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{{Use American English|date=January 2019}}{{Short description|Two-bit quantum communication protocol}}

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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. See the section below "The protocol" 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, 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''' (or dense coding) is a [[quantum communication]] protocol to transmit two classical bits of information (i.e., either 00, 01, 10 or 11) from a sender (often called Alice) to a receiver (often called Bob), by sending only one [[qubit]] from Alice to Bob, under the assumption of Alice and Bob pre-sharing an entangled state.<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 1992 and experimentally actualized in 1996 by Mattle, Weinfurter, Kwiat and [[Anton Zeilinger|Zeilinger]] using entangled photon pairs.<ref name="NielsenChuang2010" /> By performing one of four [[quantum gate]] operations on the (entangled) qubit she possesses, Alice can prearrange the measurement Bob makes. After receiving Alice's qubit, operating on the pair and measuring both, Bob has two classical bits of information. If Alice and Bob do not already share entanglement before the protocol begins, then it is impossible to send two classical bits using 1 qubit, as this would violate [[Holevo's theorem]].

In [[quantum information theory]], '''superdense coding''' (oralso referred to as ''dense coding'') is a [[quantum communication]] protocol to transmitcommunicate twoa number of classical bits of information (i.e.,by eitheronly 00,transmitting 01,a 10smaller ornumber 11)of fromqubits, aunder the assumption of sender (oftenand calledreceiver Alice)pre-sharing toan aentangled receiverresource. (oftenIn calledits Bob)simplest form, bythe sendingprotocol onlyinvolves onetwo parties, often referred to as [[qubitAlice and Bob]] fromin this context, which share a pair of maximally entangled qubits, and allows Alice to Bob,transmit undertwo thebits assumption(i.e., one of Alice00, and01, 10 or 11) to Bob pre-sharingby ansending entangledonly stateone [[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 19921970<ref>[https://orsattath.wordpress.com/2021/08/14/stephen-wiesner/ andStephen experimentallyWiesner]. actualizedMemorial inblog 1996post by Mattle,Or WeinfurterSattath, Kwiatwith andscan [[Antonof Zeilinger|Zeilinger]]Bennett's usinghandwritten entanglednotes photonfrom pairs1970.<ref name="NielsenChuang2010"See also [https:/>/www.scottaaronson.com/blog/?p=5730 ByStephen performingWiesner one(1942–2021)] of fourby [[quantumScott gateAaronson]], operationswhich onalso thediscusses this topic.</ref> (entangled)though qubitnot shepublished possesses,by Alicethem canuntil prearrange1992) theand measurementexperimentally Bobactualized makes.in After1996 receivingby Alice'sKlaus qubitMattle, operatingHarald onWeinfurter, thePaul pairG.&nbsp;Kwiat and measuring[[Anton both,Zeilinger]] Bobusing hasentangled twophoton classicalpairs.<ref bitsname="NielsenChuang2010" of/> information.Superdense Ifcoding Alicecan andbe Bobthought doof notas alreadythe shareopposite entanglementof before[[quantum theteleportation]], protocolin begins,which thenone ittransfers isone impossiblequbit from Alice to sendBob by communicating two classical bits, usingas 1 qubit,long as thisAlice wouldand violateBob [[Holevo'shave theorem]]a pre-shared Bell pair.<ref name="NielsenChuang2010" />

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">Wang, C., Deng, F.-G., Li, Y.-S., Liu, X.-S., & Long, G. L. (2005). Quantum secure direct communication with high-dimension quantum superdense coding. Physical Review A, 71(4).</ref>

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]].

It 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/>

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 b₁ and b₂ are classic Boolean, and the zeroes at the left-hand side represent the pure [[quantum state]] <math>|0\rangle</math>. See the section belownamed "[[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.

== The protocol ==

Line 24 ⟶ 23:

=== 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>and writemay thebe Bell state asomitted:

:<math>|\Phi ^{+}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle)</math>.

