Superdense coding: Difference between revisions

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{{merge|Quantum dense coding}}

'''Superdense coding''' is a technique used in [[quantum information theory]]. Itto usessend [[quantumtwo entanglement]]bits toof sendclassical data.information Theusing processonly canone bequbit, explainedwith asthe follows:aid of [[Quantum entanglement|entanglement]].

== The idea ==

If Alice wished to send information to Bob by sending qubits, it might seem at first that she can only send one bit per qubit. For example, if she wished to send the bit 0 she could send a qubit in the state <math>|0\rangle</math> and if she wished to send 1 she could send a qubit in the state <math>|1\rangle</math>. It turns out, however, that entanglement can be used to transfer information more efficiently than this. The process by which this is possible is known as superdense coding. The basic principle of the process is simple, Alice has two bits of information that she wishes to send to Bob and wants to do this by sending only one qubit. This can be done if Alice and Bob each have one half of an entangled pair of qubits. By using local operations on her qubit, Alice can cause the pair to be in any of the four [[Bell state]]s <math>|\Phi^+\rangle</math>, <math>|\Phi^-\rangle</math>, <math>|\Psi^+\rangle</math> and <math>|\Psi^-\rangle</math> of her choosing. If Alice then sends her qubit to Bob then he can use a [[Bell measurement]] to determine the state of the pair. If Alice and Bob had previously agreed that Alice would cause the pair to be in state <math>|\Phi^+\rangle</math> if she wanted to send the bit stream 00, <math>|\Phi^-\rangle</math> for 01, <math>|\Psi^+\rangle</math> for 10 and <math>|\Psi^-\rangle</math> for 11 then with the results of the Bell state measurement Bob would have received two bits of information from Alice, even though she only sent one qubit to him.

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. We next describe this prodedure.

== The result ==

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

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

:<math>

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

</math>

is 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, since entanglement can not be broken using local operations):

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

* If Alice sends her particle through the unitary gate

:<math>\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}</math>

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

:<math>( \sigma_1 \otimes I ) |\Psi^+\rangle = |\Phi^+\rangle .</math>

* 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 resulting 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.

[[Category:Quantum information science]]