Censure

From Wikipedia, the free encyclopedia

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

A censure is an expression of strong disapproval or harsh criticism. In parliamentary procedure, it is a debatable main motion that could be adopted by a majority vote. Among the forms that it can take are a stern rebuke by a legislature, a spiritual penalty imposed by a church, or a negative judgment pronounced on a theological proposition. It is usually non-binding (requiring no compulsory action from the censured party), unlike a motion of no confidence (which may require the referenced party to resign).

Parliamentary procedure

Explanation and use

Censure (main motion)

Requires second?	Yes
Debatable?	Yes
Amendable?	Yes
Vote required	Majority

The motion to censure is a main motion expressing a strong opinion of disapproval that could be debated by the assembly and adopted by a majority vote. According to Robert's Rules of Order (Newly Revised) (RONR), it is an exception to the general rule that "a motion must not use language that reflects on a member's conduct or character, or is discourteous, unnecessarily harsh, or not allowed in debate." Demeter's Manual notes, "It is a reprimand, aimed at reformation of the person and prevention of further offending acts." While there are many possible grounds for censuring members of an organization, such as embezzlement, absenteeism, drunkenness, and so on, the grounds for censuring a presiding officer are more limited:

Serious grounds for censure against presiding officers (presidents, chairmen, etc.) are, in general: arrogation or assumption by the presiding officer of dictatorial powers – powers not conferred upon him by law – by which he harasses, embarrasses and humiliates members; or, specifically: (1) he refuses to recognize members entitled to the floor; (2) he refuses to accept and to put canonical motions to vote; (3) he refuses to entertain appropriate appeals from his decision; (4) he ignores proper points of order; (5) he disobeys the bylaws and the rules of order; (6) he disobeys the assembly's will and substitutes his own; (7) he denies to members the proper exercise of their constitutional or parliamentary rights.

More serious disciplinary procedures may involve fine, suspension, or expulsion. In some cases, the assembly may declare the chair vacant and elect a new chairman for the meeting; or a motion can be made to permanently remove an officer (depending on the rules of the assembly).

Procedure

If the motion is made to censure the presiding officer, then he must relinquish the chair to the vice-president until the motion is disposed. But during this time, the vice-president is still referred to as "Mr. Vice President" in debate, since a censure is merely a warning and not a proceeding that removes the president from the chair. An officer being censured is not referred to by name in the motion, but simply as "the president", "the treasurer", etc.

After a motion to censure is passed, the chair (or the vice-president, if the presiding officer is being censured) addresses the censured member by name. He may say something to the effect of, "Brother F, you have been censured by vote of the assembly. A censure indicates the assembly's disapproval of your conduct". ([at meetings.] This phrase should not be included as the cause for censure may have occurred outside of meetings.) "A censure is a warning. It is the warning voice of suspension or expulsion. Please take due notice thereof and govern yourself accordingly." Or, if the chair is being censured, the vice-president may say, "Mr. X, you have been censured by the assembly for the reasons contained in the resolution. I now return to you the presidency."

Politics

In politics, a censure is an alternative to more serious measures against misconduct or dereliction of duty.

Canada

Censure is an action by the House of Commons or the Senate rebuking the actions or conduct of an individual. The power to censure is not directly mentioned in the constitutional texts of Canada but is derived from the powers bestowed upon both Chambers through section 18 of the Constitution Act, 1867. A motion of censure can be introduced by any Member of Parliament or Senator and passed by a simple majority for censure to be deemed to have been delivered. In addition, if the censure is related to the privileges of the Chamber, the individual in question could be summoned to the bar of the House or Senate (or, in the case of a sitting member, to that member's place in the chamber) to be censured, and could also face other sanctions from the house, including imprisonment. Normally, censure is exclusively an on-the-record rebuke — it is not equivalent to a motion of no confidence, and a prime minister can continue in office even if censured.

Louis Riel faced Parliamentary censure for his role in the Red River Rebellion, and was expelled from Parliament 16 April 1874.

Japan

In Japan, a censure motion is a motion that can be passed by the House of Councillors, the upper house of the National Diet. No-confidence motions are passed in the House of Representatives, and this generally doesn't happen as this house is controlled by the ruling party. On the other hand, censure motions have been passed by opposition parties several times during the Democratic Party of Japan (DPJ) administrations from 2009. The motions were combined with a demand from the opposition to take a certain action, and a refusal to cooperate with the ruling party on key issues unless some actions were taken. 

