Min-entropy

Suppose an agent had access to a quantum system ${\displaystyle B}$ whose state ${\displaystyle \rho _{B}^{x}}$ depends on some classical variable ${\displaystyle X}$. Furthermore, suppose that each of its elements ${\displaystyle x}$ is distributed according to some distribution ${\displaystyle P_{X}(x)}$. This can be described by the following state over the system ${\displaystyle XB}$.

${\displaystyle \rho _{XB}=\sum _{x}P_{X}(x)|x\rangle \langle x|\otimes \rho _{B}^{x},}$

where ${\displaystyle \{|x\rangle \}}$ form an orthonormal basis. We would like to know what the agent can learn about the classical variable ${\displaystyle x}$. Let ${\displaystyle p_{g}(X|B)}$ be the probability that the agent guesses ${\displaystyle X}$ when using an optimal measurement strategy

${\displaystyle p_{g}(X|B)=\sum _{x}P_{X}(x)tr(E_{x}\rho _{B}^{x}),}$

where ${\displaystyle E_{x}}$ is the POVM that maximizes this expression. It can be shown that this optimum can be expressed in terms of the min-entropy as

${\displaystyle p_{g}(X|B)=2^{-H_{\min }(X|B)}~.}$

If the state ${\displaystyle \rho _{XB}}$ is a product state i.e. ${\displaystyle \rho _{XB}=\sigma _{X}\otimes \tau _{B}}$ for some density operators ${\displaystyle \sigma _{X}}$ and ${\displaystyle \tau _{B}}$, then there is no correlation between the systems ${\displaystyle X}$ and ${\displaystyle B}$. In this case, it turns out that ${\displaystyle 2^{-H_{\min }(X|B)}=\max _{x}P_{X}(x)~.}$

The maximally entangled state ${\displaystyle |\phi ^{+}\rangle }$ on a bipartite system ${\displaystyle {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}}$ is defined as

${\displaystyle |\phi ^{+}\rangle _{AB}={\frac {1}{\sqrt {d}}}\sum _{x_{A},x_{B}}|x_{A}\rangle |x_{B}\rangle }$

where ${\displaystyle \{|x_{A}\rangle \}}$ and ${\displaystyle \{|x_{B}\rangle \}}$ form an orthonormal basis for the spaces ${\displaystyle A}$ and ${\displaystyle B}$ respectively. For a bipartite quantum state ${\displaystyle \rho _{AB}}$, we define the maximum overlap with the maximally entangled state as

${\displaystyle q_{c}(A|B)=d_{A}\max _{\mathcal {E}}F\left((I_{A}\otimes {\mathcal {E}})\rho _{AB},|\phi ^{+}\rangle \langle \phi ^{+}|\right)^{2}}$

where the maximum is over all CPTP operations ${\displaystyle {\mathcal {E}}}$ and ${\displaystyle d_{A}}$ is the dimension of subsystem ${\displaystyle A}$. This is a measure of how correlated the state ${\displaystyle \rho _{AB}}$ is. It can be shown that ${\displaystyle q_{c}(A|B)=2^{-H_{\min }(A|B)}}$. If the information contained in ${\displaystyle A}$ is classical, this reduces to the expression above for the guessing probability.

The proof is from a paper by König, Schaffner, Renner in 2008.[4] It involves the machinery of semidefinite programs.[5] Suppose we are given some bipartite density operator ${\displaystyle \rho _{AB}}$. From the definition of the min-entropy, we have

${\displaystyle H_{\min }(A|B)=-\inf _{\sigma _{B}}\inf _{\lambda }\{\lambda |\rho _{AB}\leq 2^{\lambda }(I_{A}\otimes \sigma _{B})\}~.}$

This can be re-written as

${\displaystyle -\log \inf _{\sigma _{B}}\operatorname {Tr} (\sigma _{B})}$

subject to the conditions

${\displaystyle \sigma _{B}\geq 0}$

${\displaystyle I_{A}\otimes \sigma _{B}\geq \rho _{AB}~.}$

We notice that the infimum is taken over compact sets and hence can be replaced by a minimum. This can then be expressed succinctly as a semidefinite program. Consider the primal problem

${\displaystyle {\text{min:}}\operatorname {Tr} (\sigma _{B})}$

${\displaystyle {\text{subject to: }}I_{A}\otimes \sigma _{B}\geq \rho _{AB}}$

${\displaystyle \sigma _{B}\geq 0~.}$

This primal problem can also be fully specified by the matrices ${\displaystyle (\rho _{AB},I_{B},\operatorname {Tr} ^{*})}$ where ${\displaystyle \operatorname {Tr} ^{*}}$ is the adjoint of the partial trace over ${\displaystyle A}$. The action of ${\displaystyle \operatorname {Tr} ^{*}}$ on operators on ${\displaystyle B}$ can be written as

${\displaystyle \operatorname {Tr} ^{*}(X)=I_{A}\otimes X~.}$

We can express the dual problem as a maximization over operators ${\displaystyle E_{AB}}$ on the space ${\displaystyle AB}$ as

${\displaystyle {\text{max:}}\operatorname {Tr} (\rho _{AB}E_{AB})}$

${\displaystyle {\text{subject to: }}\operatorname {Tr} _{A}(E_{AB})=I_{B}}$

${\displaystyle E_{AB}\geq 0~.}$

Using the Choi–Jamiołkowski isomorphism, we can define the channel ${\displaystyle {\mathcal {E}}}$ such that

${\displaystyle d_{A}I_{A}\otimes {\mathcal {E}}^{\dagger }(|\phi ^{+}\rangle \langle \phi ^{+}|)=E_{AB}}$

where the bell state is defined over the space ${\displaystyle AA'}$. This means that we can express the objective function of the dual problem as

${\displaystyle \langle \rho _{AB},E_{AB}\rangle =d_{A}\langle \rho _{AB},I_{A}\otimes {\mathcal {E}}^{\dagger }(|\phi ^{+}\rangle \langle \phi ^{+}|)\rangle }$

${\displaystyle =d_{A}\langle I_{A}\otimes {\mathcal {E}}(\rho _{AB}),|\phi ^{+}\rangle \langle \phi ^{+}|)\rangle }$

as desired.

Notice that in the event that the system ${\displaystyle A}$ is a partly classical state as above, then the quantity that we are after reduces to

${\displaystyle \max P_{X}(x)\langle x|{\mathcal {E}}(\rho _{B}^{x})|x\rangle ~.}$

We can interpret ${\displaystyle {\mathcal {E}}}$ as a guessing strategy and this then reduces to the interpretation given above where an adversary wants to find the string ${\displaystyle x}$ given access to quantum information via system ${\displaystyle B}$.