Coexistence of quantum operations (original) (raw)

Simultaneous measurement and joint probability distributions in quantum mechanics

Foundations of Physics, 1979

The problem of simultaneous measurement of incompatible obseruables in quantum mechanics is studied on the one hand from the uiewpoint ofan e:;illrt, i,treet*lenl of quanlum mechanic-c and on !h: ttlter 4snd :;tc.t!it:. i,.:,.: .. theory of measurement. It is argued that it is preciseiy sucit c iheory of measurement thct should prouide a meaning to the axiomatically introduced c'oncepts, especially to the concept of obseruable. Defining an obseruable as a class of measurement procedures yielding a certain prescribed result for the probability distribution of the set of talues of some quantity (to be desuibed by the set of eigenualues of some Hermitian operator), this notion is extended to joint probability distributions of incompatible obseruables. It is shown that such an extension is possible on the basis of a theory of measurement, under the profiso that in simultaneously measuring such obseruables there is a disturbance of the measurement results of the one obseroable, caused b,,-the presence of the measuring instrument of the other obseruable. This has as a consequence that the joint probability distribution cannot obey the marginal distribution laws usually imposed. This result is of great importance in exposing quantum mechanics as an axiomatized theory, since ouerlooking it seems to prohibit an axiomatic description of simultaneous measurement of inconpatible obsert:ables by quantum mechanics.

Quantum conditional probability

THEORIA. An International Journal for Theory, History and Foundations of Science, 2013

We argue that quantum theory does not allow for a generalization of the notion of classical conditional probability by showing that the probability defined by the Lüders rule, standardly interpreted in the literature as the quantum-mechanical conditionalization rule, cannot be interpreted as such.

Classical Probability and Quantum Outcomes

Axioms, 2014

There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.

Conditional Probabilities and Collapse in Quantum Measurements

International Journal of Theoretical Physics, 2008

We show that including both the system and the apparatus in the quantum description of the measurement process, and using the concept of conditional probabilities, it is possible to deduce the statistical operator of the system after a measurement with a given result, which gives the probability distribution for all possible consecutive measurements on the system. This statistical operator, representing the state of the system after the first measurement, is in general not the same that would be obtained using the postulate of collapse.

On the notion of coexistence in quantum mechanics

2010

Abstract The notion of coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables.

Discrimination of measurement contexts in quantum mechanics

Physics Letters A, 2011

We demonstrate that it is possible to discern the way that has been followed to measure a quantum observable that can be expressed in terms of different products of observables, whereas no such discrimination is possible by assigning predetermined values. Specifically we show how to distinguish different routes (contexts) to measure C = AB = A ′ B ′ , when C, A, B and C, A ′ , B ′ commute with each other, but A and B do not commute with A ′ and B ′ .

Probabilities and Epistemic Operations in the Logics of Quantum Computation

Entropy

Quantum computation theory has inspired new forms of quantum logic, called quantum computational logics, where formulas are supposed to denote pieces of quantum information, while logical connectives are interpreted as special examples of quantum logical gates. The most natural semantics for these logics is a form of holistic semantics, where meanings behave in a contextual way. In this framework, the concept of quantum probability can assume different forms. We distinguish an absolute concept of probability, based on the idea of quantum truth, from a relative concept of probability (a form of transition-probability, connected with the notion of fidelity between quantum states). Quantum information has brought about some intriguing epistemic situations. A typical example is represented by teleportation-experiments. In some previous works we have studied a quantum version of the epistemic operations “to know”, “to believe”, “to understand”. In this article, we investigate another epi...

Quantum conditional operations

Physical Review A, 2016

An essential element of classical computation is the "if-then" construct, that accepts a control bit and an arbitrary gate, and provides conditional execution of the gate depending on the value of the controlling bit. On the other hand, quantum theory prevents the existence of an analogous universal construct accepting a control qubit and an arbitrary quantum gate as its input. Nevertheless, there are controllable sets of quantum gates for which such a construct exists. Here we provide a necessary and sufficient condition for a set of unitary transformations to be controllable, and we give a complete characterization of controllable sets in the two dimensional case. This result reveals an interesting connection between the problem of controllability and the problem of extracting information from an unknown quantum gate while using it.

Quantum mechanics as a noncommutative representation of classical conditional probabilities

Journal of Mathematical Physics

The aim of this paper is to analyze the reconstructability of quantum mechanics from classical conditional probabilities representing measurement outcomes conditioned on measurement choices. We will investigate how the quantum mechanical representation of classical conditional probabilities is situated within the broader frame of noncommutative representations. To this goal, we adopt some parts of the quantum formalism and ask whether empirical data can constrain the rest of the representation to conform to quantum mechanics. We will show that as the set of empirical data grows conventional elements in the representation gradually shrink and the noncommutative representations narrow down to the unique quantum mechanical representation.