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Mathematical Theory of State Reduction in Quantum Mechanics(Quantum Stochastic Analysis and Related Fields)

Masanao Ozawa

Published Aug 1, 1996• 0 citations
OpenAlex

Abstract

A state reduct,ion is the $\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}_{1}\mathrm{e}$ change caused by a lueasurement on a quan- tum system conditional upon the outcome.A rigorous theorv of the state reduction is developed with $\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}\mathrm{a}}1$ formalism, $\mathrm{I}^{)}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}_{\mathrm{C}}\mathrm{a}1$ interpret.ation,and $\mathrm{n}\iota \mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{s}$ .A special $\mathrm{e}\mathrm{m}_{1^{J\mathrm{h}}}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{S}$ is on the pure state reduction which trans- forms a pure $1$) $\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}_{C}\mathrm{e}$ to the pure posterior state for every outcome.Mathe- $1\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}\mathrm{a}}1$ structure of general pure state reductions is discussed and it is $1$) $\mathrm{r}\mathrm{o}\mathrm{V}\mathrm{e}\mathrm{d}$ that every pure state reduction is decomposed into just two types, called tlle von Neumann-Davies $(\mathrm{N}\mathrm{D})\mathrm{t}\mathrm{y}\mathrm{l})\mathrm{e}$ and the Gordon-Louisell $(\mathrm{G}\mathrm{L})$ type; a state reduction $\psirightarrow\psi_{x}$ ' is of the ND type if the $1\mathrm{I}\mathrm{l}\mathrm{a}_{\mathrm{P}1^{\mathrm{i}\mathrm{n}}}$ ) $\mathrm{g}\psi$ ) $\mapsto P(x|\psi)1/2_{?}l' x$ ' is linear, where $P(x|\psi')$ is the probability density of the outcome $x$ , and of the GL $\mathrm{t}\mathrm{y}\mathrm{l}$ ) $\mathrm{e}$ if $\psi_{x}$ depends only on the outcome $x$ (independent of the prior state $\sqrt))$ .

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A state reduct,ion is the stat1e\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}_{1}\mathrm{e}stat1​e change caused by a lueasurement on a quan- tum system conditional upon the outcome.A rigorous theorv of the state reduction is developed with nlathematiCa1\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}\mathrm{a}}1nlathematiCa​1 formalism, I)hysiCa1\mathrm{I}^{)}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}_{\mathrm{C}}\mathrm{a}1I)hysiC​a1 interpret.ation,and nιodels\mathrm{n}\iota \mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{s}nιodels .A special em1JhasiS\mathrm{e}\mathrm{m}_{1^{J\mathrm{h}}}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{S}em1Jh​asiS is on the pure state reduction which trans- forms a pure 111) riorStatCe\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}_{C}\mathrm{e}riorStatC​e to the pure posterior state for every outcome.Mathe- 1natiCa11\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}\mathrm{a}}11natiCa​1 structure of general pure state reductions is discussed and it is 111) roVed\mathrm{r}\mathrm{o}\mathrm{V}\mathrm{e}\mathrm{d}roVed that every pure state reduction is decomposed into just two types, called tlle von Neumann-Davies (ND)tyl)e(\mathrm{N}\mathrm{D})\mathrm{t}\mathrm{y}\mathrm{l})\mathrm{e}(ND)tyl)e and the Gordon-Louisell (GL)(\mathrm{G}\mathrm{L})(GL) type; a state reduction \psirightarrowψx\psirightarrow\psi_{x}\psirightarrowψx​ ' is of the ND type if the 1IlaP1in1\mathrm{I}\mathrm{l}\mathrm{a}_{\mathrm{P}1^{\mathrm{i}\mathrm{n}}}1IlaP1in​ ) gψ\mathrm{g}\psigψ ) ↦P(x∣ψ)1/2?l′x\mapsto P(x|\psi)1/2_{?}l' x↦P(x∣ψ)1/2?​l′x ' is linear, where P(x∣ψ′)P(x|\psi')P(x∣ψ′) is the probability density of the outcome xxx , and of the GL tyl\mathrm{t}\mathrm{y}\mathrm{l}tyl ) e\mathrm{e}e if ψx\psi_{x}ψx​ depends only on the outcome xxx (independent of the prior state ))\sqrt)))​) .
A rigorous theory of quantum state reduction, the state change of the measured system caused by a measurement conditional upon the outcome of measurement, is developed fully within quantum mechanics without leading to the vicious circle relative to the von Neumann chain. For the basis of the theory, the local measurement theorem provides the joint probability distribution for the outcomes of local successive measurements on a noninteracting entangled system without assuming the projection postulate, and the quantum Bayes principle enables us to determine operationally the quantum state from a given information on the outcome of measurement.
A rigorous theory of quantum state reduction, the state change of the measured system caused by a measurement conditional upon the outcome of measurement, is developed fully within quantum mechanics without leading to the vicious circle relative to the von Neumann chain. For the basis of the theory, the local measurement theorem provides the joint probability distribution for the outcomes of local successive measurements on a noninteracting entangled system without assuming the projection postulate, and the quantum Bayes principle enables us to determine operationally the quantum state from a given information on the outcome of measurement.
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.