Short description: Encoding an n-bit string in m qubits

← Back to Quantum information and computing

Quantum random access codes (QRACs) are a quantum information theoretic primitive used to encode a string of classical bits into a quantum state of smaller dimension, such that any single bit of the original string can be retrieved with a certain probability of success. QRACs are a form of quantum data compression that exploit the properties of quantum superposition and measurement to outperform their classical counterparts, known as random access codes (RACs).^[1]

QRACs are fundamental in the study of quantum communication complexity, quantum entanglement, and the foundations of quantum mechanics, particularly in the context of contextuality and Bell inequalities.^[2]

Quantum random access code protocol. Alice encodes an n-bit string into m qubits, with m < n. Bob chooses an index i and performs a measurement to guess the requested bit x_i.

Definition

A $(n, m, p)$ -QRAC is a protocol in which $n$ classical bits, denoted by $x = x_{1} x_{2} \dots x_{n} \in {0, 1}^{n}$ , are encoded into a quantum state $ρ_{x}$ of $m$ qubits. The goal is to retrieve any bit $x_{i}$ (where $i \in {1, \dots, n}$ ) chosen by a receiver, with a success probability of at least $p$ .^[1]

Formally, the protocol consists of two maps:

Encoding: A map $ℰ : {0, 1}^{n} \to 𝒟 (ℋ_{2^{m}})$ that assigns a density matrix $ρ_{x}$ to every input string $x$ .
Decoding: A set of Positive Operator-Valued Measures (POVMs) ${M_{i}}_{i = 1}^{n}$ . Each $M_{i} = {E_{i}^{0}, E_{i}^{1}}$ is a two-outcome measurement designed to guess the value of the $i$ -th bit.

The probability of correctly guessing the $i$ -th bit of $x$ is given by:

P (guess x_{i} correctly) = Tr (E_{i}^{x_{i}} ρ_{x}) \geq p

For the code to be non-trivial, the probability $p$ must be strictly greater than $1 / 2$ .

Comparison with Classical RACs

A classical random access code (RAC) is defined similarly, but the encoding maps the $n$ bits into $m$ classical bits. A fundamental result in information theory, derived from Holevo's theorem, states that for a classical or quantum $(n, m, p)$ -RAC to exist with $p > 1 / 2$ , we must have $m \geq n (1 - H (p))$ , where $H (p)$ is the binary entropy function. Effectively, to recover any bit with high probability, the encoded size must be close to the original size.^[3]

Although quantum and classical RACs share the same aforementioned bound, QRACs can compress information more efficiently than classical ones. For example, it is possible to encode 2 bits into 1 qubit with a success probability of $p \approx 0.85$ , and 3 bits into 1 qubit with $p \approx 0.79$ . This demonstrates a quantum advantage in terms of encoding efficiency.^[1]

Nayak's Bound

Despite this advantage, quantum mechanics does not allow for infinite compression. Ashwin Nayak proved a strict lower bound for QRACs, known as Nayak's bound. For an $(n, m, p)$ -QRAC, the number of qubits $m$ must satisfy:

m \geq n (1 - H (p))

This inequality is formally identical to the classical bound, but since quantum states exist in a continuous Hilbert space rather than a discrete set, specific constructions like the 2-to-1 and 3-to-1 codes are realizable in the quantum case where they are impossible classically for the same length $m$ . However, the number of $n$ classical bits that can be encoded with $m$ qubits is at most $4^{m} - 1$ even when $p$ is infinitely close to $1 / 2$ .^[3]^[4]^[5]

Examples

2-to-1 QRAC

The $(2, 1, p)$ -QRAC encodes $n = 2$ classical bits ( $x_{1}, x_{2}$ ) into $m = 1$ qubit. The optimal strategy yields a success probability of $p = \frac{1}{2} + \frac{1}{2 \sqrt{2}} \approx 0.854$ .^[4]

The encoding maps the four possible strings $00, 01, 10, 11$ to four qubit states located on the equator of the Bloch sphere:

$00 \to | + ⟩$
$01 \to | + i ⟩$
$10 \to | - ⟩$
$11 \to | - i ⟩$

To recover $x_{1}$ , one measures in the X-basis (Pauli-X). To recover $x_{2}$ , one measures in the Y-basis (Pauli-Y).

