Colloquium: Incompatible measurements in quantum information science

Some measurements in quantum mechanics disturb each other. This has puzzled physicists since the formulation of the theory, but only in recent decades has the incompatibility of measurements been analyzed in depth and detail, using the notion of joint measurability of generalized measurements. In this Colloquium joint measurability and incompatibility are reviewed from the perspective of quantum information science. The Colloquium starts by discussing the basic definitions and concepts. An overview on applications of incompatibility, such as in measurement uncertainty relations, the characterization of quantum correlations, or information processing tasks like quantum state discrimination, is then presented. Finally, emerging directions of research, such as a resource theory of incompatibility as well as other concepts to grasp the nature of measurements in quantum mechanics, are discussed.

Figure 1

Marginal form joint measurement of two POVMs. The data of the joint measurement are presented as a grid from which marginalization gives the data of the original pair of measurements.

Figure 2

Schematic view of measurement uncertainty relations. (a) For any quantum state $\rho$ the measurement $A$ yields results with a probability distribution $p(a)$; similarly, $B$ delivers a distribution $p(b)$. (b) A possible way to study the disturbance of $B$ through $A$ is to measure in sequence and compare the distributions $\tilde{p}(a)$ and $\tilde{p}(b)$ to $p(a)$ and $p(b)$. The reverse order of measurements can also be considered. (c) Since the measurements in (b) give probability distributions other than in (a), they can be described by different POVMs $C$ and $D$. But $C$ and $D$ are then jointly measurable by construction. The question arises: What is the best approximation of $A$ and $B$ by two jointly measurable POVMs $C$ and $D$?

Figure 3

Illustration of the incompatibility robustness. JM, compatible sets; INC, incompatible sets. The POVMs $\{A_{a|x}\}$ are mixed with arbitrary noise POVMs $\{N_{a|x}\}$ such that their mixture is compatible. The incompatibility robustness is the smallest $t$ that realizes such a decomposition; cf. Eq. (25).

Figure 4

An equally weighted coin toss between the auxiliary elements $B_{+|1}(\lambda ,\lambda )$ and $B_{+|2}(\lambda ,\lambda )$ with $\lambda =1/\sqrt{2}$ results in an effect with the Bloch vector $\lambda {\overrightarrow{a}}_2=\lambda (0,0,1)=(0,0,1/\sqrt{2})$. Here ${\overrightarrow{b}}_1$ and ${\overrightarrow{b}}_2$ are the Bloch vectors of $B_{+|1}(\lambda ,\lambda )$ and $B_{+|2}(\lambda ,\lambda )$. The vectors ${\overrightarrow{a}}_1$ and ${\overrightarrow{a}}_2$ represent the directions of the original measurements.

Figure 5

Schematic of the Bell scenario. A source distributes two particles to two parties named Alice and Bob. They perform measurements on it, and the questions arises as to whether or not the observed probability distribution can be described by a local hidden variable model.

Figure 6

Quantum state discrimination in different scenarios. (a) In minimum-error state discrimination the task is to guess the correct input label of the state with high probability. (b) In state discrimination with postinformation, the label $x$ is reviled after the measurement. Here the additional information can be used only to postprocess the classical measurement result. (c) In state discrimination with prior information, the partition $x$ of the label is received before the state, and hence a different measurement can be tailored to each subset $x$ of the labels. Adapted from [39].

Figure 7

Schematic of quantum random access codes. A sender holds $n$ bits of information $\mathbf{x}$ and encodes it into a single qudit via the quantum channel $\mathcal{E}(\mathbf{x})$. The receiver wants to recover a random bit $x_j$ from $\mathbf{x}$ by performing a measurement $\{A_{x_j|j}\}$ on the qudit.

Figure 8

Illustration for a joint measurement (a parent measurement) for the target POVMs $A$ and $B$. As long as $A$ and $B$ are jointly measurable, by first performing the Lüders measurement of $A$ and then, conditioned by the outcome $a$ obtained in the $A$ measurement, measuring a conditional POVM $\{{\tilde{B}}_{b|a}{\}}_b$ after the $A$ measurement, this sequential scheme realizes a joint measurement of $A$ and $B$.

Figure 9

The “auxiliary” Naimark dilation has a particular form that enables the following physical interpretation of an instrument $\mathcal{I}$ measuring the target POVM $A$ ([144]). The system state is coupled with a probe (the auxiliary system in the dilation) in the initial state $|{\xi }_0⟩⟨{\xi }_0|$. A unitary evolution mediated by the unitary $U$ then takes place followed by a sharp pointer measurement $Z$ of the probe, after which the probe is detached from the system.