For example, on 20 April 2012 the opposition Liberal Democratic Party (LDP), Your Party and New Renaissance Party submitted censure motions against ministers of Prime Minister Yoshihiko Noda's Democratic Party of Japan-controlled cabinet. They censured Minister of Defense Naoki Tanaka and Minister of Land Takeshi Maeda, and refused to cooperate with the government on passing an increase to Japan's consumption tax from 5% to 10%. Noda had "staked his political life" on passing the consumption tax increase, so on 4 June 2012, Noda reshuffled his cabinet and replaced Tanaka and Maeda.

On 28 August 2012, a censure motion was passed by the LDP and the New Komeito Party against Prime Minister Noda himself. The opposition parties were to boycott debate in the chamber, it means that bills passed in the DPJ-controlled House of Representatives cannot be enacted.

Australia

The Senate, the upper house of the Australian Parliament, has censured two Prime Ministers in recent decades that of Paul Keating and John Howard.

The Australian Attorney General George Brandis was censured on 2 March 2015 for his treatment of Human Rights Commission President Gillian Triggs.

Senator for Queensland Fraser Anning was censured for remarks he made about the Christchurch mosque shootings.

United Kingdom

In the UK The Crown cannot be prosecuted for breaches of the law even where it has no exemption, such as from the Health and Safety at Work etc. Act. A Crown Censure is the method by which the Health and Safety Executive records, but for Crown immunity, there would be sufficient evidence to secure a H&S conviction against the Crown.

United States

Censure is the public reprimanding of a public official for inappropriate conduct or voting behavior. When the president is censured, it serves only as a condemnation and has no direct effect on the validity of presidency, nor are there any other particular legal consequences. Unlike impeachment, censure has no basis in the Constitution or in the rules of the Senate and House of Representatives. It derives from the formal condemnation of either congressional body of their own members.

Chronology

To date, Andrew Jackson is the only sitting President of the United States to be successfully censured, although his censure was subsequently expunged from official records, and James K. Polk was also censured by the House of Representatives in 1848. Since 2017, several Members of Congress have introduced motions to censure President Donald Trump for various controversies, including as a possible substitute for impeachment during the Trump-Ukraine scandal, but none have been successful.

On 2 December 1954, Republican Senator Joseph McCarthy from Wisconsin was censured by the United States Senate for failing to cooperate with the subcommittee that was investigating him, and for insulting the committee that was recommending his censure.

On 10 June 1980, Democratic Representative Charles H. Wilson from California was censured by the House of Representatives for "financial misconduct", as a result of the "Koreagate" scandal of 1976. "Koreagate" was an American political scandal involving South Koreans seeking influence with members of Congress. An immediate goal seems to have been reversing President Richard Nixon's decision to withdraw troops from South Korea. It involved the KCIA (now the National Intelligence Service) funneling bribes and favors through Korean businessman Tongsun Park in an attempt to gain favor and influence. Some 115 members of Congress were implicated.

On 20 July 1983, Representatives Dan Crane, a Republican from Illinois, and Gerry Studds, a Democrat from Massachusetts, were censured by the House of Representatives for their involvement in the 1983 Congressional page sex scandal.

On 12 July 1999, the U.S. House of Representatives censured (in a 355-to-0 vote) a scientific publication titled "A Meta-analytic Examination of Assumed Properties of Child Sexual Abuse Using College Samples", by Bruce Rind, Philip Tromovich, and Robert Bauserman; (see Rind et al. controversy) which was published in the American Psychological Association's "Psychological Bulletin (July 1998).

On 31 July 2007, retired Army General Philip Kensinger was censured by the United States Army for misleading investigators of the Pat Tillman death in 2004.

On 6 July 2009, South Carolina Governor Mark Sanford was censured by the South Carolina Republican Party executive committee for traveling overseas on taxpayer funds to visit his mistress.

On 13 October 2009, the mayor of Sheboygan, Wisconsin, Bob Ryan, was censured due to a YouTube video that showed him making sexually vulgar comments about his sister-in-law taken at a bar on a cell phone. The censure was voted 15-0 by the Sheboygan Common Council. His powers were also quickly reduced by the Common Council, and he was ultimately removed from office two and a half years later in a recall election for continued improprieties in office.

In November 2009, members of the Charleston County Republican Party censured Republican Senator Lindsey Graham of South Carolina in response to his voting to bail out banks and other Wall Street firms, and for his sentiments on immigration reform and cap-and-trade climate change legislation.

On 2 December 2010, Democratic Rep. Charlie Rangel from the State of New York was censured after an ethics panel found he violated House rules, specifically failing to pay taxes on a villa in the Dominican Republic, improperly soliciting charitable donations, and running a campaign office out of a rent-stabilized apartment meant for residential use.