3-to-1 QRAC

The $(3, 1, p)$ -QRAC encodes $n = 3$ bits into $m = 1$ qubit. The encoding states correspond to the vertices of a cube inscribed in the Bloch sphere. The optimal success probability is $p = \frac{1}{2} + \frac{1}{2 \sqrt{3}} \approx 0.789$ . This is the maximum number of bits that can be encoded into a single qubit such that every bit can be recovered with probability strictly greater than $1 / 2$ .^[1]^[6]

Entanglement-Assisted QRACs

The concept can be extended to scenarios where the sender and receiver share entanglement. In an Entanglement-Assisted Random Access Code (EARAC), the parties share a maximally entangled state. The sender performs a measurement on their half of the entangled state based on the input string $x$ and sends the classical outcome (or a quantum system) to the receiver. This setup allows for "superdense coding" variants of RACs, achieving higher success probabilities or compression rates than unentangled QRACs.^[7]

Applications

Quantum Finite Automata

QRACs were originally developed to prove lower bounds on the size of Quantum Finite Automata (QFA). If a QFA recognizes a language with high probability, its states effectively encode the history of the input string. Nayak's bound on QRACs translates to a lower bound on the number of states required for the automaton.^[1]

Foundations of Quantum Mechanics

QRACs are closely related to the study of contextuality. The inability to simultaneously retrieve all encoded bits reflects the uncertainty principle and the disturbance caused by measurement. The success probabilities of QRACs appear as bounds in various Bell inequalities and non-contextuality inequalities. For instance, the success probability of the 2-to-1 QRAC is related to the Tsirelson bound of the CHSH inequality.^[8]

Network Coding

In quantum network coding, QRACs provide a primitive for compressing information sent over bottleneck channels in a quantum network. They help in optimizing the throughput of quantum information transfer.^[4]

Table of contents (184 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing
Matter by scale

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics
Methods and tools

Full contents

1. Foundations (11)↑

2. Conceptual and interpretations (13)↑

3. Mathematical structure and systems (13)↑

4. Atomic and spectroscopy (14)↑

Physics:Quantum Atomic structure and spectroscopy
Physics:Quantum Hydrogen atom
Physics:Quantum number
Physics:Quantum Multi-electron atoms
Physics:Quantum Fine structure
Physics:Quantum Hyperfine structure
Physics:Quantum Isotopic shift
Physics:Quantum defect
Physics:Quantum Zeeman effect
Physics:Quantum Stark effect
Physics:Quantum Spectral lines and series
Physics:Quantum Selection rules
Physics:Quantum Fermi's golden rule
Physics:Quantum beats

5. Wavefunctions and modes (9)↑

Physics:Quantum Wavefunction
Physics:Quantum Superposition principle
Physics:Quantum Eigenstates and eigenvalues
Physics:Quantum Boundary conditions and quantization
Physics:Quantum Standing waves and modes
Physics:Quantum Normal modes and field quantization
Physics:Number of independent spatial modes in a spherical volume
Physics:Quantum Density of states
Physics:Quantum carpet

6. Quantum dynamics and evolution (17)↑

7. Measurement and information (9)↑

8. Quantum information and computing (10)↑

9. Quantum optics and experiments (5)↑

Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
Physics:Quantum optics beam splitter experiments
Physics:Quantum Ultra fast lasers
Physics:Quantum Experimental quantum physics
Physics:Quantum optics

Template:Quantum optics operators

10. Open quantum systems (9)↑

11. Quantum field theory (20)↑

12. Statistical mechanics and kinetic theory (9)↑

13. Condensed matter and solid-state physics (13)↑

14. Plasma and fusion physics (8)↑

Physics:Quantum Fusion reactions and Lawson criterion
Physics:Quantum Plasma (fusion context)
Physics:Quantum Magnetic confinement fusion
Physics:Quantum Inertial confinement fusion
Physics:Quantum Plasma instabilities and turbulence
Physics:Quantum Tokamak core plasma
Physics:Quantum Tokamak edge physics and recycling asymmetries
Physics:Quantum Stellarator

