum space and vanishes on precisely x ≥ 0. Thus, a ∧ b and similarly a ∧ c are false, so ( a ∧ b ) ∨ ( a ∧ c ) is false. However, a ∧ ( b ∨ c ) equals a , which is certainly not false (there are states for which it is a viable measurement outcome ). Moreover: if the relevant Hilbert space for the particle's dynamics only admits momenta no greater than 1, then

912-418: Is far from evident (albeit true ) that quantum logic is a logic , in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses. In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving

969-476: Is set complement. Moreover, this lattice is sequentially complete , in the sense that any sequence { E i } i ∈ N of elements of the lattice has a least upper bound , specifically the set-theoretic union: LUB ⁡ ( { E i } ) = ⋃ i = 1 ∞ E i . {\displaystyle \operatorname {LUB} (\{E_{i}\})=\bigcup _{i=1}^{\infty }E_{i}{\text{.}}} In

1026-461: Is the failure of the propositional distributive law : where the symbols p , q and r are propositional variables. To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck constant is 1) let We might observe that: in other words, that the state of the particle is a weighted superposition of momenta between 0 and +1/6 and positions between −1 and +3. On

1083-494: Is the indicator function of an interval [ a , b ], the operator f ( A ) is a self-adjoint projection onto the subspace of generalized eigenvectors of A with eigenvalue in [ a , b ] . That subspace can be interpreted as the quantum analogue of the classical proposition This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's Axiom VII : The space Q of quantum propositions

1140-406: Is the lattice of closed subspaces of Hilbert H , then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H . A quantum probability measure is a function P defined on Q with values in [0,1] such that P("⊥)=0, P(⊤)=1 and if { E i } i is a sequence of pairwise-orthogonal elements of Q then Every quantum probability measure on

1197-423: Is the projection operator on H {\displaystyle {\mathcal {H}}} that represents the proposition α t i {\displaystyle \alpha _{t_{i}}} at time t i {\displaystyle t_{i}} . This α ^ {\displaystyle {\hat {\alpha }}} is a projection operator on

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1254-446: Is through the recent development of quantum computing , which has engendered a proliferation of new logics for formal analysis of quantum protocols and algorithms (see also § Relationship to other logics ). The logic may also find application in (computational) linguistics. Quantum logic can be axiomatized as the theory of propositions modulo the following identities: ("¬" is the traditional notation for " not ", "∨"

1311-424: Is true and then α t 2 {\displaystyle \alpha _{t_{2}}} at time t 2 {\displaystyle t_{2}} is true and then … {\displaystyle \ldots } and then α t n {\displaystyle \alpha _{t_{n}}} at time t n {\displaystyle t_{n}}

1368-480: Is true" Not all history propositions can be represented by a sequence of single-time propositions at different times. These are called inhomogeneous history propositions . An example is the proposition α {\displaystyle \,\alpha } OR β {\displaystyle \,\beta } for two homogeneous histories α , β {\displaystyle \,\alpha ,\beta } . The key observation of

1425-818: 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 on R , the following extension of f to operators can be made: f ( A ) = ∫ R f ( λ ) d E ⁡ ( λ ) . {\displaystyle f(A)=\int _{\mathbb {R} }f(\lambda )\,d\operatorname {E} (\lambda ).} In case f

1482-513: The temporal support of the history. We shall denote the proposition α {\displaystyle \,\alpha } as ( α 1 , α 2 , … , α n ) {\displaystyle (\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})} and read it as " α t 1 {\displaystyle \alpha _{t_{1}}} at time t 1 {\displaystyle t_{1}}

1539-621: The tensor product to define a projector α ^ := α ^ t 1 ⊗ α ^ t 2 ⊗ … ⊗ α ^ t n {\displaystyle {\hat {\alpha }}:={\hat {\alpha }}_{t_{1}}\otimes {\hat {\alpha }}_{t_{2}}\otimes \ldots \otimes {\hat {\alpha }}_{t_{n}}} where α ^ t i {\displaystyle {\hat {\alpha }}_{t_{i}}}

1596-1003: The "always true" proposition at each time t i {\displaystyle \,t_{i}} can be formed. In this way the temporal supports of α , β {\displaystyle \,\alpha ,\beta } can always be joined. We shall therefore assume that all homogeneous histories share the same temporal support. We now present the logical operations for homogeneous history propositions α {\displaystyle \,\alpha } and β {\displaystyle \,\beta } such that α ^ β ^ = β ^ α ^ {\displaystyle {\hat {\alpha }}{\hat {\beta }}={\hat {\beta }}{\hat {\alpha }}} If α {\displaystyle \alpha } and β {\displaystyle \beta } are two homogeneous histories then

1653-494: The HPO formalism is to represent history propositions by projection operators on a history Hilbert space . This is where the name "History Projection Operator" (HPO) comes from. For a homogeneous history α = ( α 1 , α 2 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})} we can use

