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.
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.
We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.
We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.
The mutual information between the sender of a classical message encoded in quantum carriers and a receiver is fundamentally limited by the Holevo quantity. Using strong subadditivity of entropy, we prove that the Holevo quantity is not larger than an exchange entropy. This implies an upper bound for coherent information. Moreover, restricting our attention to classical information, we bound the transmission distance between probability distributions by their entropic distance, which is a concave function of their Hellinger distance.
Information transmission via a quantum channel is one of the most important processes considered by the quantum information theory. Quantum channels obey nonequilibrium quantum dynamics. The task of the quantum information theory investigates properties of information transmission by means of quantum channels. The von Neumann mutual information and Holevo capacity are useful entropic quantities for this purpose. This short note discusses the relation between the von Neumann mutual information and the Holevo function and also provides a useful formula for calculating these quantities. The result is applied to the quantum dense coding under the influence of an external environment, which is the generalization of the results of ref. 4 Suppose that Alice sends a signal state carrying classical information to Bob through a quantum channel described by an arbitrary trace-preserving completely positive map L. The signal states are represented by orthonormal quantum states j ki (k 1⁄4 1; 2; . . .) which are assumed to be realized with prior probabilities k (k 1⁄4 1; 2; . . .). Then the quantum state of Alice’s system, i.e., the input system of the quantum channel, is given by in 1⁄4 P k k ðkÞ in , where ðkÞ in 1⁄4 j kih kj. In this case, we have the eigenvalue equation, inj ki 1⁄4 kj ki. After receiving the signal, Bob obtains the quantum state ðkÞ out 1⁄4 Lj kih kj with the probability k and thus the quantum state of the output system becomes out 1⁄4 Lin 1⁄4 P k k ðkÞ out. Then we consider the eigenvalue problems, outj ni 1⁄4 nj ni and ðkÞ outj ðkÞ n i 1⁄4 ðkÞ n j ðkÞ n i. To obtain the von Neumann mutual information between the input and output systems of the quantum channel L, it is necessary to introduce a composite state of the input and output systems, which is not uniquely determined. Since we now consider the transmission of classical information, we use a separable state in{out as a composite state, 5,6) in{out 1⁄4 X
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (Holevo's, Shumacher-Westmoreland-Wootters', Hall's, Scutaru's bounds, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.
We study quantum channels that are close to another channel with weakly additive Holevo information, and we derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above, which measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.