Butterfly effect

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Butterfly_effect

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.

The term is closely associated with the work of the mathematician and meteorologist Edward Norton Lorenz. He noted that the butterfly effect is derived from the example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier. Lorenz originally used a seagull causing a storm but was persuaded to make it more poetic with the use of a butterfly and tornado by 1972. He discovered the effect when he observed runs of his weather model with initial condition data that were rounded in a seemingly inconsequential manner. He noted that the weather model would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome.

The idea that small causes may have large effects in weather was earlier acknowledged by the French mathematician and physicist Henri Poincaré. The American mathematician and philosopher Norbert Wiener also contributed to this theory. Lorenz's work placed the concept of instability of the Earth's atmosphere onto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoing nonlinear dynamics and deterministic chaos.

The concept of the butterfly effect has since been used outside the context of weather science as a broad term for any situation where a small change is supposed to be the cause of larger consequences.

History

See also: Chaos theory § A popular but inaccurate analogy for chaos

In The Vocation of Man (1800), Johann Gottlieb Fichte says "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole".

Chaos theory and the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidenced by the case of the three-body problem by Poincaré in 1890. He later proposed that such phenomena could be common, for example, in meteorology.

In 1898, Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908.

In 1950, Alan Turing noted: "The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping."

The idea that the death of one butterfly could eventually have a far-reaching ripple effect on subsequent historical events made its earliest known appearance in "A Sound of Thunder", a 1952 short story by Ray Bradbury in which a time traveller alters the future by inadvertently treading on a butterfly in the past.

More precisely, though, almost the exact idea and the exact phrasing —of a tiny insect's wing affecting the entire atmosphere's winds— was published in a children's book which became extremely successful and well-known globally in 1962, the year before Lorenz published:

"...whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world." -- The Princess of Pure Reason

— Norton Juster, The Phantom Tollbooth

In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.

Lorenz wrote:

At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak vacuum tube or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last [decimal] place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution.

— E. N. Lorenz, The Essence of Chaos, University of Washington Press, Seattle (1993), page 134

In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow (the calculations were performed on a Royal McBee LGP-30 computer). Elsewhere he stated:

One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.

A Battus polydamas butterfly in Brazil

Following proposals from colleagues, in later speeches and papers, Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? as a title. Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely.

The phrase refers to the effect of a butterfly's wings creating tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings can cause the tornado: in the sense that the flap of the wings is a part of the initial conditions of an interconnected complex web; one set of conditions leads to a tornado, while the other set of conditions doesn't. The flapping wing creates a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare: domino effect). Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado.

The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of ensemble forecasting, in which a number of forecasts are made from perturbed initial conditions.

Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed. David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role. Stephen Wolfram also notes that the Lorenz equations are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations. Recent studies using generalized Lorenz models that included additional dissipative terms and nonlinearity suggested that a larger heating parameter is required for the onset of chaos.

While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied to work he published in 1969 which took the idea a step further. Lorenz proposed a mathematical model for how tiny motions in the atmosphere scale up to affect larger systems. He found that the systems in that model could only be predicted up to a specific point in the future, and beyond that, reducing the error in the initial conditions would not increase the predictability (as long as the error is not zero). This demonstrated that a deterministic system could be "observationally indistinguishable" from a non-deterministic one in terms of predictability. Recent re-examinations of this paper suggest that it offered a significant challenge to the idea that our universe is deterministic, comparable to the challenges offered by quantum physics.

In the book entitled The Essence of Chaos published in 1993, Lorenz defined butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." This feature is the same as sensitive dependence of solutions on initial conditions (SDIC) in . In the same book, Lorenz applied the activity of skiing and developed an idealized skiing model for revealing the sensitivity of time-varying paths to initial positions. A predictability horizon is determined before the onset of SDIC.

Illustrations

The butterfly effect in the Lorenz attractor
time 0 ≤ t ≤ 30		z coordinate
These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, and the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that differ by only 10−5 in the x-coordinate. Initially, the two trajectories seem coincident, as indicated by the small difference between the z coordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at t = 30.
An animation of the Lorenz attractor shows the continuous evolution.

Theory and mathematical definition

See also: Chaos theory § Lorenz's pioneering contributions to chaotic modeling

A plot of Lorenz' strange attractor for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical system that, starting from any of various arbitrarily close alternative initial conditions on the attractor, the iterated points will become arbitrarily spread out from each other.

Recurrence, the approximate return of a system toward its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.

A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical. Lorenz defined sensitive dependence as follows:

The property characterizing an orbit (i.e., a solution) if most other orbits that pass close to it at some point do not remain close to it as time advances.

If M is the state space for the map ${\displaystyle f^t}$, then ${\displaystyle f^t}$ displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance d(. , .) such that ${\displaystyle 0<d(x,y)<\delta }$ and such that

${\displaystyle d(f^{\tau }(x),f^{\tau }(y))>{\mathrm{e}}^{a\tau }d(x,y)}$

for some positive parameter a. The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems.

