Quantum relative entropy

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In quantum information theory, quantum relative entropy is a measure of distinguishability between two quantum states. It is the quantum mechanical analog of relative entropy.

Motivation[edit]

For simplicity, it will be assumed that all objects in the article are finite-dimensional.

We first discuss the classical case. Suppose the probabilities of a finite sequence of events is given by the probability distribution P = {p1...pn}, but somehow we mistakenly assumed it to be Q = {q1...qn}. For instance, we can mistake an unfair coin for a fair one. According to this erroneous assumption, our uncertainty about the j-th event, or equivalently, the amount of information provided after observing the j-th event, is

The (assumed) average uncertainty of all possible events is then

On the other hand, the Shannon entropy of the probability distribution p, defined by

is the real amount of uncertainty before observation. Therefore the difference between these two quantities

is a measure of the distinguishability of the two probability distributions p and q. This is precisely the classical relative entropy, or Kullback–Leibler divergence:

Note

  1. In the definitions above, the convention that 0·log 0 = 0 is assumed, since limx → 0 x log x = 0. Intuitively, one would expect that an event of zero probability to contribute nothing towards entropy.
  2. The relative entropy is not a metric. For example, it is not symmetric. The uncertainty discrepancy in mistaking a fair coin to be unfair is not the same as the opposite situation.

Definition[edit]

As with many other objects in quantum information theory, quantum relative entropy is defined by extending the classical definition from probability distributions to density matrices. Let ρ be a density matrix. The von Neumann entropy of ρ, which is the quantum mechanical analog of the Shannon entropy, is given by

For two density matrices ρ and σ, the quantum relative entropy of ρ with respect to σ is defined by

We see that, when the states are classically related, i.e. ρσ = σρ, the definition coincides with the classical case.

Non-finite (divergent) relative entropy[edit]

In general, the support of a matrix M is the orthogonal complement of its kernel, i.e. . When considering the quantum relative entropy, we assume the convention that −s · log 0 = ∞ for any s > 0. This leads to the definition that

when

This can be interpreted in the following way. Informally, the quantum relative entropy is a measure of our ability to distinguish two quantum states where larger values indicate states that are more different. Being orthogonal represents the most different quantum states can be. This is reflected by non-finite quantum relative entropy for orthogonal quantum states. Following the argument given in the Motivation section, if we erroneously assume the state has support in , this is an error impossible to recover from.

However, one should be careful not to conclude that the divergence of the quantum relative entropy implies that the states and are orthogonal or even very different by other measures. Specifically, can diverge when and differ by a vanishingly small amount as measured by some norm. For example, let have the diagonal representation

with for and for where is an orthonormal set. The kernel of is the space spanned by the set . Next let

for a small positive number . As has support (namely the state ) in the kernel of , is divergent even though the trace norm of the difference is . This means that difference between and as measured by the trace norm is vanishingly small as even though is divergent (i.e. infinite). This property of the quantum relative entropy represents a serious shortcoming if not treated with care.

Non-negativity of relative entropy[edit]

Corresponding classical statement[edit]

For the classical Kullback–Leibler divergence, it can be shown that

and the equality holds if and only if P = Q. Colloquially, this means that the uncertainty calculated using erroneous assumptions is always greater than the real amount of uncertainty.

To show the inequality, we rewrite

Notice that log is a concave function. Therefore -log is convex. Applying Jensen's inequality, we obtain

Jensen's inequality also states that equality holds if and only if, for all i, qi = (∑qj) pi, i.e. p = q.

The result[edit]

Klein's inequality states that the quantum relative entropy

is non-negative in general. It is zero if and only ρ = σ.

Proof

Let ρ and σ have spectral decompositions

So

Direct calculation gives

where Pi j = |vi*wj|2.

Since the matrix (Pi j)i j is a doubly stochastic matrix and -log is a convex function, the above expression is

Define ri = ∑jqj Pi j. Then {ri} is a probability distribution. From the non-negativity of classical relative entropy, we have

The second part of the claim follows from the fact that, since -log is strictly convex, equality is achieved in

if and only if (Pi j) is a permutation matrix, which implies ρ = σ, after a suitable labeling of the eigenvectors {vi} and {wi}.

Joint convexity of relative entropy[edit]

The relative entropy is jointly convex. For and states we have

Monotonicity of relative entropy[edit]

The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations on density matrices,

.

This inequality is called Monotonicity of quantum relative entropy.

An entanglement measure[edit]

Let a composite quantum system have state space

and ρ be a density matrix acting on H.

The relative entropy of entanglement of ρ is defined by

where the minimum is taken over the family of separable states. A physical interpretation of the quantity is the optimal distinguishability of the state ρ from separable states.

Clearly, when ρ is not entangled

by Klein's inequality.

Computing relative entropy of entanglement is NP-complete.[1]

Relation to other quantum information quantities[edit]

One reason the quantum relative entropy is useful is that several other important quantum information quantities are special cases of it. Often, theorems are stated in terms of the quantum relative entropy, which lead to immediate corollaries concerning the other quantities. Below, we list some of these relations.

Let ρAB be the joint state of a bipartite system with subsystem A of dimension nA and B of dimension nB. Let ρA, ρB be the respective reduced states, and IA, IB the respective identities. The maximally mixed states are IA/nA and IB/nB. Then it is possible to show with direct computation that

where I(A:B) is the quantum mutual information and S(B|A) is the quantum conditional entropy.

References[edit]

  1. ^ Huang, Yichen (21 March 2014). "Computing quantum discord is NP-complete". New Journal of Physics. 16 (3): 033027. arXiv:1305.5941Freely accessible. Bibcode:2014NJPh...16c3027H. doi:10.1088/1367-2630/16/3/033027.