Super-dense coding

Fri, 23/10/2015 - 17:54 by Anonymous

'''Superdense coding''' (aka quantum dense coding, or dense coding) is a method of utilizing shared quantum [[entanglement]] to increase the rate at which information may be sent through a noiseless quantum channel. Sending a single [[qubit]] noiselessly between two parties gives a maximum rate of communication of one [[bit]] per qubit (by the [[HSW]] Theorem). If the sender's qubit is maximally entangled with a qubit in the receiver's possession, then dense coding increases the maximum rate to two bits per qubit. ==Dense Coding for Qubits== If the sender ([[Alice]]) and receiver ([[Bob]]) share a maximally entangled state :|\Psi^+_{AB}\rangle = \frac{1}{\sqrt{2}}\big( |0_A\rangle |0_B \rangle + |1_A\rangle |1_B \rangle then Alice may perform encode two bits of information into the shared state by using one of four [[unitary]] operations corresponding to the different two bit strings. The operations consist of the identity (doing nothing), a bit flip \sigma_X (where |0\rangle \rightarrow |1\rangle and |1\rangle \rightarrow |0\rangle), a phase flip \sigma_Z (where |0\rangle \rightarrow |0\rangle and |1\rangle \rightarrow -|1\rangle), or a combination of both \sigma_Y. After encoding, Alice and Bob share one of the states :00 \rightarrow (\mathbb{I}_A \otimes \mathbb{I}_B)|\Psi^+_{AB}\rangle = |\Psi^+_{AB}\rangle :01 \rightarrow (\sigma^X_A \otimes \mathbb{I}_B)|\Psi^+_{AB}\rangle = \frac{1}{\sqrt{2}}\big( |1_A\rangle |0_B \rangle + |0_A\rangle |1_B \rangle = |\Phi^+_{AB}\rangle :10 \rightarrow (\sigma^Y_A \otimes \mathbb{I}_B)|\Psi^+_{AB}\rangle = \frac{-i}{\sqrt{2}}\big( |1_A\rangle |0_B \rangle - |0_A\rangle |1_B \rangle = |\Phi^-_{AB}\rangle :11 \rightarrow (\sigma^Z_A \otimes \mathbb{I}_B)|\Psi^+_{AB}\rangle = \frac{1}{\sqrt{2}}\big( |0_A\rangle |0_B \rangle - |1_A\rangle |1_B \rangle = |\Psi^-_{AB}\rangle These resultant shared states are orthogonal, and if Alice sends her state to Bob he can undertake an orthogonal [[measurement]] to determine which of the four operators Alice used, and hence determine what the original two bits of Alice's message are. ==See Also== [[Entanglement]] [[Entanglement assisted classical capacity]] [1] C. H. Bennett and Stephen J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992). [[Category:quantum Information Theory]]