On 4 January 2010, members of the Lexington County Republican Party censured Senator Lindsey Graham of South Carolina for his support of government intervention in the private financial sector and for “debasing” longstanding Republican beliefs in economic competition.

On 22 January 2013, the Arizona Republican Party censured longtime Sen. John McCain for what it called his “long and terrible” record of voting with liberal Democrats on some issues.

Catholic Church

Canon law

In canon law, a censure is a penalty imposed primarily for the purpose of breaking contumacy and reintegrating the offender in the community.

The ecclesiastical censures are excommunication and interdict, which can be imposed on any member of the Church, and suspension, which only affects clerics.

Theological censure

In Catholic theology, a theological censure is a doctrinal judgment by which the church stigmatizes certain teachings detrimental to faith or morals.

Quantum logic

From Wikipedia, the free encyclopedia

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

 

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 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 complementary variables in quantum mechanics, such as position and momentum.

Quantum logic can be formulated either as a modified version of propositional logic or as a noncommutative and non-associative many-valued (MV) logic.

Quantum logic has 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. Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein. However, this idea had been around for some time and had been revived several years earlier by George Mackey's work on group representations and symmetry.

The more common view regarding quantum logic, however, is that it provides a formalism for relating observables, system preparation filters and states. In this view, the quantum logic approach resembles more closely the C*-algebraic 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).

Differences with classical logic

Quantum logic has some properties that 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 (using some system of units where the reduced Planck's 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]"

Note: The choice of p, q, and r in this example is intuitive but not formally valid (that is, p and (q or r) is also false here); see section "Quantum logic as the logic of observables" below for details and a valid example. 

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 each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So,

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

Thus the distributive law fails.

Introduction

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. The set of principles for manipulating these quantum propositions was called quantum logic by von Neumann and Birkhoff in their 1936 paper. George Mackey, in his 1963 book (also called Mathematical Foundations of Quantum Mechanics), attempted to provide a set of axioms for this propositional system as an orthocomplemented lattice. Mackey viewed elements of this set as potential yes or no questions an observer might ask about the state of a physical system, questions that would be settled by some measurement. Moreover, Mackey defined a physical observable in terms of these basic questions. Mackey's axiom system is somewhat unsatisfactory though, since it assumes that the partially ordered set is actually given as the orthocomplemented closed subspace lattice of a separable Hilbert space. Constantin Piron, Günther Ludwig and others have attempted to give axiomatizations that do not require such explicit relations to the lattice of subspaces. 

The axioms of an orthocomplemented lattice are most commonly stated as algebraic equations concerning the poset and its operations. A set of axioms using instead disjunction (denoted as ${\displaystyle \cup }$) and negation (denoted as ${\displaystyle ^{\perp }}$) is as follows:

• ${\displaystyle a=a^{\perp \perp }}$
• ${\displaystyle \cup }$ is commutative and associative.
• There is a maximal element ${\displaystyle 1}$, and ${\displaystyle 1=b\cup b^{\perp }}$ for any ${\displaystyle b}$.
• ${\displaystyle a\cup (a^{\perp }\cup b)^{\perp }=a}$.

An orthomodular lattice satisfies the above axioms, and additionally the following one:

• The orthomodular law: If ${\displaystyle 1=(a^{\perp }\cup b^{\perp })^{\perp }\cup (a\cup b)^{\perp }}$ then ${\displaystyle a=b}$.

Alternative formulations include sequent calculi, and tableaux systems.

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 using the finite-dimensional spectral theorem.

Quantum logic as the logic of observables

One semantics of quantum logic is that quantum logic is the logic of boolean observables in quantum mechanics, where an observable p is associated with the set of quantum states for which p (when measured) is true with probability 1 (this completely characterizes the observable). 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 are a superposition of p and q.

Thus, 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 an 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 vanishes at x ≥ 0 with P(|p|≤1) = 1. Thus, a ∧ b and similarly a ∧ c are false, so (a ∧ b) ∨ (a ∧ c) is false. However, a ∧ (b ∨ c) equals a and might be true. 

To understand more, let p₁ and p₂ be the momenta for the restriction of the particle wave function to x < 0 and x ≥ 0 respectively (with the wave function zero outside of the restriction). Let ${\displaystyle |p|\!\upharpoonright _{>1}}$ be the restriction of |p| to momenta that are (in absolute value) >1.

