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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.]]
 
[[File:Video superdense coding.ogg|right|thumb|Schematic video demonstrating individual steps of superdense coding. A message consisting of two bits (in video these are (1,&nbsp;0)) is sent from station A to station B using only a single particle. This particle is a member of an entangled pair created by source S. Station A at first applies a properly chosen operation to its particle and then sends it to station B, which measures both particles in the Bell basis. The measurement result retrieves the two bits sent by station A.]]
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]].<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" />
 
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]].<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 |bibcode=1992PhRvL..69.2881B }}</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)|Charles H.&nbsp;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 (1942-20211942–2021)] by [[Scott Aaronson]], which also discusses this topic.</ref> (though not published by them until 1992) and experimentally actualized in 1996 by Klaus Mattle, Harald Weinfurter, Paul G.&nbsp;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]].
 
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.<ref name="Wang2005">{{Cite journal |last1=Wang, C.,|first1=Chuan |last2=Deng, F.|first2=Fu-G.,Guo |last3=Li, Y.|first3=Yan-S.,Song |last4=Liu, X.|first4=Xiao-S., &Shu |last5=Long, G.|first5=Gui L.Lu (|date=2005).-04-28 |title=Quantum secure direct communication with high-dimension quantum superdense coding |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.044305 |journal=Physical Review A, |volume=71( |issue=4) |pages=044305 |doi=10.1103/PhysRevA.71.044305|url-access=subscription }}</ref>
 
== Overview ==
[[File:Superdense coding.png|right|framethumb|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 booleanBoolean, 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.]]
 
Suppose [[Alice and Bob|Alice]] wants to send two classical bits of information (00, 01, 10, or 11) to Bob using [[Qubit|qubits]] (instead of classical [[Bit|bits]]). To do this, an entangled state (e.g. a Bell state) is prepared using a Bell circuit or gate by Charlie, a third person. Charlie then sends one of these qubits (in the Bell state) to Alice and the other to Bob. Once Alice obtains her qubit in the entangled state, she applies a certain quantum gate to her qubit depending on which two-bit message (00, 01, 10 or 11) she wants to send to Bob. Her entangled qubit is then sent to Bob who, after applying the appropriate quantum gate and making a [[Measurement in quantum mechanics|measurement]], can retrieve the classical two-bit message. Observe that Alice does not need to communicate to Bob which gate to apply in order to obtain the correct classical bits from his projective measurement.
 
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=== Preparation ===
 
The protocol starts with the preparation of an entangled state, which is later shared between Alice and Bob. SupposeFor example, the following [[Bell state]]
 
:<math>|\Phi ^{+}\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B)</math>
 
is prepared, where <math>\otimes </math> denotes the [[tensor product]], is prepared. Note: weIn cancommon omitusage the tensor product symbol <math>\otimes</math> andmay writebe the Bell state asomitted:
 
:<math>|\Phi ^{+}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle)</math>.
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=== Sharing ===
 
After the preparation of the Bell state <math>|\Phi ^{+}\rangle</math>, the qubit denoted by subscript ''A'' is sent to Alice and the qubit denoted by subscript ''B'' is sent to Bob (note: this is the reason these states have subscripts). At this point, Alice and Bob may be in completely different locations, which might bean veryunlimited distantdistance from each other.
 
There may be aan longarbitrary period of time between the preparation and sharing of the entangled state <math>|\Phi ^{+}\rangle</math> and the rest of the steps in the procedure.
 
=== Encoding ===
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: <math>|B_{00}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle)</math>
In other words, the entangled state shared between Alice and Bob has not changed, i.e. it is still <math>|\Phi ^{+}\rangle</math>. The notation <math>|B_{00}\rangle</math> is also used to remind us of the factindicates 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 logic gate#X|quantum ''NOT'' (or ''bit-flip'') gate]], <math>X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} </math>, to her qubit, so that the resultant entangled quantum state becomes
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:<math>|B_{10}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle - |1_A1_B\rangle)</math>
 
4. If, instead, Alice wants to send the classical two-bit string 11 to Bob, then she applies the quantum gate <math>Z*X = iY = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} </math> to her qubit, so that the resultant entangled state becomes
 
:<math>|B_{11}\rangle = \frac{1}{\sqrt{2}}(|0_A1_B\rangle -|1_A0_B\rangle ) </math>
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==== Example ====
 
For example, if the resultant entangled state (after the operations performed by Alice) was <math>B_{01} = \fractfrac{1}{\sqrt{2}}(|1_A0_B\rangle + |0_A1_B\rangle)</math>, then a CNOT with A as control bit and B as target bit will change <math>B_{01}</math> to become <math>B_{01}' = \fractfrac{1}{\sqrt{2}}(|1_A1_B\rangle + |0_A1_B\rangle)</math>. Now, the Hadamard gate is applied only to A, to obtain
 
: <math>
<math>B_{01}'' = \frac{1}{\sqrt{2}} \left({\left(\frac{1}{\sqrt{2}}(|0 \rangle - |1 \rangle) \right) }_A \otimes
|1_B\rangle B_{01}'' += {\left(\fractfrac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle) \right) }_A \otimesleft[
{\left(\fractfrac{1}{\sqrt{2}}(|0 \rangle - |1 \rangle) \right)}_A \otimes |11_B\rangle +
|1_B\rangle\right)</math>
<math>B_{01}'' = \frac{1}{\sqrt{2}} \left({\left(\fractfrac{1}{\sqrt{2}}(|0 \rangle -+ |1 \rangle) \right) }_A \otimes |1_B\rangle
 
