Trading quantum for classical resources in quantum data compression

We study the visible compression of a source $E={|φi〉,pi}$ of pure quantum signal states or, more formally, the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity, when the compressor is given the identity of the input state sequence as classical information. According to the quantum source coding theorem, the optimal quantum rate is the von Neumann entropy $S(E)$ qubits per signal. We develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately. This leads to a trade-off curve $Q*(R),$ where $Q*(R)$ qubits per signal is the optimal quantum rate for a given classical rate of $R$ bits per signal. Our main result is an explicit characterization of this trade-off function by a simple formula in terms of only single-signal, perfect fidelity encodings of the source. We give a thorough discussion of many further mathematical properties of our formula, including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parametrized states. We also show that our result leads to a number of corollaries characterizing the trade-off between information gain and state disturbance for quantum sources. In addition, we indicate how our techniques also provide a solution to the so-called remote state preparation problem. Finally, we develop a probability-free version of our main result which may be interpreted as an answer to the question: “How many classical bits does a qubit cost?” This theorem provides a type of dual to Holevo’s theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.