Line 34 ⟶ 33:

=== 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 ===

Line 46 ⟶ 45:

1. If Alice wants to send the classical two-bit string 00 to Bob, then she applies the identity quantum gate, <math>\mathbb{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}</math>, to her qubit, so that it remains unchanged. The resultant entangled state is then

: <math>|\Phi ^{+}\rangle := |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

:<math> |\Psi ^{+}\rangle := |B_{01}\rangle = \frac{1}{\sqrt{2}}(|0_A1_B1_A0_B\rangle + |1_A0_B0_A1_B\rangle)</math>

3. If Alice wants to send the classical two-bit string 10 to Bob, then she applies the [[Quantum logic gate#Phase shift gates|quantum ''phase-flip'' gate]] <math>Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}</math> to her qubit, so the resultant entangled state becomes

:<math>|\Phi ^{-}\rangle := |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>|\Psi ^{-}\rangle := |B_{11}\rangle = \frac{1}{\sqrt{2}}(|0_A1_B\rangle -|1_A0_B\rangle ) </math>

The matrices <math>X</math> and, <math>Z</math> are two of the [[Pauli matrices]]. The quantum states <math>|\Phi ^{+}\rangle</math>, <math>|\Psi ^{+}\rangle</math>, <math>|\Phi ^{-}\rangle</math> and <math>|\Psi ^{-}\rangleY</math> (or,are respectively, <math>B_{00}, B_{01}, B_{10}</math> and <math>B_{11}</math>) areknown theas [[BellPauli statesmatrices]].

=== Sending ===

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==== Example ====

For example, if the resultant entangled state (after the operations performed by Alice) was <math>|\Psi ^{+}\rangle := B_{01} = \fractfrac{1}{\sqrt{2}}(|0_A1_B1_A0_B\rangle + |1_A0_B0_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}}(|0_A1_B1_A1_B\rangle + |1_A1_B0_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)].

</math>

For simplicity, let's get rid of the subscripts, somay webe haveremoved:

: <math>

\begin{align}

<math> B_{01}'' &=

\fractfrac{1}{\sqrt{2}}

\left(

\frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle) \otimes |1\rangle +

\fractfrac{1}{\sqrt{2}}(|0 \rangle - |1 \rangle) \otimes |1\rangle +

\fractfrac{1}{\sqrt{2}}(|01 0\rangle + |11 1\rangle) +\otimes |1\rangle

\right)

\right) \\

=

&=

\fractfrac{1}{\sqrt{2}}

\left(

\frac{1}{\sqrt{2}}(|01 \rangle + |11 \rangle) +

\fractfrac{1}{\sqrt{2}}(|01 \rangle - |11 \rangle) +

\tfrac{1}{\sqrt{2}}(|01 \rangle + |11\rangle)

\right)

\right) =

=

\fractfrac{1}{2}|01 \rangle +- \fractfrac{1}{2} |11 \rangle +

\fractfrac{1}{2}|01 \rangle -+ \fractfrac{1}{2}|11 \rangle

= |01 \rangle.

\end{align}

</math>

Now, Bob has the basis state <math>|01\rangle</math>, so he knows that Alice wanted to send the two-bit string 01.

== Security ==

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:<math>\omega \rightarrow (\Phi_x \otimes I)(\omega)</math>

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 ''Fmodelled by a [[POVM]] <submath>y\{F_y\}_y</submath>, with <math>F_y</math> positive semidefinite operators such that <math display="inline">\sum_y F_y=I</math>''. The probability that Bob's measuring apparatus registers the message ''<math>y''</math> is thus

<math display="block">p(y|x)=\langle F_y, (\Phi_x \otimes I)(\omega)\rangle\equiv \operatorname{Tr}[ F_y(\Phi_x \otimes I)(\omega)].</math>

:<math>\operatorname{Tr}\; (\Phi_x \otimes I)(\omega) \cdot F_y .</math>

Therefore, to achieve the desired transmission, we require that

:<math display="block">p(y|x)=\operatorname{Tr}\;[F_y (\Phi_x \otimes I)(\omega)] \cdot= F_y .\delta_{xy},</math>

:where <math>\operatorname{Tr}\; (\Phi_x \otimes I)(\omega) \cdot F_y = \delta_{xy}</math> is the [[Kronecker delta]].

where ''δ_xy'' is the [[Kronecker delta]].

== 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]