Superdense coding

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.


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.


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

:sigma_1 = egin{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^+ 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

:egin{bmatrix} frac{1}{n} & ; & ; \ ; & ddots ; \ ; & ; & frac{1}{n} end{bmatrix}

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 "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.


* C. Bennett and S.J. Wiesner. "Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states." Phys. Rev. Lett., 69:2881, 1992 []

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Quantum entanglement — Quantum mechanics Uncertainty principle …   Wikipedia

  • Quantum channel — In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information… …   Wikipedia

  • Quantum information — For the journal with this title, see Historical Social Research. In quantum mechanics, quantum information is physical information that is held in the state of a quantum system. The most popular unit of quantum information is the qubit, a two… …   Wikipedia

  • Qubit — A qubit is not to be confused with a cubit, which is an ancient measure of length. A quantum bit or qubit IPA| [ kju.bɪt] (sometimes also qbit) is a unit of quantum information. That information is described by a state vector in a two level… …   Wikipedia

  • Charge qubit — Circuit diagram of a Cooper pair box circuit. The island (dotted line) is formed by the superconducting electrode between the gate capacitor and the junction capacitance. In quantum computing, a charge qubit is a superconducting qubit whose basis …   Wikipedia

  • Optical lattice — Simulation of an optical lattice potential. An optical lattice is formed by the interference of counter propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the… …   Wikipedia

  • Flux qubit — In quantum computing, flux qubits (also known as persistent current qubits) are micrometer sized loops of superconducting metal interrupted by a number of Josephson junctions. The junction parameters are engineered during fabrication so that a… …   Wikipedia

  • Quantum finite automata — In quantum computing, quantum finite automata or QFA are a quantum analog of probabilistic automata. They are related to quantum computers in a similar fashion as finite automata are related to Turing machines. Several types of automata may be… …   Wikipedia

  • One-way quantum computer — The one way or measurement based quantum computer is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is one way because the… …   Wikipedia

  • Cluster state — In quantum information and quantum computing, a cluster state is a type of highly entangled state of multiple qubits. Cluster states are generated in lattices of qubits with Ising type interactions. A cluster C is a connected subset of a d… …   Wikipedia