Resource Theory of Coherence Based on Positive-Operator-Valued Measures

Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis; i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurements, i.e., to arbitrary positive-operator-valued measures (POVMs). We establish POVM-based coherence measures and POVM-incoherent operations that coincide for the case of von Neumann measurements with their counterparts in standard coherence theory. We provide a semidefinite program that allows us to characterize interconversion properties of resource states and exemplify our framework by means of the qubit trine POVM, for which we also show analytical results.

Figure 1

We introduce a resource theory of POVM-based coherence by making use of the Naimark construction. Quantum states $\rho$ are embedded as $\mathcal{E}[\rho ]=\rho \otimes |1⟩⟨1|$ to act on a higher-dimensional Hilbert space (Naimark space). The POVM $\mathbf{E}$ is extended to a projective measurement $\mathbf{P}$ on the Naimark space, which defines a set of block-incoherent states $\mathcal{I}$. The POVM-coherence measure $C(\rho ,\mathbf{E})$ is the distance between $\mathcal{E}[\rho ]$ and its projection $\mathrm{\Delta }[\mathcal{E}[\rho ]]$ onto block-incoherent states.

Figure 2

POVM-based coherence theory for qubit states with respect to the trine POVM ${\mathbf{E}}^{\text{trine}}$ in the Bloch sphere representation. Gray lines indicate the three measurement directions. (Left) POVM-based coherence of pure qubits (surface of sphere). The states $|0⟩$ and $|1⟩$ have maximal coherence of $C=\log 3$. The Bloch vectors of the three states with the lowest pure-state coherence $C=1$ are antipodal to the measurement directions. (Right) Maximally achievable conversion fidelity $F_{\max }(\rho ,\sigma )={\max }_{{\mathrm{\Lambda }}_{\mathrm{MPI}}}F({\mathrm{\Lambda }}_{\mathrm{MPI}}[\rho ],\sigma )$ between a pure initial state $\rho$ (red dot) subjected to POVM-incoherent operations ${\mathrm{\Lambda }}_{\mathrm{MPI}}$ and a pure target state $\sigma$ on the sphere surface. Here, $\rho =|\psi ⟩⟨\psi |$ with $|\psi ⟩=\cos (\pi /8)|0⟩+\sin (\pi /8)|1⟩$. Only states in the orbit of $|\psi ⟩$ under the six POVM-incoherent unitaries can be reached with unit fidelity, as depicted by the yellow spots.