Positive Operator Valued Measures and Feller Markov kernels

Abstract

A Positive Operator Valued Measure (POVM) is a map F : B(X ) → L + s (H ) from the Borel σ -algebra of a topological space X to the space of positive selfadjoint operators on a Hilbert space H . We assume X to be Hausdorff, locally compact and second countable and prove that a POVM F is commutative if and only if it is the smearing of a spectral measure E by means of a Feller Markov kernel. Moreover, we prove that the smearing can be realized by means of a strong Feller Markov kernel if and only if F is uniformly continuous. Finally, we prove that a POVM which is norm bounded by a finite measure ν admits a strong Feller Markov kernel.That provides a characterization of the smearing which connects a commutative POVM F to a spectral measure E and is relevant both from the mathematical and the physical viewpoint since smearings of spectral measures form a large and very relevant subclass of POVMs: they are paradigmatic for the modeling of certain standard forms of noise in quantum measurements, they provide optimal approximators as marginals in joint measurements of incompatible observables [21], they are important for a range of quantum information processing protocols, where classical post-processing plays a role [30].The mathematical and physical relevance of the results is discussed and particular emphasis is given to the connections between the Markov kernel and the imprecision of the measurement process.

Joint measurability through Naimark’s dilation theorem

“…are the unsharp position and momentum observables respectively ( [18,36,15,8,13,9]). Notice that the maps µ ∆ (q) := (1 ∆ * |g| 2 )(q) andμ ∆ (p) := (1 ∆ * |ĝ| 2 )(p) define two Markov kernels ( [8,9,13]). Now, we can define the isometry…”

“…[13,5]). A POVM F : B(X) → L + s (H) is commutative if and only if there exists a bounded self-adjoint operator A = λ dE λ with spectrum σ(A) ⊂ [0, 1], a subset Γ ⊂ σ(A), E(Γ) = 1, a ring R which generates B(X) and a Feller Markov Kernel µ : Γ × B(X) → [0, 1] such that…”

Two-moment characterization of spectral measures on the real line

“…∎ Remark 4. 5 If the element a is invertible, then in this case we can give another proof of the implication (ii)⇒(i) of Theorem 4.4, which relies on a result of Petz. Without loss of generality, we can assume that 0 < p < q.…”

“…In particular, this is the case in quantum information theory and in quantum optics (to represent measurement statistics). Among the papers undertaking this line of research, the following are noteworthy [37,22,31,14,3,4,30,25,5,6].…”

Two-moment characterization of spectral measures on the real line

“…In view of above it is important from the mathematical and the physical view-point to study the relationships between semispectral measures and spectral measures. The papers [27,15,2,9,3,4,22,5,6,18] are some examples from this line of research. In particular, in [22] J. Kiukas, P. Lahti and K. Ylinen asked the following question: When is a positive operator measure projection valued?…”