Various forms of optimality for quantum observables described as normalized positive-operator-valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables that are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient conditions are imposed on informationally complete POVMs within the set of pure states. We also discuss joint and sequential measurements of optimal quantum observables.
Two different approaches to a characterization of the degree of (in)compatibility of quantum observables are investigated. First, recent examples of the (partial) commutativity of spectral measures of incompatible observables are proved to be generic. The analysis is extended to the case of compatible or incompatible unsharp, or stochastic observables, leading to a general criterion for commutativity of position and momentum effects. Further, a recently proposed information theoretic quantification of the (in)compatibility of noncommuting observables is generalized, and the relation between ‘‘maximal information,’’ ‘‘minimal uncertainty,’’ partial commutativity, and strict correlation is further clarified. Both approaches are illustrated in a number of examples.
We give an overview of joint unsharp measurements
of non-commuting observables using positive operator
valued measures (POVMs). We exemplify the role
played by joint measurability of POVMs in entropic
uncertainty relation for Alice’s pair of non-commuting
observables in the presence of Bob’s entangled quantum
memory. We show that Bob should record the
outcomes of incompatible (non-jointly measurable)
POVMs in his quantum memory so as to beat the entropic
uncertainty bound. In other words, in addition
to the presence of entangled Alice–Bob state, implementing
incompatible POVMs at Bob’s end is necessary
to beat the uncertainty bound and hence predict
the outcomes of non-commuting observables with
improved precision. We also explore the implications
of joint measurability to validate a moment matrix
constructed from average pairwise correlations of
three dichotomic non-commuting qubit observables.
We prove that a classically acceptable moment matrix
– which ascertains the existence of a legitimate
joint probability distribution for the outcomes of all
the three dichotomic observables – could be realized if
and only if compatible POVMs are employed.
For any quantum observable (positive operator valued measure (POVM)), we show that its compatible quantum instruments can be identified with certain channels and can be seen as combinations of Lüders operations and channels. We prove that a POVM is rank-1 exactly when its compatible instruments are nuclear and their associate channels are entanglement-breaking. Any instrument can be maximally refined into a rank-1 instrument. We present a characterization for (minimal) pure realizations of instruments and characterize completely the minimal pure measurement models of POVMs. The standard model of quantum measurement theory is generalized for arbitrary observables. Finally, we study post measurement states.
One of the basic lessons of quantum theory is that one cannot obtain information on an unknown quantum state without disturbing it. Hence, by performing a certain measurement, we limit the other possible measurements that can be effectively implemented on the original input state. It has been recently shown by two of the authors of the present article [T. Heinosaari and T. Miyadera, Phys. Rev. A 91, 022110 (2015)] that one can implement sequentially any device, either channel or observable, which is compatible with the first measurement. In this work, we prove that this can be done, apart from some special cases, only when the succeeding device is implemented on a larger system than just the input system. This means that some part of the still available quantum information has been flown to the environment and cannot be gathered by accessing the input system only. We characterize the size of the post-measurement system by determining the class of measurements for the observable in question that allow the subsequent realization of any measurement process compatible with the said observable. We also study the class of measurements that allow the subsequent realization of any observable jointly measurable with the first one and show that these two classes coincide when the first observable is extreme.
This article introduces the concepts of multi-observables and multi-instruments in quantum mechanics. A multi-observable
A (multi-instrument
I) has an outcome space of the form
Ω=Ω1×⋯×Ωn and is denoted by
Ax1⋯xn (
Ix1⋯xn) where
(x1,…,xn)∈Ω. We also call
A (
I) an
n-observable (
n-instrument) and when
n=2 we call
A (
I) a bi-observable (bi-instrument). We point out that bi-observables
A (
I) and bi-instruments have been considered in past literature, but the more general case appears to be new. In particular, two observables (instruments) have been defined to coexist or be compatible if they possess a joint bi-observable (bi-instrument). We extend this definition to
n observables and
n instruments by considering joint marginals of
n-observables and joint reduced marginals of
n-instruments. We show that a
n-instrument measures a unique
n-observable and if a finite umber of instruments coexist, then their measured observables coexist. We prove that there is a close relationship between a nontrivial
n-observable and its parts. Moreover, a similar result holds for instruments. We next show that a natural definition for the tensor product of a finite number of instruments exist and possess reasonable properties. We then discuss sequential products of a finite number of observables and instruments. We present various examples such as Kraus, Holevo and Lüders instruments.