Quantum stochastic calculus

From formulasearchengine
Revision as of 01:56, 29 January 2014 by en>Monkbot (Computational considerations: Fix CS1 deprecated date parameter errors)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

The min entropy is a conditional information measure. It is a one-shot analogue of the conditional quantum entropy.

To interpret a conditional information measure, suppose Alice and Bob were to share a bipartite quantum state ρAB. 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 ρAB be a bipartite density operator on the space AB. The min-entropy of A conditioned on B is defined to be

Hmin(A|B)ρinfσBDmax(ρAB||IAσB)

where the infimum ranges over all density operators σB on the space B. The measure Dmax is the maximum relative entropy defined as

Dmax(ρ||σ)=infλ{λ:ρ2λσ}

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

Hminϵ(A|B)ρ=supρHmin(A|B)ρ

where the sup and inf range over density operators ρ'AB which are ϵ-close to ρAB. This measure of ϵ-close is defined in terms of the purified distance

P(ρ,σ)=1F(ρ,σ)2

where F(ρ,σ) 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)ρ=limϵ0limn1nHminϵ(An|Bn)ρn.

This is called the fully asymptotic equipartition theorem.[2] The smoothed entropies share many interesting properties with the von Neumann entropy. For example, the smooth min entropy is strongly subadditive[3]

Hminϵ(A|B)ρHminϵ(A|BC)ρ.

Operational interpretation of smoothed min entropy

Henceforth, we shall drop the subscript ρ 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 ρBx depends on some classical variable X. Furthermore, suppose that each of its elements x is distributed according to some distribution PX(x). This can be described by the following state over the system XB.

ρXB=xPX(x)|xx|ρBx

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

pg(X|B)=xPX(x)tr(ExρBx)

where Ex is the POVM that maximizes this expression. It can be shown that this optimum can be expressed in terms of the min-entropy as

pg(X|B)=2Hmin(X|B).

If the state ρXB is a product state i.e. ρXB=σXτB for some density operators σX and τB, then there is no correlation between the systems X and B. In this case, it turns out that 2Hmin(X|B)=maxxPX(x).

Min-entropy as distance from maximally entangled state

The maximally entangled state |ϕ+ on a bipartite system AB is defined as

|ϕ+AB=1dxA,xB|xA|xB

where {|xA} and {|xB} form an orthonormal basis for the spaces A and B respectively. For a bipartite quantum state ρAB, we define the maximum overlap with the maximally entangled state as

qc(A|B)=dmaxF((IA)ρAB,|ϕ+ϕ+|)

where the maximum is over all CPTP operations . This is a measure of how correlated the state ρAB is. It can be shown that qc(A|B)=2Hmin(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 Konig, Schaffner, Renner '08.[4] It involves the machinery of semidefinite programs,.[5] Suppose we are given some bipartite density operator ρAB. From the definition of the min-entropy, we have

Hmin(A|B)=infσBinfλ{λ|ρAB2λ(IAσB)}.

This can be re-written as

loginfσBTr(σB)

subject to the conditions

σB0
IAσBρAB.

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

min:Tr(σB)
subject to: IAσBρAB
σB0.

This primal problem can also be fully specified by the matrices (ρAB,IB,Tr*) where Tr* is the adjoint of the partial trace over A. The action of Tr* on operators on B can be written as

Tr*(X)=IAX.

We can express the dual problem as a maximization over operators EAB on the space AB as

max:Tr(ρABEAB)
subject to: TrA(EAB)=IB
IB0.

Using the Choi Jamiolkowski isomorphism, we can define the channel such that

IA(|ϕ+ϕ+|)=EAB

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

ρAB,EAB=ρAB,IA(|ϕ+ϕ+|)
=IA(ρAB),|ϕ+ϕ+|)

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

maxPX(x)x|(ρBx)|x.

We can interpret 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.[6]

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  1. Vazirani, Umesh, and Thomas Vidick. "Fully device independent quantum key distribution." Template:Arxiv (2012)
  2. Beaudry, Normand J., and Renato Renner. "An intuitive proof of the data processing inequality." Quantum Information & Computation 12.5-6 (2012): 432-441. Template:Arxiv
  3. Beaudry, Normand J., and Renato Renner. "An intuitive proof of the data processing inequality." Quantum Information & Computation 12.5-6 (2012): 432-441. Template:Arxiv
  4. Konig, R., Renato Renner, and Christian Schaffner. "The operational meaning of min-and max-entropy." Information Theory, IEEE Transactions on 55.9 (2009): 4337-4347. Template:Arxiv
  5. John Watrous, Theory of quantum information, Fall 2011, course notes, https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/07.pdf
  6. Konig, R., Renato Renner, and Christian Schaffner. "The operational meaning of min-and max-entropy." Information Theory, IEEE Transactions on 55.9 (2009): 4337-4347. Template:Arxiv