15. Timeline (8)↑

16. Advanced and frontier topics (16)↑

References

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 Ambainis, A.; Nayak, A.; Ta-Shma, A.; Vazirani, U. (1999). "Dense quantum coding and a lower bound for 1-way quantum automata". Proceedings of the thirty-first annual ACM symposium on Theory of Computing. pp. 376–383. doi:10.1145/301250.301347. ISBN 1-58113-067-8.
↑ Galvão, E. F. (2002). "Foundations of quantum theory and quantum information applications". PhD Thesis, Oxford University. Bibcode: 2002quant.ph.12124G.
↑ ^3.0 ^3.1 Nayak, A. (1999). "Optimal lower bounds for quantum automata and random access codes". 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039). pp. 369–376. doi:10.1109/SFFCS.1999.814608. ISBN 0-7695-0409-4.
↑ ^4.0 ^4.1 ^4.2 Hayashi, Masahito; Iwama, Kazuo; Nishimura, Harumichi; Raymond, Rudy; Yamashita, Shigeru (2007). "Quantum Network Coding". in Thomas, Wolfgang; Weil, Pascal (in en). Stacs 2007. Lecture Notes in Computer Science. 4393. Berlin, Heidelberg: Springer. pp. 610–621. doi:10.1007/978-3-540-70918-3_52. ISBN 978-3-540-70918-3.
↑ Iwama, Kazuo; Nishimura, Harumichi; Raymond, Rudy; Yamashita, Shigeru (2007). Arge, Lars; Cachin, Christian; Jurdziński, Tomasz et al.. eds. "Unbounded-Error One-Way Classical and Quantum Communication Complexity" (in en). Automata, Languages and Programming (Berlin, Heidelberg: Springer): 110–121. doi:10.1007/978-3-540-73420-8_12. ISBN 978-3-540-73420-8. https://link.springer.com/chapter/10.1007/978-3-540-73420-8_12?error=cookies_not_supported&code=2e698b6e-4033-42ef-90f9-dded9e106dcf.
↑ Hayashi, M; Iwama, K; Nishimura, H; Raymond, R; Yamashita, S (2006-08-04). "(4,1)-Quantum random access coding does not exist—one qubit is not enough to recover one of four bits". New Journal of Physics 8 (8): 129–129. doi:10.1088/1367-2630/8/8/129. ISSN 1367-2630. https://iopscience.iop.org/article/10.1088/1367-2630/8/8/129.
↑ Pawłowski, M.; Żukowski, M. (2010). "Entanglement-assisted random access codes". Physical Review A 81 (4). doi:10.1103/PhysRevA.81.042326. Bibcode: 2010PhRvA..81d2326P.
↑ Tavakoli, A.; Hameedi, A.; Marques, B.; Bourennane, M. (2015). "Quantum random access codes using single d-level systems". Physical Review Letters 114 (17). doi:10.1103/PhysRevLett.114.170502. PMID 25978213. Bibcode: 2015PhRvL.114q0502T.

Author: Harold Foppele

0.00

(0 votes)

[Ambainis2002-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 Ambainis, A.; Nayak, A.; Ta-Shma, A.; Vazirani, U. (1999). "Dense quantum coding and a lower bound for 1-way quantum automata". Proceedings of the thirty-first annual ACM symposium on Theory of Computing. pp. 376–383. doi:10.1145/301250.301347. ISBN 1-58113-067-8.

[Galvao2002-2] Galvão, E. F. (2002). "Foundations of quantum theory and quantum information applications". PhD Thesis, Oxford University. Bibcode: 2002quant.ph.12124G.

[Nayak1999-3] 3.0 ^3.1 Nayak, A. (1999). "Optimal lower bounds for quantum automata and random access codes". 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039). pp. 369–376. doi:10.1109/SFFCS.1999.814608. ISBN 0-7695-0409-4.

[Hayashi2006-4] 4.0 ^4.1 ^4.2 Hayashi, Masahito; Iwama, Kazuo; Nishimura, Harumichi; Raymond, Rudy; Yamashita, Shigeru (2007). "Quantum Network Coding". in Thomas, Wolfgang; Weil, Pascal (in en). Stacs 2007. Lecture Notes in Computer Science. 4393. Berlin, Heidelberg: Springer. pp. 610–621. doi:10.1007/978-3-540-70918-3_52. ISBN 978-3-540-70918-3.

[5] Iwama, Kazuo; Nishimura, Harumichi; Raymond, Rudy; Yamashita, Shigeru (2007). Arge, Lars; Cachin, Christian; Jurdziński, Tomasz et al.. eds. "Unbounded-Error One-Way Classical and Quantum Communication Complexity" (in en). Automata, Languages and Programming (Berlin, Heidelberg: Springer): 110–121. doi:10.1007/978-3-540-73420-8_12. ISBN 978-3-540-73420-8. https://link.springer.com/chapter/10.1007/978-3-540-73420-8_12?error=cookies_not_supported&code=2e698b6e-4033-42ef-90f9-dded9e106dcf.

[6] Hayashi, M; Iwama, K; Nishimura, H; Raymond, R; Yamashita, S (2006-08-04). "(4,1)-Quantum random access coding does not exist—one qubit is not enough to recover one of four bits". New Journal of Physics 8 (8): 129–129. doi:10.1088/1367-2630/8/8/129. ISSN 1367-2630. https://iopscience.iop.org/article/10.1088/1367-2630/8/8/129.

[Pawlowski2010-7] Pawłowski, M.; Żukowski, M. (2010). "Entanglement-assisted random access codes". Physical Review A 81 (4). doi:10.1103/PhysRevA.81.042326. Bibcode: 2010PhRvA..81d2326P.

[Tavakoli2015-8] Tavakoli, A.; Hameedi, A.; Marques, B.; Bourennane, M. (2015). "Quantum random access codes using single d-level systems". Physical Review Letters 114 (17). doi:10.1103/PhysRevLett.114.170502. PMID 25978213. Bibcode: 2015PhRvL.114q0502T.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

Anonymous

Search

Physics:Quantum random access code

Namespaces

More

Page actions

Contents

Definition