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{{Short description|Two-bit quantum communication protocol}}

'''Superdense coding''' is a technique used in [[quantum information theory]] to send two bits of classical information using only one qubit, with the aid of [[Quantum entanglement|entanglement]].

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{{Use American English|date=January 2019}}

[[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''' (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]] 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 Klaus Mattle, Harald Weinfurter, Paul G.&nbsp;Kwiat and [[Anton 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 |first1=Chuan |last2=Deng |first2=Fu-Guo |last3=Li |first3=Yan-Song |last4=Liu |first4=Xiao-Shu |last5=Long |first5=Gui 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|thumb|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 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.

== The protocol ==

The protocol can be split into five different steps: preparation, sharing, encoding, sending, and decoding.

=== Preparation ===

The protocol starts with the preparation of an entangled state, which is later shared between Alice and Bob. For 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]]. In common usage the tensor product symbol <math>\otimes</math> may be omitted:

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

=== 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. Alice and Bob may be in different locations, an unlimited distance from each other.

There may be an arbitrary period between the preparation and sharing of the entangled state <math>|\Phi ^{+}\rangle</math> and the rest of the steps in the procedure.

=== Encoding ===

By applying a [[Quantum logic gate|quantum gate]] to her qubit locally, Alice can transform the entangled state <math>|\Phi ^{+}\rangle</math> into any of the four [[Bell states]] (including, of course, <math>|\Phi ^{+}\rangle</math>). Note that this process cannot "break" the entanglement between the two qubits.

Let's now describe which operations Alice needs to perform on her entangled qubit, depending on which classical two-bit message she wants to send to Bob. We'll later see why these specific operations are performed. There are four cases, which correspond to the four possible two-bit strings that Alice may want to send.

Suppose Alice would like to send classical information to Bob using [[qubit]]s, 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 in quantum mechanics|measurement]]. The question is: how much classical information can be transmitted per qubit? Since non-orthogonal [[quantum state]]s can not be distinguished reliably, one would guess that Alice can do no better than one classical bit per qubit. Indeed this bound on efficiency has been proven formally. 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.

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

== Details ==

: <math>|B_{00}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle)</math>

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.

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> indicates that Alice wants to send the two-bit string 00.

Suppose parts of a [[Bell state]], say

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

</math>

:<math>|B_{01}\rangle = \frac{1}{\sqrt{2}}(|1_A0_B\rangle + |0_A1_B\rangle)</math>

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):

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

* Obviously, if Alice does nothing, the system remains in the state <math>|\Psi^+\rangle</math>.

:<math>|B_{10}\rangle = \frac{1}{\sqrt{2}}(|0_A0_B\rangle - |1_A1_B\rangle)</math>

* If Alice sends her particle through the unitary gate

:4. If, instead, Alice wants to send the classical two-bit string 11 to Bob, then she applies the quantum gate <math>\sigma_1Z*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>

(notice this is one of the [[Pauli matrices]]), the total two-particle system now is in state

The matrices <math>X</math>, <math>Z</math>, and <math>Y</math> are known as [[Pauli matrices]].

:<math>( \sigma_1 \otimes I ) |\Psi^+\rangle = |\Phi^+\rangle .</math>

=== Sending ===

* If <math>\sigma_1</math> is replaced by <math>\sigma_3</math>, the initial state <math>|\Psi^+\rangle </math> is transformed into <math>|\Psi^-\rangle </math>.

After having performed one of the operations described above, Alice can send her entangled qubit to Bob using a [[quantum network]] through some conventional physical medium.

* Similarly, if Alice applies <math>i \sigma_2 \otimes I</math> to the system, the resultant state is <math>|\Phi^-\rangle </math>

=== Decoding ===

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.

In order for Bob to find out which classical bits Alice sent he will perform the [[Controlled NOT gate|CNOT]] unitary operation, with A as control qubit and B as target qubit. Then, he will perform <math>H\otimes I</math> unitary operation on the entangled qubit A. In other words, the Hadamard quantum gate H is only applied to A (see the figure above).

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.

