The adaptive classical capacity of a quantum channel, or Information capacities of three symmetric pure states in three dimensions

Abstract

We investigate the capacity of three symmetric quantum states in three real dimensions to carry classical information. Several such capacities have already been defined, depending on what operations are allowed in the protocols that the sender uses to encode classical information into these quantum states, and that the receiver uses to decode it. These include the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,1</inf> capacity, which is the capacity achievable if separate measurements must be used for each of the received states, and the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,∞</inf> capacity, which is the capacity achievable if joint measurements are allowed on the tensor product of all of the received states. We discover a new classical information capacity of quantum channels, the adaptive capacity C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,A</inf> , which lies strictly between the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,1</inf> and the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,∞</inf> capacities. The adaptive capacity allows the use of what is known as the LOCC (local operations and classical communication) model of quantum operations for decoding the channel outputs. This model requires each of the signals to be measured by a separate apparatus, but allows the quantum states of these signals to be measured in stages, with the first stage partially reducing their quantum states; measurements in subsequent stages may depend on the results of a classical computation taking as input the outcomes of the first round of measurements. We also show that even in three dimensions, with the information carried by an ensemble containing three pure states, achieving the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,1</inf> capacity may require a positive operator valued measure (POVM) with six outcomes.