We consider pairs of discrete quantum observables (POVMs) and analyze the relation between the notions of nondisturbance, joint measurability, and commutativity. We specify conditions under which these properties coincide or differ—depending, for instance, on the interplay between the number of outcomes and the Hilbert space dimension or on algebraic properties of the effect operators. We also show that (non-)disturbance is, in general, not a symmetric relation and that it can be decided and quantified by means of a semidefinite program.
A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to
MMs on finite-dimensional Hilbert spaces. Suppose we want to measure an observable
A whose outcomes
Ax are represented by positive operators (effects) on a Hilbert Space
H. We call
H the base or object system. We interact
H with a probe system on another Hilbert space
K by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable
F whose outcomes
Fx are measured by an apparatus that is frequently (but need not be) classical in practice. The
MM protocol gives a method for determining the probability of an outcome
Ax for any state of
H in terms of the outcome
Fx. The interaction channel usually entangles this state with an initial probe state of
K that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing
MMs. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a
d-dimensional Hilbert space to each of the
n possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when
n>d, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
In the general theory of quantum measurement, one associates a positive semidefinite operator on a
d-dimensional Hilbert space to each of the
n possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert--Schmidt orthogonal, but when
n>d, orthogonality is restricted by positivity. This restriction has consequences which are not present from a classical perspective; in particular, we find it allows us to more precisely state the old quantum adage that it is impossible to extract information from a system without disturbance. Restricting attention throughout to the Luders state updating rule, we consider three properties of a measurement: its disturbance, a measure of how the expected post-measurement state deviates from the input state, its purity gain, a measure of the expected reduction of uncertainty resulting from measurement, and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. We show that these quantities satisfy a simple algebraic relation amounting to an expression of an information-disturbance trade-off. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
The incompatibility between quantum measurements (as mathematically represented by positive operator-valued measures, i.e., POVMs) is a key feature of quantum mechanics and is intrinsically related to the noncommutativity of operators. For both theoretical and practical considerations, it is desirable to quantify the degree of incompatibility between quantum measurements, and considerable effort has been devoted to this issue. In this paper, we provide a novel approach to measurement incompatibility by exploiting the Lüders channels derived from POVMs and employing the measurement disturbance to quantify incompatibility. This is achieved by constructing an approximately joint measurement for a pair of POVMs, which is an enlarged POVM with the correct marginal property for one of the two POVMs but not necessarily for the other. The degree of failure of the marginal property for the other POVM is a kind of measurement disturbance and can be naturally interpreted as a quantifier of the incompatibility between the two POVMs. We reveal basic properties of this quantifier of measurement incompatibility, identify its maximal value in some cases, compare it with several popular measures in the literature, and illustrate it with some typical examples. Some related open issues are also discussed.
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for simultaneous measurements of their observables. For this purpose, all the possible state changes caused by measurements of an observable are characterized, and the notion of disturbance in measurement is formalized in terms of operations derived by the measuring interaction.
We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. Specifically, for a pair of observables
A and
B with purely discrete spectra, we compare the following two probability distributions: one resulting from a measurement of
A followed by a measurement of
B on a given state and the other obtained from a measurement of
B alone on the same state. We show that maximizing the distance between these two distributions over all states yields a valid measure of the incompatibility of observables
A and
B, which is zero if and only if they commute and is strictly greater than zero (and less than or equal to one) otherwise. For finite-dimensional systems, we obtain a tight upper bound on the incompatibility of any pair of observables and show that the bound is attained when the observables are totally nondegenerate and associated with mutually unbiased bases. In the process, we also establish an important relation between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of nonprojective measurements. In particular, we obtain a nontrivial upper bound on the incompatibility of one L\"uders instrument with another.