Haar-average fidelity versus entanglement fidelity

PDF p. 3: Opening paragraph

Assume the Axiom of Choice. Let $d_L\ge 1$, let $H_L={\mathbb{C}}^{d_L}$, and let $\mathcal{E}:\mathcal{L}(H_L)\to \mathcal{L}(H_L)$ be a quantum channel, meaning a complex-linear, completely positive, trace-preserving map. Let $U$ range over the unitary group $U(d_L)$ with normalized Haar probability, and put $\mathrm{\mid }{\psi }_U⟩=U\mathrm{\mid }0⟩$. The Haar-average pure-state fidelity and entanglement fidelity at the maximally mixed input are respectively

$\bar{F}(\mathcal{E})={\int }_{U(d_L)}⟨{\psi }_U\mathrm{\mid }\mathcal{E}(\mathrm{\mid }{\psi }_U⟩⟨{\psi }_U\mathrm{\mid })\mathrm{\mid }{\psi }_U⟩dU,F_e(\mathcal{E})=⟨\mathrm{\Phi }\mathrm{\mid }(\mathrm{i}\mathrm{d}\otimes \mathcal{E})(\mathrm{\mid }\mathrm{\Phi }⟩⟨\mathrm{\Phi }\mathrm{\mid })\mathrm{\mid }\mathrm{\Phi }⟩,$

where $\mathrm{\mid }\mathrm{\Phi }⟩=d_L^{-1\mathrm{/}2}{\sum }_{j=0}^{d_L-1}\mathrm{\mid }j⟩\otimes \mathrm{\mid }j⟩$ and $\mathrm{i}\mathrm{d}$ is the identity map on operators on the reference copy of $H_L$. Then

$\bar{F}(\mathcal{E})=\frac{d_L{F}_e(\mathcal{E})+1}{d_L+1}\mathrm{.}$

In particular, let $V:H_L\to H_P$ be an isometry between nonzero finite-dimensional complex coordinate spaces, let $\mathcal{N}:\mathcal{L}(H_P)\to \mathcal{L}(H_O)$ be a channel with a finite nonempty Kraus family $(E_l)$, and let $P=V{V}^{†}$. Put $Q=\mathcal{N}(P)$ and let $Q^{-1\mathrm{/}2}$ be the inverse of its positive square root on its support and zero on its kernel. A transpose recovery $\mathcal{T}$ is a channel obtained by completing the map with Kraus operators $P{E}_l^{†}{Q}^{-1\mathrm{/}2}$ outside the support of $Q$ using a code state. Let ${\mathcal{E}}_{\mathrm{t}\mathrm{r}}$ be the logical channel obtained by encoding with $V$, applying $\mathcal{N}$ and $\mathcal{T}$, and decoding the code-supported output with $V^{†}$. If $M_{[\mu ,l],[\nu ,k]}=⟨\bar{\mu }\mathrm{\mid }{E}_l^{†}{E}_k\mathrm{\mid }\bar{\nu }⟩$, where $\mathrm{\mid }\bar{\mu }⟩=V\mathrm{\mid }\mu ⟩$, and if ${\mathrm{Tr}}_L$ contracts the logical indices, define $F_{\mathrm{t}\mathrm{r}}=d_L^{-2}\mathrm{\parallel }{\mathrm{Tr}}_L\sqrt{M}{\mathrm{\parallel }}_F^2$, with $\sqrt{M}$ the positive square root and $\mathrm{\parallel }B{\mathrm{\parallel }}_F^2={\sum }_{l,k}\mathrm{\mid }{B}_{l,k}{\mathrm{\mid }}^2$. Then $F_e({\mathcal{E}}_{\mathrm{t}\mathrm{r}})=F_{\mathrm{t}\mathrm{r}}$ and $\bar{F}({\mathcal{E}}_{\mathrm{t}\mathrm{r}})=(d_L{F}_{\mathrm{t}\mathrm{r}}+1)\mathrm{/}(d_L+1)$. This includes singular $M$ and $Q$ and is independent of the completion choice.

Finite operator and ket conventions

Finite-dimensional quantum channel

Entanglement fidelity on a finite logical space

Normalized Haar probability on a compact group

The Axiom of Choice

QEC matrix and logical partial trace

Transpose (Petz) recovery on a code

Transpose fidelity from the QEC matrix