1710-531: The Hilbert space. Thus the projector used to represent the proposition ¬ α {\displaystyle \neg \alpha } (i.e. "not α {\displaystyle \alpha } ") is ¬ α ^ := I − α ^ . {\displaystyle {\widehat {\neg \alpha }}:=\mathbb {I} -{\hat {\alpha }}.} As an example, consider

1767-483: The OR operation, include all the possibilities for how the proposition " α 1 {\displaystyle \,\alpha _{1}} and then α 2 {\displaystyle \,\alpha _{2}} " can be false. We therefore see that the definition of ¬ α ^ {\displaystyle {\widehat {\neg \alpha }}} agrees with what

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1824-444: The case of the lattice of projections there are infinitely many solutions to the above equations (any closed, algebraic complement of p solves it; it need not be the orthocomplement). More generally, propositional valuation has unusual properties in quantum logic. An orthocomplemented lattice admitting a total lattice homomorphism to {⊥,⊤} must be Boolean. A standard workaround is to study maximal partial homomorphisms q with

1881-528: The closed subspaces of a Hilbert space is induced by a density matrix  — a nonnegative operator of trace 1. Formally, Quantum logic embeds into linear logic and the modal logic B . Indeed, modern logics for the analysis of quantum computation often begin with quantum logic, and attempt to graft desirable features of an extension of classical logic thereonto; the results then necessarily embed quantum logic. The orthocomplemented lattice of any set of quantum propositions can be embedded into

1938-455: The correct logic for propositional inference generally, most notably by the philosopher Hilary Putnam , at least at one point in his career. This thesis was an important ingredient in Putnam's 1968 paper " Is Logic Empirical? " in which he analysed the epistemological status of the rules of propositional logic. Modern philosophers reject quantum logic as a basis for reasoning, because it lacks

1995-553: The damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one. Their work proved fruitless, and now lies in poor repute. Most philosophers find quantum logic an unappealing competitor to classical logic . It

2052-470: The disjuncts is true. For example, consider a simple one-dimensional particle with position denoted by x and momentum by p , and define observables: Now, position and momentum are Fourier transforms of each other, and the Fourier transform of a square-integrable nonzero function with a compact support is entire and hence does not have non-isolated zeroes. Therefore, there is no wave function that

2109-483: The history Hilbert space naturally encodes the logical structure of history propositions. The lattice operations on the set of projection operations on the history Hilbert space H {\displaystyle H} can be applied to model the lattice of logical operations on history propositions. If two homogeneous histories α {\displaystyle \,\alpha } and β {\displaystyle \,\beta } don't share

2166-818: The history proposition " α {\displaystyle \,\alpha } and β {\displaystyle \,\beta } " is also a homogeneous history. It is represented by the projection operator α ∧ β ^ := α ^ β ^ {\displaystyle {\widehat {\alpha \wedge \beta }}:={\hat {\alpha }}{\hat {\beta }}} ( = β ^ α ^ ) {\displaystyle (={\hat {\beta }}{\hat {\alpha }})} If α {\displaystyle \alpha } and β {\displaystyle \beta } are two homogeneous histories then

2223-652: The history proposition " α {\displaystyle \,\alpha } or β {\displaystyle \,\beta } " is in general not a homogeneous history. It is represented by the projection operator α ∨ β ^ := α ^ + β ^ − α ^ β ^ {\displaystyle {\widehat {\alpha \vee \beta }}:={\hat {\alpha }}+{\hat {\beta }}-{\hat {\alpha }}{\hat {\beta }}} The negation operation in

2280-406: The lattice of projection operators takes P ^ {\displaystyle {\hat {P}}} to ¬ P ^ := I − P ^ {\displaystyle \neg {\hat {P}}:=\mathbb {I} -{\hat {P}}} where I {\displaystyle \mathbb {I} } is the identity operator on

2337-406: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=HPO&oldid=1100995316 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages HPO formalism In standard quantum mechanics a physical system

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2394-1581: The negation of the two-time homogeneous history proposition α = ( α 1 , α 2 ) {\displaystyle \,\alpha =(\alpha _{1},\alpha _{2})} . The projector to represent the proposition ¬ α {\displaystyle \neg \alpha } is ¬ α ^ = I ⊗ I − α ^ 1 ⊗ α ^ 2 {\displaystyle {\widehat {\neg \alpha }}=\mathbb {I} \otimes \mathbb {I} -{\hat {\alpha }}_{1}\otimes {\hat {\alpha }}_{2}} = ( I − α ^ 1 ) ⊗ α ^ 2 + α ^ 1 ⊗ ( I − α ^ 2 ) + ( I − α ^ 1 ) ⊗ ( I − α ^ 2 ) {\displaystyle =(\mathbb {I} -{\hat {\alpha }}_{1})\otimes {\hat {\alpha }}_{2}+{\hat {\alpha }}_{1}\otimes (\mathbb {I} -{\hat {\alpha }}_{2})+(\mathbb {I} -{\hat {\alpha }}_{1})\otimes (\mathbb {I} -{\hat {\alpha }}_{2})} The terms which appear in this expression: can each be interpreted as follows: These three homogeneous histories, joined with

