Superdense coding: Difference between revisions

Revision as of 01:16, 9 March 2007

Revision as of 16:15, 9 April 2007

Line 1:

'''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 [[Quantum entanglement|entanglement]].

== The ideaOverview ==

Suppose Alice would like to send classical information to Bob using qubits. 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 can not 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. We next describe this prodedure.

== The resultResultant ==

Crucial to thethis 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 such state via local manipulation.

Suppose parts of a [[Bell state]], say

Line 15:

</math>

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 so surprising, sincefor entanglement can notcannot be broken using local operations):

* Obviously, if Alice does nothing, the system remains in the state <math>|\Psi^+\rangle</math>.

Line 29:

* If <math>\sigma_1</math> is replaced by <math>\sigma_3</math>, the initial state <math>|\Psi^+\rangle </math> is transformed into <math>|\Psi^-\rangle </math>.

* Similarly, if Alice applies <math>i \sigma_2 \otimes I</math> to the system, the resultingresultant state is <math>|\Phi^-\rangle </math>

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. NoTherefore, no useful information whatsoever is gained by Eve, unless she can interact with Bob's qubit, that is, establish an entangled state.

== General dense coding scheme ==

Line 64:

==References==

* C. Bennett and S.J. Wiesner. ''Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states.'' Phys. Rev. Lett., 69:2881, 1992[http://prola.aps.org/abstract/PRL/v69/i20/p2881_1]