superdense coding

In quantum information theory, superdense coding (also referred to as dense coding) is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assumption of sender and receiver pre-sharing an entangled resource. In its simplest form, the protocol involves two parties, often referred to as Alice and Bob in this context, which share a pair of maximally entangled qubits, and allows Alice to transmit two bits (i.e., one of 00, 01, 10 or 11) to Bob by sending only one qubit. This protocol was first proposed by Charles H. Bennett and Stephen Wiesner in 1970 (though not published by them until 1992¹) and experimentally actualized in 1996 by Klaus Mattle, Harald Weinfurter, Paul G. Kwiat and Anton Zeilinger using entangled photon pairs. Superdense coding can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair. The transmission of two bits via a single qubit is made possible by the fact that Alice can choose among four quantum gate operations to perform on her share of the entangled state. Alice determines which operation to perform accordingly to the pair of bits she wants to transmit. She then sends Bob the qubit state evolved through the chosen gate.…

In quantum information theory, superdense coding (also referred to as dense coding) is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assumption of sender and receiver pre-sharing an entangled resource. In its simplest form, the protocol involves two parties, often referred to as Alice and Bob in this context, which share a pair of maximally entangled qubits, and allows Alice to transmit two bits (i.e., one of 00, 01, 10 or 11) to Bob by sending only one qubit. This protocol was first proposed by Charles H. Bennett and Stephen Wiesner in 1970 (though not published by them until 1992) and experimentally actualized in 1996 by Klaus Mattle, Harald Weinfurter, Paul G. Kwiat and Anton Zeilinger using entangled photon pairs. Superdense coding can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair. The transmission of two bits via a single qubit is made possible by the fact that Alice can choose among four quantum gate operations to perform on her share of the entangled state. Alice determines which operation to perform accordingly to the pair of bits she wants to transmit. She then sends Bob the qubit state evolved through the chosen gate. Said qubit thus encodes information about the two bits Alice used to select the operation, and this information can be retrieved by Bob thanks to pre-shared entanglement between them. After receiving Alice's qubit, operating on the pair and measuring both, Bob obtains two classical bits of information. It is worth stressing that if Alice and Bob do not pre-share entanglement, then the superdense protocol is impossible, as this would violate Holevo's theorem. Superdense coding is the underlying principle of secure quantum secret coding. The necessity of having both qubits to decode the information being sent eliminates the risk of eavesdroppers intercepting messages. |\Phi ^{+}\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B) is prepared, where \otimes denotes the tensor product. In common usage the tensor product symbol \otimes may be omitted: |\Phi ^{+}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle) . 1. If Alice wants to send the classical two-bit string 00 to Bob, then she applies the identity quantum gate, \mathbb{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} , to her qubit, so that it remains unchanged. The resultant entangled state is then |B_{00}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle) In other words, the entangled state shared between Alice and Bob has not changed, i.e. it is still |\Phi ^{+}\rangle . The notation |B_{00}\rangle indicates that Alice wants to send the two-bit string 00. 2. If Alice wants to send the classical two-bit string 01 to Bob, then she applies the quantum NOT (or bit-flip) gate, X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , to her qubit, so that the resultant entangled quantum state becomes |B_{01}\rangle = \frac{1}{\sqrt{2}}(|1_A0_B\rangle + |0_A1_B\rangle) 3. If Alice wants to send the classical two-bit string 10 to Bob, then she applies the quantum phase-flip gate Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} to her qubit, so the resultant entangled state becomes |B_{10}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle - |1_A1_B\rangle) 4. If, instead, Alice wants to send the classical two-bit string 11 to Bob, then she applies the quantum gate Z*X = iY = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} to her qubit, so that the resultant entangled state becomes |B_{11}\rangle = \frac{1}{\sqrt{2}}(|0_A1_B\rangle -|1_A0_B\rangle ) The matrices X , Z , and Y are known as Pauli matrices. For example, if the resultant entangled state (after the operations performed by Alice) was B_{01} = \tfrac{1}{\sqrt{2}}(|1_A0_B\rangle + |0_A1_B\rangle) , then a CNOT with A as control bit and B as target bit will change B_{01} to become B_{01}' = \tfrac{1}{\sqrt{2}}(|1_A1_B\rangle + |0_A1_B\rangle) . Now, the Hadamard gate is applied only to A, to obtain B_{01} = \tfrac{1}{\sqrt{2}} \left[ {\left(\tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle)\right)}_A \otimes |1_B\rangle + {\left(\tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\right)}_A \otimes |1_B\rangle \right]. For simplicity, subscripts may be removed: \begin{align} B_{01} &= \tfrac{1}{\sqrt{2}} \left( \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle) \otimes |1\rangle + \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes |1\rangle \right) \\ &= \tfrac{1}{\sqrt{2}} \left( \tfrac{1}{\sqrt{2}}(|01 \rangle - |11\rangle) + \tfrac{1}{\sqrt{2}}(|01 \rangle + |11\rangle) \right) = \tfrac{1}{2}|01\rangle - \tfrac{1}{2}|11\rangle + \tfrac{1}{2}|01\rangle + \tfrac{1}{2}|11\rangle = |01\rangle. \end{align} Now, Bob has the basis state |01\rangle , so he knows that Alice wanted to send the two-bit string 01. 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 Bob's measurement be modelled by a POVM \{F_y\}_y , with F_y positive semidefinite operators such that \sum_y F_y=I . The probability that Bob's measuring apparatus registers the message y is thus p(y|x)=\langle F_y, (\Phi_x \otimes I)(\omega)\rangle\equiv \operatorname{Tr}[ F_y(\Phi_x \otimes I)(\omega)]. Therefore, to achieve the desired transmission, we require that p(y|x)=\operatorname{Tr}[F_y (\Phi_x \otimes I)(\omega)] = \delta_{xy}, where \delta_{xy} is the Kronecker delta. ==References== ==External links== Qubits, Quantum Mechanics and Computers Course Notes by Michael Nielsen

Text derived from https://en.wikipedia.org/w/index.php?oldid=1316144647 under CC BY-SA 4.0; modifications have been made.