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[[File:Superdense coding.png|right|frame|When the sender and receiver share a Bell state, two classical bits can be packed into one qubit. In the diagram, lines carry [[qubit]]s, while the doubled lines carry classic [[bit]]s. The variables b<sub>1</sub> and b<sub>2</sub> are classic boolean, and the zeroes at the left hand side represent the pure [[quantum state]] <math>|0\rangle</math>. See the section named "[[Superdense coding#The protocol|The protocol]]" below for more details regarding this picture.]]
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 received 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]].<ref name="bennett1992communication">{{cite journal|last1=Bennett|first1=C.|last2=Wiesner|first2=S.|year=1992|title=Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states|journal=Physical Review Letters|volume=69|issue=20|pages=2881–2884|doi=10.1103/PhysRevLett.69.2881|pmid=10046665}}</ref><ref name="NielsenChuang2010">{{cite book|url=https://books.google.com/books?id=-s4DEy7o-a0C|title=Quantum Computation and Quantum Information: 10th Anniversary Edition|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac L.|date=9 December 2010|publisher=Cambridge University Press|isbn=978-1-139-49548-6|page=97|section=2.3 Application: superdense coding}}</ref> This protocol was first proposed by [[Charles H. Bennett (physicist)|Bennett]] and [[Stephen Wiesner|Wiesner]] in 1970<ref>[https://orsattath.wordpress.com/2021/08/14/stephen-wiesner/ Stephen Wiesner] Memorial blog post by Or Sattath, with scan of Bennett's handwritten notes from 1970. See also [https://www.scottaaronson.com/blog/?p=5730 Stephen Wiesner (1942-2021)] by [[Scott Aaronson]], which also discusses this topic.</ref> (though not published by them until 1992) and experimentally actualized in 1996 by Mattle, Weinfurter, Kwiat and [[Anton Zeilinger|Zeilinger]] using entangled photon pairs.<ref name="NielsenChuang2010" /> 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.<ref name="NielsenChuang2010" />
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]].
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