1. Introduction
2. Background
2.1. Block-coherence theoretical framework
2.2. POVM coherence theoretical framework
2.3. Max-relative entropy and the coherent rank
3. Block-coherence measures
The block-coherence measure ${{ \mathcal C }}_{\max }(\rho ,{\mathbf{P}})$ of a quantum state ρ with respect to the projective measurement ${\mathbf{P}}$ is defined as
The block-coherence measure ${{ \mathcal C }}_{\max }(\rho ,{\mathbf{P}})$ is a block-coherence monotone under $\mathrm{MBI}$ operations and quasi-convex.
First, we show that ${{ \mathcal C }}_{\max }(\rho ,{\mathbf{P}})\geqslant 0$ with the equality if and only if $\rho \in {{ \mathcal I }}_{B}({ \mathcal H })$.
Let $\mathrm{MBI}$ denote the set of the MBI operations. For a given state ρ and $\epsilon \geqslant 0$, the one-shot block-coherence cost under the MBI operations is defined as
For $\epsilon \gt 0$, the coherence measures satisfy
Now we prove the first inequality of equation (
A block-coherence measure based on coherent rank is defined as
The quantity ${{ \mathcal C }}_{0}(\rho ,{\mathbf{P}})$ is a coherence monotone under the block-incoherent operation.
Apparently ${{ \mathcal C }}_{0}(\rho ,{\mathbf{P}})\geqslant 0$. Next we prove that the quantity ${{ \mathcal C }}_{0}(\rho ,{\mathbf{P}})$ satisfies ${{ \mathcal C }}_{0}(\rho ,{\mathbf{P}})=0\iff \rho \in {{ \mathcal I }}_{B}({ \mathcal H })$.
If $| \psi \rangle =\tfrac{{K}_{i}| \phi \rangle }{\sqrt{\mathrm{Tr}[{K}_{i}| \phi \rangle \langle \phi | {K}_{i}^{\dagger }]}}$, where $\{{K}_{i}\}$ is a set of incoherent-preserving Kraus operators, then ${{ \mathcal C }}_{0}(| \psi \rangle \langle \psi | )\leqslant {{ \mathcal C }}_{0}(| \phi \rangle \langle \phi | )$.
For $\epsilon \gt 0$, the value of the one-shot block-coherence cost under $\mathrm{MBI}$ is equal to ${{ \mathcal C }}_{0}^{\epsilon }(\rho ,{\mathbf{P}})$, namely,
We begin with the lower bound on ${{ \mathcal C }}_{\mathrm{MBI}}^{\epsilon }(\rho ,{\mathbf{P}})$. Let ${\mathrm{log}}_{2}N={{ \mathcal C }}_{\mathrm{MBI}}^{\epsilon }(\rho ,{\mathbf{P}})$, then there exists an operation ${{\rm{\Lambda }}}_{\mathrm{MBI}}$ such that $F[{{\rm{\Lambda }}}_{\mathrm{MBI}}({\psi }_{N}),\rho ]\geqslant 1-\epsilon $. Thus, we obtain
4. POVM-based coherence measures
Let ${\mathbf{E}}=\{{E}_{i}={A}_{i}^{\dagger }{A}_{i}\}{}_{i=1}^{n}$ be a POVM on the Hilbert space ${ \mathcal H }$, the quantity based on the max-relative entropy
We first prove that ${{ \mathcal C }}_{\max }(\varepsilon (\rho ),\bar{{\mathbf{P}}})$ is invariant under unitary transformation. Note that
Let ${\mathbf{E}}=\{{E}_{i}={A}_{i}^{\dagger }{A}_{i}\}{}_{i=1}^{n}$ be a POVM on the d-dimensional Hilbert space ${ \mathcal H }$, and ${\mathbf{P}}=\{{P}_{i}={V}^{\dagger }{\mathbb{I}}\otimes | i\rangle \langle i| V\}{}_{i=1}^{n}$ be a canonical Naimark extension of ${\mathbf{E}}={\{{E}_{i}\}}_{i=1}^{n}$. We use ${ \mathcal O }$ to denote the set of the maximally POVM-incoherent operations. For a state ρ and $\epsilon \geqslant 0$, the one-shot block-coherence cost under ${ \mathcal O }$ is defined as
For quantum state ρ and $\epsilon \gt 0$, we have