2451-404: The notation for " or ", and "∧" the notation for " and ".) Some authors restrict to orthomodular lattices , which additionally satisfy the orthomodular law: ("⊤" is the traditional notation for truth and ""⊥" the traditional notation for falsity .) Alternative formulations include propositions derivable via a natural deduction , sequent calculus or tableaux system. Despite

2508-438: The orthomodular law falsifies the deduction theorem . Quantum logic admits no reasonable material conditional ; any connective that is monotone in a certain technical sense reduces the class of propositions to a Boolean algebra . Consequently, quantum logic struggles to represent the passage of time. One possible workaround is the theory of quantum filtrations developed in the late 1970s and 1980s by Belavkin . It

2565-472: The other hand, the propositions " p and q " and " p and r " each assert tighter restrictions on simultaneous values of position and momentum than are allowed by the uncertainty principle (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So there are no states that can support either proposition, and In his classic 1932 treatise Mathematical Foundations of Quantum Mechanics , John von Neumann noted that projections on

2622-542: The philosopher Hilary Putnam popularized Mackey's work in two papers in 1968 and 1975, in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist David Finkelstein . Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement , but Gleason's theorem presents severe difficulties for this goal. Later, Putnam retracted his views, albeit with much less fanfare, but

2679-549: The physics problems. Tim Maudlin writes that quantum "logic 'solves' the [measurement] problem by making the problem impossible to state." Quantum logic remains in limited use among logicians as an extremely pathological counterexample (Dalla Chiara and Giuntini: "Why quantum logics? Simply because 'quantum logics are there!'"). Although the central insight to quantum logic remains mathematical folklore as an intuition pump for categorification , discussions rarely mention quantum logic. Quantum logic's best chance at revival

2736-496: The proposition ¬ α {\displaystyle \neg \alpha } should mean. Quantum logic#Projections as propositions In the mathematical study of logic and the physical analysis of quantum foundations , quantum logic is a set of rules for manip­ulation of propositions inspired by the structure of quantum theory . The formal system takes as its starting point an obs­ervation of Garrett Birkhoff and John von Neumann , that

2793-438: The relatively developed proof theory , quantum logic is not known to be decidable . The remainder of this 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 in the finite-dimensional case. The Hamiltonian formulations of classical mechanics have three ingredients: states , observables and dynamics . In

2850-450: The same temporal support they can be modified so that they do. If t i {\displaystyle \,t_{i}} is in the temporal support of α {\displaystyle \,\alpha } but not β {\displaystyle \,\beta } (for example) then a new homogeneous history proposition which differs from β {\displaystyle \,\beta } by including

2907-539: The set operations of union and intersection corresponding to the Boolean conjunctives and subset inclusion corresponding to material implication . In fact, a stronger claim is true: they must obey the infinitary logic L ω 1 ,ω . 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

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2964-438: The simplest case of a single particle moving in R , the state space is the position–momentum space R . 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 propositions concerning

3021-428: The standard semantics, it is not the characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic. 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

3078-576: The structure of an ortho­complemented lattice , and recognized that a physical observable could be defined in terms of quantum propositions. Although Mackey's presentation still assumed that the ortho­complemented lattice is the lattice of closed linear subspaces of a separable Hilbert space, Constantin Piron , Günther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space. Inspired by Hans Reichenbach 's then-recent defence of general relativity ,

3135-566: The structure of experimental tests in classical mechanics forms a Boolean algebra , but the structure of experimental tests in quantum mechanics forms a much more complicated structure. A number of other logics have also been proposed to analyze quantum-mechanical phenomena, unfortunately also under the name of "quantum logic(s)". They are not the subject of this article. For discussion of the similarities and differences between quantum logic and some of these competitors, see § Relationship to other logics . Quantum logic has been proposed as

3192-399: The system at a fixed time can be represented by an orthogonal projection operator P ^ {\displaystyle {\hat {P}}} on H {\displaystyle {\mathcal {H}}} (See quantum logic ). This representation links together the lattice operations in the lattice of logical propositions and the lattice of projection operators on

3249-679: The tensor product "history Hilbert space" H = H ⊗ H ⊗ … ⊗ H {\displaystyle H={\mathcal {H}}\otimes {\mathcal {H}}\otimes \ldots \otimes {\mathcal {H}}} Not all projection operators on H {\displaystyle H} can be written as the sum of tensor products of the form α ^ {\displaystyle {\hat {\alpha }}} . These other projection operators are used to represent inhomogeneous histories by applying lattice operations to homogeneous histories. Representing history propositions by projectors on

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