  The communication cost of a classical protocol is typically measured in terms
of the number of bits communicated for this determines the time required for
communication during the protocol. Similarly, for quantum communication
protocols, which use finite-dimensional quantum states, the communication cost
is measured in terms of the number of qubits communicated. However, in quantum
physics, one can also use infinite-dimensional states, like optical quantum
states, for communication protocols. Communication cost measures based on
counting the (equivalent) number of qubits transmitted during communication
cannot be directly used to measure the cost of such protocols, which use
infinite-dimensional states. Moreover, one cannot infer any physical property
of infinite-dimensional protocols using such qubit based communication costs.
In this paper, we provide a framework to understand the growth of physical
resources in infinite-dimensional protocols. We focus on optical protocols for
the sake of concreteness. The time required for communication and the energy
expended during communication are identified as the important physical
resources of such protocols. In an optical protocol, the time required for
communication is determined by the number of time-bin modes that are
transmitted from one party to another. The mean photon number of the messages
sent determines the energy required during communication in the protocol. We
prove a lower bound on the tradeoff between the growth of these two quantities
with the growth of the problem size. We call such tradeoff relations optical
quantum communication complexity relations.
