In
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
, superdense coding (also referred to as ''dense coding'') is a
quantum communication
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum mechanics, qu ...
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
Alice and Bob are fictional characters commonly used as placeholders in discussions about cryptographic systems and protocols, and in other science and engineering literature where there are several participants in a thought experiment. The A ...
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
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
.
This protocol was first proposed by
Charles H. Bennett and
Stephen Wiesner in 1970
Stephen Wiesner
Memorial blog post by Or Sattath, with scan of Bennett's handwritten notes from 1970. See als
Stephen Wiesner (1942-2021)
by Scott Aaronson
Scott Joel Aaronson (born May 21, 1981) is an American theoretical computer scientist and David J. Bruton Jr. Centennial Professor of Computer Science at the University of Texas at Austin. His primary areas of research are quantum computing an ...
, which also discusses this topic. (though not published by them until 1992) and experimentally actualized in 1996 by Klaus Mattle, Harald Weinfurter, Paul G. Kwiat and Anton Zeilinger
Anton Zeilinger (; born 20 May 1945) is an Austrian quantum physicist and Nobel laureate in physics of 2022. Zeilinger is professor of physics emeritus at the University of Vienna and senior scientist at the Institute for Quantum Optics and Qua ...
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
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, lik ...
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.[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).]
Overview
Suppose Alice wants to send two classical bits of information (00, 01, 10, or 11) to Bob using qubits
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
(instead of classical 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
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
, 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
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
:
is prepared, where denotes the tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
. In common usage the tensor product symbol may be omitted:
:.
Sharing
After the preparation of the Bell state , 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 and the rest of the steps in the procedure.
Encoding
By applying a quantum gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, lik ...
to her qubit locally, Alice can transform the entangled state into any of the four Bell states
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
(including, of course, ). 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.
1. If Alice wants to send the classical two-bit string 00 to Bob, then she applies the identity quantum gate, , to her qubit, so that it remains unchanged. The resultant entangled state is then
:
In other words, the entangled state shared between Alice and Bob has not changed, i.e. it is still . The notation 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, , to her qubit, so that the resultant entangled quantum state becomes
:
3. If Alice wants to send the classical two-bit string 10 to Bob, then she applies the quantum ''phase-flip'' gate to her qubit, so the resultant entangled state becomes
:
4. If, instead, Alice wants to send the classical two-bit string 11 to Bob, then she applies the quantum gate to her qubit, so that the resultant entangled state becomes
:
The matrices , , and are known as Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
.
Sending
After having performed one of the operations described above, Alice can send her entangled qubit to Bob using a quantum network
Quantum networks form an important element of quantum computing and quantum communication systems. Quantum networks facilitate the transmission of information in the form of quantum bits, also called qubits, between physically separated quantum ...
through some conventional physical medium.
Decoding
In order for Bob to find out which classical bits Alice sent he will perform the CNOT
In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-''X'' gate'','' controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based ...
unitary operation, with A as control qubit and B as target qubit. Then, he will perform unitary operation on the entangled qubit A. In other words, the Hadamard quantum gate H is only applied to A (see the figure above).
* If the resultant entangled state was then after the application of the above unitary operations the entangled state will become
* If the resultant entangled state was then after the application of the above unitary operations the entangled state will become
* If the resultant entangled state was then after the application of the above unitary operations the entangled state will become
* If the resultant entangled state was then after the application of the above unitary operations the entangled state will become
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 or (as you can see from the outcomes and the example below).
Example
For example, if the resultant entangled state (after the operations performed by Alice) was , then a CNOT with A as control bit and B as target bit will change to become . Now, the Hadamard gate is applied only to A, to obtain
For simplicity, subscripts may be removed:
Now, Bob has the basis state , so he knows that Alice wanted to send the two-bit string 01.
Security
Superdense coding is a form of secure quantum communication. 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
General dense coding schemes can be formulated in the language used to describe quantum channel
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information i ...
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
:
on subsystem 1. On the combined system, this is effected by
:
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
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and ...
, with positive semidefinite operators such that . The probability that Bob's measuring apparatus registers the message is thus
Therefore, to achieve the desired transmission, we require that
where is the Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
.
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.[Schaetz, T., Barrett, M. D., Leibfried, D., Chiaverini, J., Britton, J., Itano, W. M., … Wineland, D. J. (2004). Quantum Dense Coding with Atomic Qubits. Physical Review Letters, 93(4).] In 2017, a channel capacity of 1.665 was achieved with a fidelity of 0.87 through optical fibers.[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).] High dimensional ququarts (states formed in photon pairs by non-degenerate spontaneous parametric down-conversion
Spontaneous parametric down-conversion (also known as SPDC, parametric fluorescence or parametric scattering) is a nonlinear instant optical process that converts one photon of higher energy (namely, a pump photon), into a pair of photons (namely, ...
) were used to reach a channel capacity of 2.09 (with a limit of 2.32) with a fidelity of 0.98.[Hu, X.-M., Guo, Y., Liu, B.-H., Huang, Y.-F., Li, C.-F., & Guo, G.-C. (2018). Beating the channel capacity limit for superdense coding with entangled ququarts. Science Advances, 4(7), eaat9304.] Nuclear Magnetic Resonance (NMR) has also been used to share among three parties.[Wei, D., Yang, X., Luo, J., Sun, X., Zeng, X., & Liu, M. (2004). NMR experimental implementation of three-parties quantum superdense coding. Chinese Science Bulletin, 49(5), 423–426.]
References
* Wilde, Mark M., 2017,
Quantum Information Theory, Cambridge University Press
Also available a
eprint arXiv:1106.1145
External links
Qubits, Quantum Mechanics and Computers Course Notes
* by Michael Nielsen
Michael Aaron Nielsen (born January 4, 1974) is a quantum physicist, science writer, and computer programming researcher living in San Francisco.
Work
In 1998, Nielsen received his PhD in physics from the University of New Mexico. In 2004, he wa ...
{{quantum computing
Quantum information science