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.

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

:$|Psi^+\; angle\; =\; frac\{1\}\{sqrt\{2\; (|0\; angle\_A\; otimes\; |1\; angle\_B\; +\; |1\; angle\_A\; otimes\; |0\; angle\_B)$

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):

* Obviously, if Alice does nothing, the system remains in the state $|Psi^+\; angle$.

* If Alice sends her particle through the unitary gate

:

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

:$(\; sigma\_1\; otimes\; I\; )\; |Psi^+\; angle\; =\; |Phi^+\; angle\; .$

* If $sigma\_1$ is replaced by $sigma\_3$, the initial state $|Psi^+\; angle$ is transformed into $|Psi^-\; angle$.

* Similarly, if Alice applies $i\; sigma\_2\; otimes\; I$ to the system, the resultant state is $|Phi^-\; angle$

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", i.e.

:$omega\; in\; H\; otimes\; H$

has the maximally mixed state

:

as its partial trace. 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

:$omega\; ightarrow\; (Phi\_x\; otimes\; Id)(omega)$

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 "F_y". 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 "δ_xy" is the Kronecker delta.

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]