The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the logistic map:

${\displaystyle x_{n+1}=4x_n(1-x_n),0\le x_0\le 1,}$

which, unlike most chaotic maps, has a closed-form solution:

${\displaystyle x_n={\sin }^2(2^n\theta \pi )}$

where the initial condition parameter ${\displaystyle \theta }$ is given by ${\displaystyle \theta =\frac{1}{\pi }{\sin }^{-1}(x_0^{1/2})}$. For rational ${\displaystyle \theta }$, after a finite number of iterations ${\displaystyle x_n}$ maps into a periodic sequence. But almost all ${\displaystyle \theta }$ are irrational, and, for irrational ${\displaystyle \theta }$, ${\displaystyle x_n}$ never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2ⁿ shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps ${\displaystyle x_n}$ folded within the range [0, 1].

In physical systems

In weather

Overview

The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example. The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive to initial conditions. They add the caveat: "Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat wrong."

Differentiating types of butterfly effects

The concept of the butterfly effect encompasses several phenomena. The two kinds of butterfly effects, including the sensitive dependence on initial conditions, and the ability of a tiny perturbation to create an organized circulation at large distances, are not exactly the same. In Palmer et al., a new type of butterfly effect is introduced, highlighting the potential impact of small-scale processes on finite predictability within the Lorenz 1969 model. Additionally, the identification of ill-conditioned aspects of the Lorenz 1969 model points to a practical form of finite predictability. These two distinct mechanisms suggesting finite predictability in the Lorenz 1969 model are collectively referred to as the third kind of butterfly effect. The authors in  have considered Palmer et al.'s suggestions and have aimed to present their perspective without raising specific contentions.

The third kind of butterfly effect with finite predictability, as discussed in, was primarily proposed based on a convergent geometric series, known as Lorenz's and Lilly's formulas. Ongoing discussions are addressing the validity of these two formulas for estimating predictability limits in.

A comparison of the two kinds of butterfly effects and the third kind of butterfly effect has been documented. In recent studies, it was reported that both meteorological and non-meteorological linear models have shown that instability plays a role in producing a butterfly effect, which is characterized by brief but significant exponential growth resulting from a small disturbance.

Recent debates on butterfly effects

The first kind of butterfly effect (BE1), known as SDIC (Sensitive Dependence on Initial Conditions), is widely recognized and demonstrated through idealized chaotic models. However, opinions differ regarding the second kind of butterfly effect, specifically the impact of a butterfly flapping its wings on tornado formation, as indicated in two 2024 articles. In more recent discussions published by Physics Today, it is acknowledged that the second kind of butterfly effect (BE2) has never been rigorously verified using a realistic weather model. While the studies suggest that BE2 is unlikely in the real atmosphere, its invalidity in this context does not negate the applicability of BE1 in other areas, such as pandemics or historical events.

For the third kind of butterfly effect, the limited predictability within the Lorenz 1969 model is explained by scale interactions in one article and by system ill-conditioning in another more recent study.

Finite predictability in chaotic systems

According to Lighthill (1986), the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review, it was found that Lorenz's perspective on the predictability limit can be condensed into the following statement:

• (A). The Lorenz 1963 model qualitatively revealed the essence of a finite predictability within a chaotic system such as the atmosphere. However, it did not determine a precise limit for the predictability of the atmosphere.
• (B). In the 1960s, the two-week predictability limit was originally estimated based on a doubling time of five days in real-world models. Since then, this finding has been documented in Charney et al. (1966) and has become a consensus.

Recently, a short video has been created to present Lorenz's perspective on predictability limit.

A recent study refers to the two-week predictability limit, initially calculated in the 1960s with the Mintz-Arakawa model's five-day doubling time, as the "Predictability Limit Hypothesis." Inspired by Moore's Law, this term acknowledges the collaborative contributions of Lorenz, Mintz, and Arakawa under Charney's leadership. The hypothesis supports the investigation into extended-range predictions using both partial differential equation (PDE)-based physics methods and Artificial Intelligence (AI) techniques.

In quantum mechanics

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics, including atoms in strong fields and the anisotropic Kepler problem. Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller and John B. Delos and co-workers. The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist.

Other authors suggest that the butterfly effect can be observed in quantum systems. Zbyszek P. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians. David Poulin et al. presented a quantum algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected to slightly different dynamics". They consider fidelity decay to be "the closest quantum analog to the (purely classical) butterfly effect". Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to initial conditions are known as quantum chaos.

In popular culture

Main article: Butterfly effect in popular culture

The butterfly effect has appeared across media such as literature (for instance, A Sound of Thunder), films and television (such as The Simpsons), video games (such as Life Is Strange), webcomics (such as Homestuck), musical references (such as "Butterfly Effect" by Travis Scott), AI-driven expansive language models, and more.

