Quantum resource theories

Quantum resource theories (QRTs) offer a highly versatile and powerful framework for studying different phenomena in quantum physics. From quantum entanglement to quantum computation, resource theories can be used to quantify a desirable quantum effect, develop new protocols for its detection, and identify processes that optimize its use for a given application. Particularly, QRTs have revolutionized the way we think about familiar properties of physical systems such as entanglement, elevating them from being just interesting fundamental phenomena to being useful in performing practical tasks. The basic methodology of a general QRT involves partitioning all quantum states into two groups, one consisting of free states and the other consisting of resource states. Accompanying the set of free states is a collection of free quantum operations arising from natural restrictions placed on the physical system, restrictions that force the free operations to act invariantly on the set of free states. The QRT then studies what information processing tasks become possible using the restricted operations. Despite the large degree of freedom in how one defines the free states and free operations, unexpected similarities emerge among different QRTs in terms of resource measures and resource convertibility. As a result, objects that appear quite distinct on the surface, such as entanglement and quantum reference frames, appear to have great similarity on a deeper structural level. This article reviews the general framework of a quantum resource theory, focusing on common structural features, operational tasks, and resource measures. To illustrate these concepts, an overview is provided on some of the more commonly studied QRTs in the literature.

Figure 1

In a quantum resource theory, the precious commodity is some physical property or phenomenon that emerges according to the principles of quantum mechanics. The paradigmatic example is quantum entanglement.

Figure 2

Quantum entanglement is a quantum resource in the “distant-lab” scenario where the free operations are LOCC. From [264].

Figure 3

A physically implementable CPTP map is one that can be realized by a sequence of channels, each having the form depicted in the figure. The four steps—(i) appending ancilla state ${\omega }_i$, (ii) applying a unitary $U_i$, (iii) performing a projective measurement, and (iv) classically postprocessing the measurement outcome—all must be free operations in the QRT.

Figure 4

A heuristic diagram of QRTs, classified according to the properties of their set of free states. Non-Gaussianity is an example of a QRT with a nonconvex set of free states. Entanglement theory is an example of a QRT that is convex but not affine. Real (vs complex) quantum mechanics is an example of an affine QRT that does not have a resource-destroying map, and athermality, asymmetry, and coherence, are examples of QRTs with a resource-destroying map. From [167].

Figure 5

Realization of a superchannel.

Figure 6

Quantum teleportation. )Single (respectively, double) line arrows corresponds to quantum (respectively, classical) communication. The static resource, a maximally entangled state, is converted to a dynamical resource via LOCC. From [27].

Figure 7

Heuristic description of $G$-covariant operations. The channel $\mathrm{\Phi }$ is $G$ covariant if the blue and purple pathways commute for all group elements.

Figure 8

Resource theory of Bell nonlocality.

Figure 9

Free superchannels.

Figure 10

(a) Multi-instrument: Quantum input and output in purple (double line) and classical input and output in black (single line). (b) Free (compatible) multi-instrument.

Figure 11

Testing region of a pair $(\mathbf{p},\mathbf{q})$.

Figure 12

Inclusion of two testing regions.