(a ∧ b) ∨ (a ∧ c) corresponds to states with ${\displaystyle |p_{1}|\!\upharpoonright _{>1}=0}$ and ${\displaystyle |p_{2}|\!\upharpoonright _{>1}=0}$ (this holds even if we defined p differently so as to make such states possible; also, a ∧ b corresponds to ${\displaystyle |p_{1}|\!\upharpoonright _{>1}=0}$ and ${\displaystyle p_{2}=0}$). As an operator, ${\displaystyle p=p_{1}+p_{2}}$, and nonzero ${\displaystyle |p_{1}|\!\upharpoonright _{>1}}$ and ${\displaystyle |p_{2}|\!\upharpoonright _{>1}}$ might interfere to produce zero ${\displaystyle |p|\!\upharpoonright _{>1}}$. Such interference is key to the richness of quantum logic and quantum mechanics.

The propositional lattice of a classical system

The so-called Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest case of a single particle moving in R³, the state space is the position-momentum space R⁶. We will merely note here that an observable is some real-valued function f on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the value f(x), that is the value of f for some particular system state x, is obtained by a process of measurement of f. 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."

It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to that of some Boolean algebra of subsets of the state space. By logic in this context we mean the rules that relate set operations and ordering relations, such as de Morgan's laws. These are analogous to the rules relating boolean conjunctives and material implication in classical propositional logic. For technical reasons, we will also assume that the algebra of subsets of the state space is that of all Borel sets. The set of propositions is ordered by the natural ordering of sets and has a complementation operation. In terms of observables, the complement of the proposition {f ≥ a} is {f < a}. 

We summarize these remarks as follows: 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 of elements of the lattice has a least upper bound, specifically the set-theoretic union:

${\displaystyle \operatorname {LUB} (\{E_{i}\})=\bigcup _{i=1}^{\infty }E_{i}.}$

The 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:

${\displaystyle f(A)=\int _{\mathbb {R} }f(\lambda )\,d\operatorname {E} (\lambda ).}$

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].

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

• The orthocomplemented lattice Q of propositions of a quantum mechanical system is the lattice of closed subspaces of a complex Hilbert space H where orthocomplementation of V is the orthogonal complement V^⊥.

Q is also sequentially complete: any pairwise disjoint sequence{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. 

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

${\displaystyle I=p\vee q}$

${\displaystyle 0=p\wedge q}$

have exactly one solution, namely the set-theoretic complement of p. In these equations I refers to the atomic proposition that is identically true and 0 the atomic proposition that is identically false. 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).

Having made these preliminary remarks, we turn everything around and attempt to define observables within the projection lattice framework and using this definition establish the correspondence between self-adjoint operators and observables: A Mackey observable is a countably additive homomorphism from the orthocomplemented lattice of the Borel subsets of R to Q. To say the mapping φ is a countably additive homomorphism means that for any sequence {S_i}_i of pairwise disjoint Borel subsets of R, {φ(S_i)}_i are pairwise orthogonal projections and

${\displaystyle \varphi \left(\bigcup _{i=1}^{\infty }S_{i}\right)=\sum _{i=1}^{\infty }\varphi (S_{i}).}$

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

Theorem. There is a bijective correspondence between Mackey observables and densely defined self-adjoint operators on H. 

This is the content of the spectral theorem as stated in terms of spectral measures.

Statistical structure

Imagine a forensics lab that 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_i}_i is a sequence of pairwise orthogonal elements of Q then

${\displaystyle \operatorname {P} \!\left(\sum _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\operatorname {P} (E_{i}).}$

The following highly non-trivial theorem is due to Andrew Gleason: 

Theorem. Suppose Q is 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

${\displaystyle \operatorname {P} (E)=\operatorname {Tr} (SE)}$

for any self-adjoint projection E in Q.

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 that preserves the orthocomplemented structure of Q, that is

${\displaystyle \alpha \!\left(\sum _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\alpha (E_{i})}$

for any sequence {E_i}_i of pairwise orthogonal self-adjoint projections. Note that this property implies monotonicity of α. If P is a quantum probability measure on Q, then E → α(E) is also a quantum probability measure on Q. By the Gleason theorem characterizing quantum probability measures quoted above, any automorphism α induces a mapping α* on the density operators by the following formula:

${\displaystyle \operatorname {Tr} (\alpha ^{*}(S)E)=\operatorname {Tr} (S\alpha (E)).}$

The mapping α* is bijective and preserves convex combinations of density operators. This means

${\displaystyle \alpha ^{*}(r_{1}S_{1}+r_{2}S_{2})=r_{1}\alpha ^{*}(S_{1})+r_{2}\alpha ^{*}(S_{2})\quad }$

whenever 1 = r₁ + r₂ and r₁, r₂ are non-negative real numbers. Now we use a theorem of Richard V. Kadison: 

Theorem. Suppose β is a bijective map from density operators to density operators that is convexity preserving. Then there is an operator U on the Hilbert space that is either linear or conjugate-linear, preserves the inner product and is such that

${\displaystyle \beta (S)=USU^{*}}$

for every density operator S. In the first case we say U is unitary, in the second case U is anti-unitary.

