Min-entropy

The min-entropy, in information theory, is the smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the most likely outcome. The various Rényi entropies are all equal for a uniform distribution, but measure the unpredictability of a nonuniform distribution in different ways. The min-entropy is never greater than the ordinary or Shannon entropy (which measures the average unpredictability of the outcomes) and that in turn is never greater than the Hartley or max-entropy, defined as the logarithm of the number of outcomes with nonzero probability.

As with the classical Shannon entropy and its quantum generalization, the von Neumann entropy, one can define a conditional version of min-entropy. The conditional quantum min-entropy is a one-shot, or conservative, analog of conditional quantum entropy.

To interpret a conditional information measure, suppose Alice and Bob were to share a bipartite quantum state $\rho _{AB}$. Alice has access to system $A$ and Bob to system $B$. The conditional entropy measures the average uncertainty Bob has about Alice's state upon sampling from his own system. The min-entropy can be interpreted as the distance of a state from a maximally entangled state.

This concept is useful in quantum cryptography, in the context of privacy amplification (See for example ).

Definition for classical distributions

If $P=(p_{1},...,p_{n})$ is a classical finite probability distribution, its min-entropy can be defined as $$H_{\rm {min}}({\boldsymbol {P}})=\log {\frac {1}{P_{\rm {max}}}},\qquad P_{\rm {max}}\equiv \max _{i}p_{i}.$$One way to justify the name of the quantity is to compare it with the more standard definition of entropy, which reads $H({\boldsymbol {P}})=\sum _{i}p_{i}\log(1/p_{i})$, and can thus be written concisely as the expectation value of $\log(1/p_{i})$ over the distribution. If instead of taking the expectation value of this quantity we take its minimum value, we get precisely the above definition of $H_{\rm {min}}({\boldsymbol {P}})$.

From an operational perspective, the min-entropy equals the negative logarithm of the probability of successfully guessing the outcome of a random draw from $P$. This is because it is optimal to guess the element with the largest probability and the chance of success equals the probability of that element.

Definition for quantum states

A natural way to generalize "min-entropy" from classical to quantum states is to leverage the simple observation that quantum states define classical 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 $\rho$, to still define $H_{\rm {min}}(\rho )$ as $\log(1/P_{\rm {max}})$, but this time defining $P_{\rm {max}}$ as the maximum possible probability that can be obtained measuring $\rho$, maximizing over all possible projective measurements. Using this, one gets the operational definition that the min-entropy of $\rho$ equals the negative logarithm of the probability of successfully guessing the outcome of any measurement of $\rho$.

Formally, this leads to the definition $$H_{\rm {min}}(\rho )=\max _{\Pi }\log {\frac {1}{\max _{i}\operatorname {tr} (\Pi _{i}\rho )}}=-\max _{\Pi }\log \max _{i}\operatorname {tr} (\Pi _{i}\rho ),$$where we are maximizing over the set of all projective measurements $\Pi =(\Pi _{i})_{i}$, $\Pi _{i}$ represent the measurement outcomes in the POVM formalism, and $\operatorname {tr} (\Pi _{i}\rho )$ is therefore the probability of observing the $i$-th outcome when the measurement is $\Pi$.

A more concise method to write the double maximization is to observe that any element of any POVM is a Hermitian operator such that $0\leq \Pi \leq I$, and thus we can equivalently directly maximize over these to get $$H_{\rm {min}}(\rho )=-\max _{0\leq \Pi \leq I}\log \operatorname {tr} (\Pi \rho ).$$In fact, this maximization can be performed explicitly and the maximum is obtained when $\Pi$ is the projection onto (any of) the largest eigenvalue(s) of $\rho$. We thus get yet another expression for the min-entropy as: $$H_{\rm {min}}(\rho )=-\log \|\rho \|_{\rm {op}},$$remembering that the operator norm of a Hermitian positive semidefinite operator equals its largest eigenvalue.

Conditional entropies

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

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

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

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

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

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

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

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

where $F(\rho ,\sigma )$ 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

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

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

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

Operational interpretation of smoothed min-entropy

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

Min-entropy as uncertainty about classical information

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

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

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

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

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

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

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

Since the conditional min-entropy is always smaller than the conditional Von Neumann entropy, it follows that

$p_{g}(X|B)\geq 2^{-S(A|B)_{\rho }}~.$

Min-entropy as overlap with the maximally entangled state

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

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

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

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

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

Proof of operational characterization of min-entropy

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

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

This can be re-written as

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

subject to the conditions

$\sigma _{B}\geq 0$

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

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

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

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

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

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

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

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

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

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

$E_{AB}\geq 0~.$

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

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

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

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

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

as desired.

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

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

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

See also

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