* If the resultant entangled state was <math>B_{00}</math> then after the application of the above unitary operations the entangled state will become <math>|00\rangle</math>

== General dense coding scheme ==

* If the resultant entangled state was <math>B_{01}</math> then after the application of the above unitary operations the entangled state will become <math>|01\rangle</math>

* If the resultant entangled state was <math>B_{10}</math> then after the application of the above unitary operations the entangled state will become <math>|10\rangle</math>

* If the resultant entangled state was <math>B_{11}</math> then after the application of the above unitary operations the entangled state will become <math>|11\rangle</math>

These operations performed by Bob can be seen as a measurement which projects the entangled state onto one of the four two-qubit basis vectors <math>|00 \rangle, |01 \rangle, |10 \rangle</math> or <math>|11 \rangle</math> (as you can see from the outcomes and the example below).

General dense coding schemes can be formulated in the language used to describe [[quantum channel]]s. Alice and Bob share a maximally entangled state ''ω'', i.e.

==== Example ====

:<math>\omega \in H \otimes H </math>

For example, if the resultant entangled state (after the operations performed by Alice) was <math>B_{01} = \tfrac{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}' = \tfrac{1}{\sqrt{2}}(|1_A1_B\rangle + |0_A1_B\rangle)</math>. Now, the Hadamard gate is applied only to A, to obtain

has the maximally mixed state

: <math>

:<math>\begin{bmatrix} \frac{1}{n} & \; & \; \\ \; & \ddots \; \\ \; & \; & \frac{1}{n} \end{bmatrix}</math>

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].

</math>

For simplicity, subscripts may be removed:

as its [[partial trace]]. 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

: <math>

\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}

</math>

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

:<math>\; \Phi_x</math>

== Security ==

on subsytem 1. On the combined system, this is effected by

Superdense coding is a form of secure quantum communication.<ref name="Wang2005" /> If an eavesdropper, commonly called Eve, intercepts Alice's qubit en route to Bob, all that is obtained by Eve is part of an entangled state. Without access to Bob's qubit, Eve is unable to get any information from Alice's qubit. A third party is unable to eavesdrop on information being communicated through superdense coding and an attempt to measure either qubit would collapse the state of that qubit and alert Bob and Alice.

== General dense coding scheme ==

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

General dense coding schemes can be formulated in the language used to describe [[quantum channel]]s. 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

where ''Id'' 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 ''F_y''. The probability that Bob's measuring apparatus registers the message ''y'' is

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

on subsystem 1. On the combined system, this is effected by

Therefore, to achieve the desired transmission, we require that

:<math>\operatorname{Tr}omega \;rightarrow (\Phi_x \otimes IdI)(\omega) \cdot F_y = \delta_{xy}</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 Bob's measurement be modelled by a [[POVM]] <math>\{F_y\}_y</math>, 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

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

<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>

Therefore, to achieve the desired transmission, we require that

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

where <math>\delta_{xy}</math> 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 |first1=Xiao-Min |last2=Guo |first2=Yu |last3=Liu |first3=Bi-Heng |last4=Huang |first4=Yun-Feng |last5=Li |first5=Chuan-Feng |last6=Guo |first6=Guang-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 magnetic resonance]] (NMR) has also been used to share among three parties.<ref name="Wei2004">{{Cite journal |last1=Wei |first1=Daxiu |last2=Yang |first2=Xiaodong |last3=Luo |first3=Jun |last4=Sun |first4=Xianping |last5=Zeng |first5=Xizhi |last6=Liu |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]

==External links==

* [http://www-inst.eecs.berkeley.edu/~cs191/fa09/lectures/lecture6_fa09.pdf Qubits, Quantum Mechanics and Computers Course Notes]

* {{YouTube | w5rCn593Dig| Superdense coding: how to send two bits using one qubit }} by [[Michael Nielsen]]

{{quantum computing}}

* C. Bennett and S.J. Wiesner. ''Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states.'' Phys. Rev. Lett., 69:2881, 1992[http://prola.aps.org/abstract/PRL/v69/i20/p2881_1]

[[Category:Quantum information science]]