Draft:Quantum Chernoff Distance

• Comment: Not enough independent, significant coverage. WikiOriginal-9 (talk) 17:51, 7 November 2023 (UTC)

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The quantum Chernoff distance generalizes the classical Chernoff distance, which measures the dissimilarity between probability distributions. This follows since probability distributions can be seen as particular cases of quantum states, specifically as quantum states with diagonal density matrices. Both metrics have a clear interpretation as the optimal decay rate of error probability in classical and quantum hypothesis testing

Interpretation

The problem of quantum hypothesis testing consists of identifying an unknown state $\sigma \in {\mathcal {D}}({\mathcal {H}})$ that belongs to a discrete set of quantum states $\{\sigma _{i}\}_{i=1}^{K}$, given $N$ copies of the state. Each hypothesis or state has a prior probability denoted by $\{\pi _{i}\}_{i=1}^{K}$. For the case of binary quantum hypothesis testing, the optimal probability of error is obtained using the Helstrom measurement, which yields the following probability of error

$p_{e}={\frac {1}{2}}\left(1+\left\|\pi _{1}\sigma _{1}^{\otimes N}-\pi _{2}\sigma _{2}^{\otimes N}\right\|_{1}\right)$

Finally, this formula can be used to define the Chernoff distance as

$C_{Q}(\sigma _{1},\sigma _{2})=\lim _{N\rightarrow \infty }{\frac {-\log p_{e}}{N}}$

Definition

A close-form expression for the quantum Chernoff distance is given by

$C_{Q}(\sigma _{1},\sigma _{2}):=-\log \left(\inf _{0\leq s\leq 1}\mathrm {Tr} \left(\sigma _{1}^{s}\sigma _{2}^{1-s}\right)\right)$

which as mention before, generalizes the classical concept for distributions,

$C(p_{1},p_{2}):=-\log \left(\inf _{0\leq s\leq 1}\sum _{i}p_{1}(i)^{s}p_{2}(i)^{1-s}\right)$

Properties

When at least one of the states is pure, i.e., its rank is equal to 1, then the Chernoff distance becomes $C_{Q}(\sigma ,\rho ):=-\log \left(\mathrm {Tr} \left(\sigma \rho \right)\right)$.

This metric is related to other commonly used metrics for quantum states, such as the trace distance or Ulhmann fidelity. Specifically,

$F(\sigma ,\rho )^{2}\leq e^{-C_{Q}(\sigma ,\rho )}\leq F(\sigma ,\rho )$

where $F(\sigma ,\rho ):=\left\|{\sqrt {\sigma }}{\sqrt {\rho }}\right\|_{1}$ denotes the Uhlmann fidelity. Similarly,

$1-{\frac {1}{2}}\left\|\rho -\sigma \right\|_{1}\leq e^{-C_{Q}(\sigma ,\rho )}\leq {\sqrt {1-{\frac {1}{4}}\left\|\rho -\sigma \right\|_{1}^{2}}}$