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.

As with the classical Shannon entropy and its quantum generalization, the von Neumann entropy, one can define a conditional versions 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 ^[1]).

Definitions

Definition: 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.^[2] The smoothed entropies share many interesting properties with the von Neumann entropy. For example, the smooth min entropy satisfy a data-processing inequality: ^[3]

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)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 $2^{-H_{\min}(X|B)} = \max_x P_X(x)~.$

Min-entropy as distance from 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 '08.^[4] It involves the machinery of semidefinite programs,.^[5] 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 Jamiolkowski 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.

References

↑ Vazirani, Umesh, and Thomas Vidick. "Fully device independent quantum key distribution." arXiv:1210.1810 (2012)
↑ Marco Tomamichel, Roger Colbeck, Renato Renner. "The Fully Quantum Asymptotic Equipartition Property." Information Theory, IEEE Transactions on 55 (2009), p. 5840-5847
↑ Renato Renner, "Security of Quantum Key Distribution", Ph.D. Thesis, Diss. ETH No. 16242 arXiv:quant-ph/0512258
↑ König, R., Renato Renner, and Christian Schaffner. "The operational meaning of min-and max-entropy." Information Theory, IEEE Transactions on 55.9 (2009): 4337-4347. arXiv:0807.1338
↑ John Watrous, Theory of quantum information, Fall 2011, course notes, https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/07.pdf

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