Quantifying superpositions of quantum evolutions

Quantum mechanics allows coherent superposition of different states of matter. This quality is responsible for major nonclassical phenomena that occur in quantum systems. Beyond states, coherent superpositions are also possible among quantum evolutions. We characterize such superpositions here. A resource theoretic framework is developed to quantify superposition present in an arbitrary quantum evolution. In addition to characterization, the framework considers superposition as a quantum resource. This resource can be exploited to perform certain quantum tasks that are otherwise impossible. We identify maximally resourceful evolutions and demonstrate how these could enable one to implement arbitrary quantum operations and superoperations. We also discuss the roles of superposition to exhibit nonclassical behaviors present in evolutions; for example, acausality, temporal Bell correlations, and indefinite temporal and causal orders.

Figure 1

A quantum operation on a quantum state ${\eta }_S$, say $\mathrm{\Phi }$, is expressed using operator-sum-representation [42], as ${\eta }_S⟶{\eta }_S^{\prime }=\mathrm{\Phi }\left({\eta }_S\right)={\sum }_m{E}_m{\eta }_S{E}_m^{†}$, where the $E_m$'s are known as operation elements (also Kraus operators) and satisfy the condition ${\sum }_m{E}_m^{†}{E}_m\le \mathbb{I}$. It can be implemented using a quantum circuit, as depicted above, by (i) attaching system ${\eta }_S$ with an ancilla in a state ${|0\rangle \langle 0|}_A$, (ii) applying a global unitary operation $U_{SA}$ on the joint system, then (iii) projecting or tracing out the ancilla, such that $\mathrm{\Phi }\left({\eta }_S\right)={Tr}_A\left[U_{SA}({\eta }_S\otimes |0\rangle \langle 0{|}_A)U_{SA}^{†}\right]$. Note that the operation elements are $E_m={\langle m|}_A{U}_{SA}{|0\rangle }_A$. Deterministic implementation of the operation, ${\sum }_m{E}_m^{†}{E}_m=\mathbb{I}$, corresponds to completely tracing out the ancilla $A$. On the other hand, selective projection on the $A$ part leads to probabilistic (or selective) implantation of the operation, for which ${\sum }_m{E}_m^{†}{E}_m<\mathbb{I}$. The selective implementation of operation element $E_m$ results in a quantum state $E_m{\eta }_S{E}_m^{†}/p_m$ with the probability $p_m=Tr\left(E_m{\eta }_S{E}_m^{†}\right)$.

Figure 2

A quantum superoperation [43] can be implemented using quantum isometry operations [48] in a quantum circuit. A superoperation $\tilde{\mathrm{\Omega }}$, that transforms an initial $\mathrm{\Phi }$ to a final $\mathrm{\Lambda }$ quantum operation, is designated as $\mathrm{\Lambda }\left({\eta }_S\right)=\tilde{\mathrm{\Omega }}\left(\mathrm{\Phi }\right)\left({\eta }_S\right)={Tr}_B\left[V(\mathrm{\Phi }\otimes {\mathbb{I}}_B)\left(U{\eta }_S{U}^{†}\right)V^{†}\right]$. It is realized by two operations $\mathcal{U}=U·U^{†}$ and $\mathcal{V}=V·V^{†}$, where $U$ and $V$ are the isometries applied before and after the operation $\mathrm{\Phi }$. At the end of the circuit, the ancillary system $B$ is either measured (using projective measurements), where each outcome results in a probabilistic (or selective) superoperation, or simply discarded (traced out), thereby realizing a deterministic superoperation.