Superdense coding

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

Overview[edit]

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. Holevo's theorem discusses this bound on efficiency. 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. Also, it can be proved that the maximum amount of classical information that can be sent (even while using entangled state) using one qubit is 2 bits.

Details[edit]

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

|B_{00}\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\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):

  • Suppose, Alice wants to sent the classical bits 00

Then she will perform Identity unitary operation on her particle. Obviously, her entangled qubit remains unchanged. The resultant tangled qubit would be |B_{00}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle)

  • Suppose, Alice wants to sent 01 bits. Then she will perform X unitary operation.


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

X is like a quantum NOT gate. After the application of X unitary gate the resultant entangled quantum state would be 
  |B_{01}\rangle = \frac{1}{\sqrt{2}}(|1_A0_B\rangle + |0_A1_B\rangle)

  • Suppose, Alice wants to sent 10 bits. Then she will perform Z unitary operation.


  Z = 
  \begin{bmatrix} 
    1 & 0 \\
    0 & -1 \\
  \end{bmatrix}
After the application of Z unitary gate the resultant entangled quantum state would be 
  |B_{10}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle - |1_A1_B\rangle)

  • Suppose, Alice wants to sent 11 bits. Then she will perform XZ unitary operation.

After the application of XZ unitary gate the resultant entangled quantum state would be 
  |B_{11}\rangle = \frac{1}{\sqrt{2}}(|1_A0_B\rangle - |0_A1_B\rangle)


  X,Z,I,XZ(=Y)
are Pauli gates. 
  B_{00}, B_{01}, B_{10}, B_{11}
are called Bell states.

Now, if Bob wants to find which classical bits did Alice wants to send he will perform the CNOT unitary operation followed by H\otimes Iunitary operation on the entangled qubit.

  • If the resultant entangled qubit was B_{00} then after the application of the above unitary operations the entangled qubit will become |00\rangle
  • If the resultant entangled qubit was B_{01} then after the application of the above unitary operations the entangled qubit will become |01\rangle
  • If the resultant entangled qubit was B_{10} then after the application of the above unitary operations the entangled qubit will become |10\rangle
  • If the resultant entangled qubit was B_{11} then after the application of the above unitary operations the entangled qubit will become |11\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[edit]

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 I)(\omega)

where I 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 I)(\omega) \cdot F_y .

Therefore, to achieve the desired transmission, we require that

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

where δxy is the Kronecker delta.

References[edit]

  • C. Bennett and S.J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69:2881, 1992 [1]
  • Birgitta Whaley. Qubits, Quantum Mechanics and Computers.[2]