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Analysis Seminar

Wednesday, 19 September, 2012

SPEAKER: Professor Paul Bourdon

TITLE: Dense Coding with Partially Entangled  Quantum Particles

ABSTRACT: By sliding a coin across a dining table to your friend "Bob", you can convey one of two messages: "heads" (which might mean "pass the salt") or "tails" (which might mean "pass the pepper"). However, if you and Bob share two fully entangled "quantum coins" (one in Bob's possession and one in yours), you can send one of four different messages with your quantum coin. When Bob looks at your quantum coin he will see either heads (H) or tails (T), but you can prepare your coin in such a way that when he looks at both your coin and his, he will see exactly that sequence in {HH, HT, TH, TT} that you wish him to see, and thus receive one of four messages from your two-sided quantum coin. This

is a simple example of the magic of dense coding, in which the quantum coins might be photons (with, e.g., heads = vertically polarized; tails = horizontally polarized). In general, dense coding involves two

parties, customarily called Alice and Bob, with each assumed to possess one of a pair of entangled d-dimensional quantum particles known as qudits. If the pair of qudits that Alice and Bob share is fully

entangled, then in theory Alice can convey one of d-squared messages to Bob via her qudit. These messages are prepared/encoded by Alice by applying to her qudit a physical operation modeled by a d x d unitary matrix; moreover, in order that the messages encoded by Alice never be misinterpreted by Bob, the messages must correspond to orthogonal vectors in the state space of the two-qudit system that Alice and Bob share. The mathematics of this situation will be the focus of the talk, with the principal research-level issue being: how is the number of distinct messages that Alice may encode related to the degree of entanglement of the two-qudit system she and Bob use as the basis for their communication?




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