Remark. This note is included for technical accuracy only, and should not concern most readers. The result quoted above is not directly stated in Kadison's paper, but can be reduced to it by noting first that β extends to a positive trace preserving map on the trace class operators, then applying duality and finally applying a result of Kadison's paper.

The operator U is not quite unique; if r is a complex scalar of modulus 1, then r U will be unitary or anti-unitary if U is and will implement the same automorphism. In fact, this is the only ambiguity possible. 

It follows that automorphisms of Q are in bijective correspondence to unitary or anti-unitary operators modulo multiplication by scalars of modulus 1. Moreover, we can regard automorphisms in two equivalent ways: as operating on states (represented as density operators) or as operating on Q.

Non-relativistic dynamics

In non-relativistic physical systems, there is no ambiguity in referring to time evolution since there is a global time parameter. Moreover, an isolated quantum system evolves in a deterministic way: if the system is in a state S at time t then at time s > t, the system is in a state F_s,t(S). Moreover, we assume

• The dependence is reversible: The operators F_s,t are bijective.
• The dependence is homogeneous: F_s,t = F_s − t,0.
• The dependence is convexity preserving: That is, each F_s,t(S) is convexity preserving.
• The dependence is weakly continuous: The mapping R→ R given by t → Tr(F_s,t(S) E) is continuous for every E in Q.

By Kadison's theorem, there is a 1-parameter family of unitary or anti-unitary operators {U_t}_t such that

${\displaystyle \operatorname {F} _{s,t}(S)=U_{s-t}SU_{s-t}^{*}}$

In fact.

Theorem. Under the above assumptions, there is a strongly continuous 1-parameter group of unitary operators {U_t}_t such that the above equation holds. 

Note that it follows easily from uniqueness from Kadison's theorem that

${\displaystyle U_{t+s}=\sigma (t,s)U_{t}U_{s}}$

where σ(t,s) has modulus 1. Now the square of an anti-unitary is a unitary, so that all the U_t are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by modifying each U_t by a scalar of modulus 1.)

Pure states

A convex combination of statistical states S₁ and S₂ is a state of the form S = p₁ S₁ +p₂ S₂ where p₁, p₂ are non-negative and p₁ + p₂ =1. Considering the statistical state of system as specified by lab conditions used for its preparation, the convex combination S can be regarded as the state formed in the following way: toss a biased coin with outcome probabilities p₁, p₂ and depending on outcome choose system prepared to S₁ or S₂.

Density operators form a convex set. The convex set of density operators has extreme points; these are the density operators given by a projection onto a one-dimensional space. To see that any extreme point is such a projection, note that by the spectral theorem S can be represented by a diagonal matrix; since S is non-negative all the entries are non-negative and since S has trace 1, the diagonal entries must add up to 1. Now if it happens that the diagonal matrix has more than one non-zero entry it is clear that we can express it as a convex combination of other density operators. 

The extreme points of the set of density operators are called pure states. If S is the projection on the 1-dimensional space generated by a vector ψ of norm 1 then

${\displaystyle \operatorname {Tr} (SE)=\langle E\psi |\psi \rangle }$

for any E in Q. In physics jargon, if

${\displaystyle S=|\psi \rangle \langle \psi |,}$

where ψ has norm 1, then

${\displaystyle \operatorname {Tr} (SE)=\langle \psi |E|\psi \rangle .}$

Thus pure states can be identified with rays in the Hilbert space H.

The measurement process

Consider a quantum mechanical system with lattice Q that 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

${\displaystyle {\frac {1}{\operatorname {Tr} (ES)}}ESE.}$

• If the result of the measurement is F

${\displaystyle {\frac {1}{\operatorname {Tr} ((I-E)S)}}(I-E)S(I-E).}$

(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:

${\displaystyle \operatorname {M} _{E}(S)=ESE+(I-E)S(I-E).}$

We see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as described above, is a special case of quantum operations.

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 Belavkin  (see also Bouten et al.). A similar approach is provided by the consistent histories formalism. On the other hand, quantum logics derived from many-valued logic extend its range of applicability to irreversible quantum processes 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).

In 2004, Prakash Panangaden described how to capture the kinematics of quantum causal evolution using System BV, a deep inference logic originally developed for use in structural proof theory. Alessio Guglielmi, Lutz Straßburger, and Richard Blute have also done work in this area.