Quantum Measurement Formalism (my notes on density operators and POVMs)

Jean‐Pierre Gazeau, Bárbara Heller

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple ormore elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched.

Haṡok Chang

Z. Kurucz

Positive operator valued measurements (POVMs) are the most general measurements that can be performed on quantum systems.A POVM is described by set of positive operators instead of orthogonal projections, and it can always be traced back to joint von Neumann measurements on the system and on an ancilla attached to it (Busch et. al, 1991;Peres, 1993).* Note on nomenclature: subscript letters denote the appropriate state spaces for vectors and density operators throughout the document.The same symbol with different subscripts intends to mean that the same quantum information is represented on different systems.

Jay Lawrence

I.F. Staricek

Kenji Nakahira

Quantum theory combines density matrices, Born probabilities, tensor-product composites, positive-operator-valued measures (POVMs), and quantum channels. In a finite-dimensional causal operational theory, we prove that two postulates suffice: local input-output statistics identify channels, and every state admits an equivalent-system purification, unique up to reversible dynamics. The full complex quantum formalism follows; every consistent probability rule is realized as a POVM, so measurement no-restriction is derived rather than assumed.