Superdense coding

Superdense coding is a technique used in quantum information theory to send two bits of classical information using only one qubit, with the aid of entanglement.

Contents

Overview

Suppose Alice would like to send classical information to Bob using qubits, instead of classical bits. Alice would encode the classical information in a qubit and send it to Bob. After receiving the qubit, Bob recovers the classical information via measurement. The question is: how much classical information can be transmitted per qubit? Since non-orthogonal quantum states cannot be distinguished reliably, one would guess that Alice can do no better than one classical bit per qubit. Indeed this bound on efficiency has been proven formally. Thus there is no advantage gained in using qubits instead of classical bits. However, with the additional assumption that Alice and Bob share an entangled state, two classical bits per qubit can be achieved. The term superdense refers to this doubling of efficiency.

Details

Crucial to this procedure is the shared entangled state between Alice and Bob, and the property of entangled states that a (maximally) entangled state can be transformed into another state via local manipulation.

Suppose parts of a Bell state, say

|\Psi^%2B\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B %2B |1\rangle_A \otimes |0\rangle_B)

are distributed to Alice and Bob. The first subsystem, denoted by subscript A, belongs to Alice and the second, B, system to Bob. By only manipulating her particle locally, Alice can transform the composite system into any one of the Bell states (this is not entirely surprising, for entanglement cannot be broken using local operations):

\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

(notice this is one of the Pauli matrices), the total two-particle system now is in state

( \sigma_1 \otimes I ) |\Psi^%2B\rangle = |\Phi^%2B\rangle .

So, depending on the message she would like to send, Alice performs one of the four local operations given above and sends her qubit to Bob. By performing a projective measurement in the Bell basis on the two particle system, Bob decodes the desired message.

Notice, however, that if some mischievous person, Eve, intercepts Alice's qubit en route to Bob, all that is obtained by Eve is part of an entangled state. Therefore, no useful information whatsoever is gained by Eve unless she can interact with Bob's qubit.

General dense coding scheme

General dense coding schemes can be formulated in the language used to describe quantum channels. Alice and Bob share a maximally entangled state ω. Let the subsystems initially possessed by Alice and Bob be labeled 1 and 2, respectively. To transmit the message x, Alice applies an appropriate channel

\; \Phi_x

on subsystem 1. On the combined system, this is effected by

\omega \rightarrow (\Phi_x \otimes Id)(\omega)

where Id denotes the identity map on subsystem 2. Alice then sends her subsystem to Bob, who performs a measurement on the combined system to recover the message. Let the effects of Bob's measurement be Fy. The probability that Bob's measuring apparatus registers the message y is

\operatorname{Tr}\; (\Phi_x \otimes Id)(\omega) \cdot F_y .

Therefore, to achieve the desired transmission, we require that

\operatorname{Tr}\; (\Phi_x \otimes Id)(\omega) \cdot F_y = \delta_{xy}

where δxy is the Kronecker delta.

References

General
Quantum communication
Quantum algorithms
Quantum complexity theory
Quantum computing models
Decoherence prevention
Physical implementations
Quantum optics
Ultracold atoms
Spin-based
Superconducting quantum computing