\right)].
For simplicity, let's get rid of the subscripts, so we have
</math>
 
For simplicity, let's get rid of the subscripts, somay webe haveremoved:
<math>B_{01}'' =
: <math>
\frac{1}{\sqrt{2}}
\begin{align}
\left(
<math> B_{01}'' &=
\frac{1}{\sqrt{2}}(|0 \rangle - |1 \rangle) \otimes |1\rangle +
\fractfrac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle) \otimes |1\rangle
\left(
\right)
\fractfrac{1}{\sqrt{2}}(|01 0\rangle - |11 1\rangle) \otimes |1\rangle +
=
\fractfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes |1\rangle
\right) \\
\left(
&=
\frac{1}{\sqrt{2}}(|01 \rangle - |11 \rangle) +
\fractfrac{1}{\sqrt{2}}(|01 \rangle + |11 \rangle)
\left(
\right)
\fractfrac{1}{\sqrt{2}}(|01 \rangle +- \frac{1}{2}|11 \rangle) +
=
\fractfrac{1}{\sqrt{2}}(|01 \rangle - \frac{1}{2}+ |11 \rangle +)
\right) =
\frac{1}{2}|01 \rangle + \frac{1}{2}|11 \rangle
= \tfrac{1}{2}|01\rangle - \tfrac{1}{2}|11\rangle +
\tfrac{1}{2}|01\rangle + \tfrac{1}{2}|11\rangle
= |01\rangle.
\end{align}
</math>
 
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== Experimental ==
The protocol of superdense coding has been actualized in several experiments using different systems to varying levels of channel capacity and fidelities. In 2004, trapped [[beryllium -9]] ions were used in a maximally entangled state to achieve a channel capacity of 1.16 with a fidelity of 0.85.<ref name="Schaetz2004">{{Cite journal |last1=Schaetz, |first1=T., |last2=Barrett, |first2=M. D., |last3=Leibfried, |first3=D., |last4=Chiaverini, |first4=J., |last5=Britton, |first5=J., |last6=Itano, |first6=W. M., |last7=Jost |first7=J. D. |last8=Langer |first8=C. |last9=Wineland, |first9=D. J. (|date=2004).-07-22 |title=Quantum Dense Coding with Atomic Qubits |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.93.040505 |journal=Physical Review Letters, |volume=93( |issue=4) |pages=040505 |doi=10.1103/PhysRevLett.93.040505|pmid=15323743 |url-access=subscription }}</ref> In 2017, a channel capacity of 1.665 was achieved with a fidelity of 0.87 through optical fibers.<ref name="William2017">{{Cite journal |last1=Williams |first1=Brian P. |last2=Sadlier |first2=Ronald J. |last3=Humble |first3=Travis S. |date=2017-02-01 |title=Superdense Coding over Optical Fiber Links with Complete Bell-State Measurements |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.050501 |journal=Physical Review Letters |volume=118 |issue=5 |pages=050501 |doi=10.1103/PhysRevLett.118.050501|pmid=28211745 |arxiv=1609.00713 }}Williams, B. P., Sadlier, R. J., & Humble, T. S. (2017). Superdense Coding over Optical Fiber Links with Complete Bell-State Measurements. Physical Review Letters, 118(5).</ref> High -dimensional [[ququarts]] (states formed in photon pairs by non-degenerate [[spontaneous parametric down-conversion]]) were used to reach a channel capacity of 2.09 (with a limit of 2.32) with a fidelity of 0.98.<ref name="Hu2018">{{Cite journal |last1=Hu, X.|first1=Xiao-M.,Min |last2=Guo, Y.,|first2=Yu |last3=Liu, B.|first3=Bi-H.,Heng |last4=Huang, Y.|first4=Yun-F.,Feng |last5=Li, C.|first5=Chuan-F., &Feng |last6=Guo, G.|first6=Guang-C.Can (|date=2018).-07-06 |title=Beating the channel capacity limit for superdense coding with entangled ququarts. |journal=Science Advances, |language=en |volume=4( |issue=7), |pages=eaat9304 |doi=10.1126/sciadv.aat9304 |issn=2375-2548 |pmc=6054506 |pmid=30035231}}</ref> [[Nuclear Magneticmagnetic Resonanceresonance]] (NMR) has also been used to share among three parties.<ref name="Wei2004">{{Cite journal |last1=Wei, D.,|first1=Daxiu |last2=Yang, X.,|first2=Xiaodong |last3=Luo, J.,|first3=Jun |last4=Sun, X.,|first4=Xianping |last5=Zeng, X., &|first5=Xizhi |last6=Liu, M.|first6=Maili (|date=2004).-03-01 |title=NMR experimental implementation of three-parties quantum superdense coding |url=https://link.springer.com/article/10.1007/BF02900957 |journal=Chinese Science Bulletin, |language=en |volume=49( |issue=5), |pages=423–426 |doi=10.1007/BF02900957 |issn=1861-9541|url-access=subscription }}</ref>
 
==References==
{{reflist}}
 
== Further reading ==
* Wilde, Mark M., 2017, [http://www.cambridge.org/us/academic/subjects/computer-science/cryptography-cryptology-and-coding/quantum-information-theory-2nd-edition Quantum Information Theory, Cambridge University Press], Also available at [https://arxiv.org/abs/1106.1445 eprint arXiv:1106.1145]