Modelling quantum measurements without superposition

Abstract

Superposition is the core feature that sets quantum theory apart from classical physics. Here, we investigate whether sets of quantum measurements can be modelled by using only devices that are operationally classical, in the sense that they have no superposition properties. This leads us to propose classical measurement models, which we show to be stronger than commutative measurements but weaker than joint measurability. We determine both the exact depolarisation noise rate and the measurement loss rate at which the all projective measurements in $d$-dimensional quantum theory admit a classical model. For finite sets of quantum measurements we develop methods both for constructing classical models and for falsifying the existence of such model via prepare-and-measure setups. Furthermore, we show that this concept also has operational implications. For that, we consider whether quantum measurements with classical side-information can be implemented in sequence without causing a disturbance and we show that classical models imply an affirmative answer. Our work sheds light on superposition as a resource for quantum measurement devices.

Figures (3)

• Figure 1: Classifying measurement models. Commuting observables, joint measurability and POVMs represent increasingly broad notions of measurement. We introduce classical models based on measurement devices without superposition features. These are intermediate between commutation and joint measurability.
• Figure 2: Classical measurement model. A random variable, $\lambda$, is used to direct the incoming quantum state, $\rho$, to a measurement device called $\mathcal{M}_\lambda$. This device performs a measurement in a fixed basis, $\{E_{k|\lambda}\}_k$. The outcome $k$, the device label $\lambda$ and the label of the quantum measurement $x$, are post-processed into the output $a$.
• Figure 3: Sequential measurements. A state $\rho$ is measured twice in sequence by devices that share a classical common cause $\lambda$. They statistically implement specific POVMs $\{A_a\}$ and $\{B_b\}$. The measurements are non-disturbing if there exists a way to implement $\{A_a\}$ without altering the outcome distribution of $\{B_b\}$.