Quantum logic

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_logic 

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 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 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 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.

Introduction

The most notable difference between quantum logic and classical logic is the failure of the propositional distributive law:

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 (using some system of units where the reduced Planck constant is 1) 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]"

We might observe that:

p and (q or r) = true

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 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

(p and q) or (p and r) = false

History and Philosophical Debate

In his classic 1932 treatise Mathematical Foundations of Quantum Mechanics, John von Neumann noted that projections on 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 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, 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 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 would agree that quantum logic is not a competitor to classical logic. It 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, some modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems. Tim Maudlin writes that quantum "logic 'solves' the [measurement] problem by making the problem impossible to state."

Quantum logic remains in use among logicians and interests are expanding 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.

Algebraic structure

Quantum logic can be axiomatized as the theory of propositions modulo the following identities:

• a = ¬¬a
• ∨ is commutative and associative.
• There is a maximal element ⊤, and ⊤ = b∨¬b for any b.
• a∨¬(¬a∨b) = a.

("¬" is the traditional notation for "not", "∨" the notation for "or", and "∧" the notation for "and".)

Some authors restrict to orthomodular lattices, which additionally satisfy the orthomodular law:

• If ⊤ = ¬(¬a∨¬b)∨¬(a∨b) then a = b.

("⊤" 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 the relatively developed proof theory, quantum logic is not known to be decidable.

Quantum logic as the logic of observables

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.

Logic of classical mechanics

The Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest case of a single particle moving in R³, the state space is the position–momentum space R⁶. 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 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."

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 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_ω1,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 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: $${\displaystyle \mathrm{LUB}(\{E_i\})=\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\text{.}}$$

Propositional lattice of a quantum mechanical system

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 on R, the following extension of f to operators can be made: $${\displaystyle f(A)={\int }_{\mathbb{R}}f(\lambda )d\mathrm{E}(\lambda ).}$$

In case f 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

• Measurement of A yields a value in the interval [a, b].

This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's Axiom VII:

• The propositions of a quantum mechanical system correspond to the lattice of closed subspaces of H; the negation of a proposition V is the orthogonal complement V^⊥.

The space Q of quantum propositions is also sequentially complete: any pairwise-disjoint sequence {V_i}_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 {V_i}_i is the closed internal direct sum.

Standard semantics

The standard semantics of quantum logic is that quantum logic is the logic of projection operators in 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,

• ¬p is the orthogonal complement of p (since for those states, the probability of observing p, P(p) = 0),
• p∧q is the intersection of p and q, and
• p∨q = ¬(¬p∧¬q) refers to states that superpose p and q.

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

Differences with classical 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

⊤ = p∨q and

⊥ = p∧q

have exactly one solution, namely the set-theoretic complement of p. In 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 a filtering property:

if a≤b and q(a) = ⊤, then q(b) = ⊤.

Failure of distributivity

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 the disjuncts is true.

For example, consider a simple one-dimensional particle with position denoted by x and momentum by p, and define observables:

• a — |p| ≤ 1 (in some units)
• b — x ≤ 0
• c — x ≥ 0

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 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 a is true.

To understand more, let p₁ and p₂ 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 |p₁|↾_≥1 = |p₂|↾_≥1 = 0 (this holds even if we defined p differently so as to make such states possible; also, a ∧ b corresponds to |p₁|↾_≥1=0 and p₂=0). Meanwhile, a corresponds to states with |p|↾_≥1 = 0. As an operator, p = p₁ + p₂, and nonzero |p₁|↾_≥1 and |p₂|↾_≥1 might interfere to produce zero |p|↾_≥1. Such interference is key to the richness of quantum logic and quantum mechanics.

Relationship to quantum measurement

Mackey observables

Given a orthocomplemented lattice Q, a Mackey observable φ 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

${\displaystyle \phi \left(\bigcup _{i=1}^{\mathrm{\infty }}{S}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\phi (S_i).}$

Equivalently, a Mackey observable is a projection-valued measure on R.

Theorem (Spectral theorem). If Q 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.

Quantum probability measures

Main articles: Gleason's theorem and Quantum statistical mechanics

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

${\displaystyle \mathrm{P}\left(\underset{i=1}{\overset{\mathrm{\infty }}{\bigvee }}{E}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\mathrm{P}(E_i).}$

Every quantum probability measure on the closed subspaces of a Hilbert space is induced by a density matrix — a nonnegative operator of trace 1. Formally,

Theorem. Suppose Q is the lattice of closed subspaces of a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure P on Q there exists a unique trace class operator S such that $${\displaystyle \mathrm{P}(E)=\mathrm{Tr}(SE)}$$ for any self-adjoint projection E in Q.

Relationship to other logics

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 a Boolean algebra, which is then amenable to classical logic.

Limitations

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 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 is known, however, that System BV, a deep inference fragment of linear logic that is very close to quantum logic, can handle arbitrary discrete spacetimes.