If ${\displaystyle P=(p_{1},...,p_{n})}$  is a classical finite probability distribution, its min-entropy can be defined as^[2]

A natural way to define a "min-entropy" for quantum states is to leverage the simple observation that quantum states result in probability distributions when measured in some basis. There is however the added difficulty that a single quantum state can result in infinitely many possible probability distributions, depending on how it is measured. A natural path is then, given a quantum state ${\displaystyle \rho }$ , to still define ${\displaystyle H_{\rm {min}}(\rho )}$  as ${\displaystyle \log(1/P_{\rm {max}})}$ , but this time defining ${\displaystyle P_{\rm {max}}}$  as the maximum possible probability that can be obtained measuring ${\displaystyle \rho }$ , maximizing over all possible projective measurements.

Formally, this would provide the definition

A more concise method to write the double maximization is to observe that any element of any POVM is a Hermitian operator such that ${\displaystyle 0\leq \Pi \leq I}$ , and thus we can equivalently directly maximize over these to get

Let ${\displaystyle \rho _{AB}}$  be a bipartite density operator on the space ${\displaystyle {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}}$ . The min-entropy of ${\displaystyle A}$  conditioned on ${\displaystyle B}$  is defined to be

${\displaystyle H_{\min }(A|B)_{\rho }\equiv -\inf _{\sigma _{B}}D_{\max }(\rho _{AB}\|I_{A}\otimes \sigma _{B})}$ 

where the infimum ranges over all density operators ${\displaystyle \sigma _{B}}$  on the space ${\displaystyle {\mathcal {H}}_{B}}$ . The measure ${\displaystyle D_{\max }}$  is the maximum relative entropy defined as

${\displaystyle D_{\max }(\rho \|\sigma )=\inf _{\lambda }\{\lambda :\rho \leq 2^{\lambda }\sigma \}}$ 

The smooth min-entropy is defined in terms of the min-entropy.

${\displaystyle H_{\min }^{\epsilon }(A|B)_{\rho }=\sup _{\rho '}H_{\min }(A|B)_{\rho '}}$ 

where the sup and inf range over density operators ${\displaystyle \rho '_{AB}}$  which are ${\displaystyle \epsilon }$ -close to ${\displaystyle \rho _{AB}}$ . This measure of ${\displaystyle \epsilon }$ -close is defined in terms of the purified distance

${\displaystyle P(\rho ,\sigma )={\sqrt {1-F(\rho ,\sigma )^{2}}}}$ 

where ${\displaystyle F(\rho ,\sigma )}$  is the fidelity measure.

These quantities can be seen as generalizations of the von Neumann entropy. Indeed, the von Neumann entropy can be expressed as

${\displaystyle S(A|B)_{\rho }=\lim _{\epsilon \rightarrow 0}\lim _{n\rightarrow \infty }{\frac {1}{n}}H_{\min }^{\epsilon }(A^{n}|B^{n})_{\rho ^{\otimes n}}~.}$ 

This is called the fully quantum asymptotic equipartition theorem.^[3] The smoothed entropies share many interesting properties with the von Neumann entropy. For example, the smooth min-entropy satisfy a data-processing inequality:^[4]

${\displaystyle H_{\min }^{\epsilon }(A|B)_{\rho }\geq H_{\min }^{\epsilon }(A|BC)_{\rho }~.}$ 

Henceforth, we shall drop the subscript ${\displaystyle \rho }$  from the min-entropy when it is obvious from the context on what state it is evaluated.

Min-entropy as uncertainty about classical information edit

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^{[citation needed]} 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 ${\displaystyle 2^{-H_{\min }(X|B)}=\max _{x}P_{X}(x)~.}$ 

Min-entropy as overlap with the maximally entangled state edit

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.

Proof of operational characterization of min-entropy edit

The proof is from a paper by König, Schaffner, Renner in 2008.^[5] It involves the machinery of semidefinite programs.^[6] 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 ${\displaystyle B}$ .

• von Neumann entropy
• Generalized relative entropy
• max-entropy