# Applied Math Colloquium

### Superadditivity in communication capacity via collective quantum measurements

### Abstract

The overarching theme of this talk will be to explore how to attain the quantum limit of the maximum rate of reliable transmission of (classical) information over a quantum channel. We will consider arguably the simplest setting to develop this problem, that of using two non-orthogonal pure states — of a given inner product — as the symbols of a binary alphabet to encode information, where each “use" of a channel comprises of transmitting one of those two pure states. We will first consider the quantum optimal measurement to distinguish between those two states with the minimum average probability of error, and evaluate the Shannon limit to the capacity (bits per use) when this optimal measurement is used on each symbol by the receiver. We will then evaluate the Holevo limit to the capacity, C_\infty, and quantify the gap between that and the Shannon capacity C_1 of the aforesaid optimal symbol-by-symbol measurement. Next, we will consider an example of a collective (projective) measurement on the two-symbol Hilbert space, whose per-symbol Shannon capacity exceeds C_1. This effect — that a quantum measurement on an n-symbol codeword, one which cannot be expressed as individually measuring each of the n symbols, can attain a higher communication capacity, even though the codeword state on which the measurement is being performed is an n-symbol product state — is known as superadditivity of capacity. Even for this binary pure-state communication problem, C_2, the maximum capacity attainable with a general two-symbol quantum measurement, is not known. We will consider some more explicit examples of superaddivity, and then discuss open problems that underlie our understanding, or lack thereof, of the relationship between the geometry of codeword states and their relationship with optimal collective quantum measurements.