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We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob by using a quantum channel while Bob is performing some projective measurements. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.

Holevo’s theorem [

Suppose Alice prepares a state

The Holevo bound states that [

where S is the von Neumann entropy and

Consider the following trace distance between two probability distributions

We can extend the definition to density matrices

where we use

We prove a Holevo-type upper bound on the mutual information of X and Y, where the mutual information is written this time in terms of the HS norm instead of the Kullback-Leibler divergence. It is recently suggested by Ellerman [

The question whether

where

In addition, the HS norm was suggested as an entanglement measure [

In the following, we will prove a Holevo-type bound on the above HS distance between the probability

where

and where

are the partial traces of

where

also known as the linear entropy, purity [

All the above is proved for the case of projective measurements. However, we expect similar results in the general case of POVM, in light of Naimark’s dilation theorem (see [

In the next section we review some basic properties of quantum logical divergence and then use these properties to demonstrate the new Holevo-type bound.

Let

In what follows we recall some basic properties of the HS distance measure, then we state and prove the main result of this paper.

Theorem 2.1. Contractivity of the HS norm with respect to projective measurements

Let

where the projections

Proof: We now write

Theorem 2.2. The joint convexity of the HS norm

The logical divergence

Proof: First observe that

where the inequality is due to the convexity of

Theorem 2.3. The monotonicity of the HS norm with respect to partial trace

Let

where b is the dimension of B.

Proof: One can find a set of unitary matrices

(see [

Observe now that the divergence is invariant under unitary conjugation, and therefore the sum in the right hand side of the above inequality is

We can now state the main result:

Theorem 2.4. A Holevo-type bound for the HS trace distance between

Suppose Alice is using a distribution

where the vectors

where q is the dimension of the space Q.

Proof: First we consider one more auxiliary quantum system, namely M for the measurement outcome for Bob. Initially the system M is in the state

as in Theorem 2.1 above: let

One can easily extend

This can be done by choosing a set of operators, conjugating

Moreover,

If we trace out Q we arrive at

Finally, we can extend

If we trace out Q we get

We can now use the properties stated in the above theorems, Equation (27) and Equation (29) to deduce

where in the first inequality we have used Theorem 2.1 and in the second inequality Theorem 2.3. The final equality is an easy consequence of the definition of the HS norm.

Corollary: Suppose Alice is sending classical information to Bob using a quantum channel Q, Bob measures the quantum state using a projective measurement defined above (having results in space Y). Under all the above assumptions

where

Proof: Clearly (see also [

It is easy to see (by a matrix representation) that for

therefore

However,

Combining this with Theorem 2.4 we find

Example: Suppose Alice sends the state

where

and

Also

The left hand side of the above inequality is a measure of the classical mutual information according to the HS norm between X and Y. The very fact that it is smaller than the Tsallis information measure of X (which is 1/2) means that the quantum channel restricts the rate of classical information transfer, where the mutual information is measured by the HS norm and the source of information X is measured by Tsallis entropy. This is analogous to Holevo’s upper bound in the framework of Tsalis/linear entropy. We find this result similar in spirit to the well-known limitation on the rate of classical information transmission via a quantum channel (without utilizing entanglement): one cannot send more than one bit for each use of the channel using a one qubit channel.

In the above example, if

This gives a bound on the classical mutual information using the quantum “logical entropy” (the Tsallis entropy).

We proved a Holevo-type bound employing the Hilbert-Schmidt distance between the density matrices on the product space

It seems that by utilizing Naimark’s dilation [

As was claimed in [

E.C. was supported by Israel Science Foundation Grant No. 1311/14 and by ERC AdG NLST.

BoazTamir,EliahuCohen,11, (2015) A Holevo-Type Bound for a Hilbert Schmidt Distance Measure. Journal of Quantum Information Science,05,127-133. doi: 10.4236/jqis.2015.54015