Incompatibility as a Resource for Programmable Quantum Instruments

Quantum instruments represent the most general type of quantum measurement, as they incorporate processes with both classical and quantum outputs. In many scenarios, it may be desirable to have some “on-demand” device that is capable of implementing one of many possible instruments whenever the experimenter desires. We refer to such objects as programmable instrument devices (PIDs), and this paper studies PIDs from a resource-theoretic perspective. A physically important class of PIDs are those that do not require quantum memories to implement, and these are naturally “free” in this resource theory. Additionally, these free objects correspond precisely to the class of unsteerable channel assemblages in the study of channel steering. The traditional notion of measurement incompatibility emerges as a resource in this theory since any PID controlling an incompatible family of instruments requires a quantum memory to build. We identify an incompatibility preorder between PIDs based on whether one can be transformed into another using processes that do not require additional quantum memories. Necessary and sufficient conditions are derived for when such transformations are possible based on how well certain guessing games can be played using a given PID. Ultimately our results provide an operational characterization of incompatibility, and they offer semi-device-independent tests for incompatibility in the most general types of quantum instruments.

Popular Summary

Quantum memories are a crucial resource for quantum information processing, as they can store quantum systems for a considerable length of time without causing them to degenerate into classical signals. In this work, we conduct a theoretical analysis on the role of quantum memories in programmable quantum devices. A programmable quantum device is a device capable of implementing a particular quantum process according to classical instructions. We find that a programmable device requires a quantum memory (with a nonnegligible lifetime) to build if and only if it can implement “incompatible” processes. These processes include and generalize the well-known notion of incompatible observables in quantum mechanics. The connections established in this paper show how the real-world resource of quantum memories can be related to fundamental phenomena that lie at the very foundations of quantum mechanics, such as measurement incompatibility and quantum steering. With a new design of guessing games, we propose a scheme to experimentally test when a given programmable device can implement incompatible processes, even when the specific inner workings of the device are unknown to the experimenter. The guessing games demonstrate a strict quantum advantage, so that every programmable device that requires a quantum memory to build can be operationally distinguished from those that can be simulated with a classical memory.

Figure 1

A general controllable quantum device applies an instrument $\{{\mathrm{\Lambda }}_{x_1|x_0}{\}}_{x_1}$ to the quantum input whenever a particular program $x_0$ is chosen. The characteristic time $\mathrm{\Delta }{t}_{\mathrm{D}}:=t_1-t_0$ is known as the quantum delay time of the device, and it measures how quickly the device functions as a q-to-q channel. The device is said to be fully programmable if the program is free to arrive at any time $t_2$, even outside the interval $[t_0,t_1)$, and we refer to such a device as a programmable instrument device. Given that classical memories are freely available, the full programmability of a PID is essentially its ability to withstand an arbitrarily late arrival of the program, termed the late-program assumption. In the framework of Ref. [44], a PID represents a so-called “multiprocess.”

Figure 2

The control device could also be split between spatially separated parties Alice and Evan. The nonsignaling constraint from Evan’s control signal to Alice’s channel output naturally arises if the spatial separation is so large that Evan’s signal propagation time exceeds the anticipated quantum delay time between Alice’s local input and output.

Figure 3

Decomposition of a general PID [(a) and (b)] and that of a simple PID [(c)]. Solid arrows stand for quantum systems and hollow arrows for classical systems. Time flows from left to right. The opaque rips indicate that the quantum (left) and classical (right) parts of the devices are temporally separated under the late-program assumption. (a) A general PID $\mathbb{\Lambda }$ can be realized by connecting one output system $E$ of a broadcast channel ${\mathcal{E}}^{A_0\to A_{1}E}$ to a PMD $\mathbb{M}$ using a quantum memory channel ${\mathrm{id}}^E$. As such, the inner working of the PID can be understood as a process of channel steering (see Sec. 4a). (b) The general PID can be represented in a different configuration, with the system $A_1$ displaced downwards and the quantum memory channel subsumed within the PMD $\mathbb{M}$. (c) A simple PID can be realized with a “mother” instrument $\mathcal{G}$ and a classical channel $p$. In “steering” terms, the simple PID implements an unsteerable channel assemblage. Note that the simple PID has the same quantum delay time $\mathrm{\Delta }{t}_{\mathrm{D}}\approx 0$ between $A_0$ and $A_1$ as the general PID does. However, the quantum memory of the general PID needs a much longer lifetime $\mathrm{\Delta }{t}_{\mathrm{QM}}\gg \mathrm{\Delta }{t}_{\mathrm{D}}$ in general to accept a late arriving classical program $x_0$, unlike the simple PID.

Figure 4

The free simulation (blue) of a PMD $\mathbb{N}$ (green) using another PMD $\mathbb{M}$ (yellow) according to Eq. (5). It has been shown that nonsimple PMDs cannot be freely simulated by simple PMDs [20]. Simulations in this form are precisely those that can be realized without additional quantum memories under the late-program assumption.

Figure 5

The free simulation (blue) of a PID $\mathbb{\Gamma }$ (green) using another PID $\mathbb{\Lambda }$ (yellow) according to Eq. (6). The simulation is composed of (i) pre-, post-, and side processing of the quantum part, (ii) pre-, post-, and side processing of the classical part, and (iii) an external classical memory connecting the quantum and classical parts. Under the late-program assumption, free simulations of PIDs represent the most general transformations given that using any quantum memory with a lifetime exceeding $\mathrm{\Delta }{t}_{\mathrm{D}}\approx 0$ is forbidden. This figure reduces to Fig. 4 when the quantum output systems $A_1$ and $B_1$ are trivial.

Figure 6

A nontransient guessing game between Alice (the player) and Bob (the referee). The setting of the game is specified by Bob’s POVM $\mathcal{M}\equiv \{M_{m,n}{\}}_{m,n}$. Alice’s strategy to the game is represented by a PID ${\mathbb{\Lambda }}^{\prime }$ (green), which is freely simulated by an actual PID $\mathbb{\Lambda }$ held in her hand. She wins the game whenever she makes a correct guess at one of Bob’s outcome indices $n$ given the other index $m$, i.e., whenever $n^{\prime }=n$.

Figure 7

A postinformation guessing game between Alice (the player) and Bob (the referee). The setting of the game is specified by Bob’s state ensemble $\varsigma \equiv \{{\sigma }_{m,n,l}{\}}_{m,n,l}$ and POVM $\mathcal{L}\equiv \{L_{l^{\prime }}{\}}_{l^{\prime }}$. Alice’s strategy to the game is represented by a PID ${\mathbb{\Lambda }}^{\prime }$ (green), which is freely simulated by an actual PID $\mathbb{\Lambda }$ held in her hand. She wins the game whenever she makes a correct guess at one of Bob’s indices $n$ while not altering the index $l$ given the index $m$, i.e., whenever $n^{\prime }=n$ and $l^{\prime }=l$.