Physics:Quantum random access code - Revision history

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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|~~Random~~ access code\|1}}		{{Sourceattribution\|Physics:Quantum random access code\|1}}

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	QRACs are fundamental in the study of quantum communication complexity, [[Quantum entanglement\|quantum entanglement]], and the foundations of quantum mechanics, particularly in the context of contextuality and Bell inequalities.<ref name="Galvao2002">{{cite journal \|last1=Galvão \|first1=E. F. \|title=Foundations of quantum theory and quantum information applications \|journal=PhD Thesis, Oxford University \|year=2002 \|arxiv=quant-ph/0212124 \|bibcode=2002quant.ph.12124G }}</ref>		QRACs are fundamental in the study of quantum communication complexity, [[Quantum entanglement\|quantum entanglement]], and the foundations of quantum mechanics, particularly in the context of contextuality and Bell inequalities.<ref name="Galvao2002">{{cite journal \|last1=Galvão \|first1=E. F. \|title=Foundations of quantum theory and quantum information applications \|journal=PhD Thesis, Oxford University \|year=2002 \|arxiv=quant-ph/0212124 \|bibcode=2002quant.ph.12124G }}</ref>
	~~<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">~~
	<div style="font-size:90%;">Quantum random access code protocol. Alice encodes an <i>n</i>-bit string into <i>m</i> qubits, with <i>m</i> < <i>n</i>. Bob chooses an index <i>i</i> and performs a measurement to guess the requested bit <i>x</i><sub>i</sub>.</div>
	~~</div>~~
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	<div style="width:300px;">		<div style="width:300px;">
	[[File:Quantum_random_access_code_yellow.png\|thumb\|280px\|Quantum random access code.]]		[[File:Quantum_random_access_code_yellow.png\|thumb\|280px\|Quantum random access code protocol. Alice encodes an <i>n</i>-bit string into <i>m</i> qubits, with <i>m</i> < <i>n</i>. Bob chooses an index <i>i</i> and performs a measurement to guess the requested bit <i>x</i><sub>i</sub>.]]
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			__TOC__
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	'''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).<ref name="Ambainis2002">{{cite book \|last1=Ambainis \|first1=A. \|last2=Nayak \|first2=A. \|last3=Ta-Shma \|first3=A. \|last4=Vazirani \|first4=U. \|chapter=Dense quantum coding and a lower bound for 1-way quantum automata \|title=Proceedings of the thirty-first annual ACM symposium on Theory of Computing \|year=1999 \|pages=376–383 \|doi=10.1145/301250.301347 \|arxiv=quant-ph/9804043 \|isbn=1-58113-067-8 }}</ref>		'''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).<ref name="Ambainis2002">{{cite book \|last1=Ambainis \|first1=A. \|last2=Nayak \|first2=A. \|last3=Ta-Shma \|first3=A. \|last4=Vazirani \|first4=U. \|chapter=Dense quantum coding and a lower bound for 1-way quantum automata \|title=Proceedings of the thirty-first annual ACM symposium on Theory of Computing \|year=1999 \|pages=376–383 \|doi=10.1145/301250.301347 \|arxiv=quant-ph/9804043 \|isbn=1-58113-067-8 }}</ref>

	QRACs are fundamental in the study of quantum communication complexity, [[Quantum entanglement\|quantum entanglement]], and the foundations of quantum mechanics, particularly in the context of contextuality and Bell inequalities.<ref name="Galvao2002">{{cite journal \|last1=Galvão \|first1=E. F. \|title=Foundations of quantum theory and quantum information applications \|journal=PhD Thesis, Oxford University \|year=2002 \|arxiv=quant-ph/0212124 \|bibcode=2002quant.ph.12124G }}</ref>		QRACs are fundamental in the study of quantum communication complexity, [[Quantum entanglement\|quantum entanglement]], and the foundations of quantum mechanics, particularly in the context of contextuality and Bell inequalities.<ref name="Galvao2002">{{cite journal \|last1=Galvão \|first1=E. F. \|title=Foundations of quantum theory and quantum information applications \|journal=PhD Thesis, Oxford University \|year=2002 \|arxiv=quant-ph/0212124 \|bibcode=2002quant.ph.12124G }}</ref>
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	~~[[File:Quantum_random_access_code_yellow.png\|400px]]~~
	<div style="font-size:90%;">Quantum random access code protocol. Alice encodes an <i>n</i>-bit string into <i>m</i> qubits, with <i>m</i> < <i>n</i>. Bob chooses an index <i>i</i> and performs a measurement to guess the requested bit <i>x</i><sub>i</sub>.</div>		<div style="font-size:90%;">Quantum random access code protocol. Alice encodes an <i>n</i>-bit string into <i>m</i> qubits, with <i>m</i> < <i>n</i>. Bob chooses an index <i>i</i> and performs a measurement to guess the requested bit <i>x</i><sub>i</sub>.</div>
	</div>		</div>
			</div>

			<div style="width:300px;">
			[[File:Quantum_random_access_code_yellow.png\|thumb\|280px\|Quantum random access code.]]
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			</div>

	== Definition ==		== Definition ==

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	'''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).<ref name="Ambainis2002">{{cite book \|last1=Ambainis \|first1=A. \|last2=Nayak \|first2=A. \|last3=Ta-Shma \|first3=A. \|last4=Vazirani \|first4=U. \|chapter=Dense quantum coding and a lower bound for 1-way quantum automata \|title=Proceedings of the thirty-first annual ACM symposium on Theory of Computing \|year=1999 \|pages=376–383 \|doi=10.1145/301250.301347 \|arxiv=quant-ph/9804043 \|isbn=1-58113-067-8 }}</ref>		'''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).<ref name="Ambainis2002">{{cite book \|last1=Ambainis \|first1=A. \|last2=Nayak \|first2=A. \|last3=Ta-Shma \|first3=A. \|last4=Vazirani \|first4=U. \|chapter=Dense quantum coding and a lower bound for 1-way quantum automata \|title=Proceedings of the thirty-first annual ACM symposium on Theory of Computing \|year=1999 \|pages=376–383 \|doi=10.1145/301250.301347 \|arxiv=quant-ph/9804043 \|isbn=1-58113-067-8 }}</ref>

	QRACs are fundamental in the study of quantum communication complexity, ~~[[Quantum entanglement\|~~quantum entanglement]], and the foundations of quantum mechanics, particularly in the context of contextuality and Bell inequalities.<ref name="Galvao2002">{{cite journal \|last1=Galvão \|first1=E. F. \|title=Foundations of quantum theory and quantum information applications \|journal=PhD Thesis, Oxford University \|year=2002 \|arxiv=quant-ph/0212124 \|bibcode=2002quant.ph.12124G }}</ref>		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.<ref name="Galvao2002">{{cite journal \|last1=Galvão \|first1=E. F. \|title=Foundations of quantum theory and quantum information applications \|journal=PhD Thesis, Oxford University \|year=2002 \|arxiv=quant-ph/0212124 \|bibcode=2002quant.ph.12124G }}</ref>

	</div>		</div>
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	Formally, the protocol consists of two maps:		Formally, the protocol consists of two maps:
	# '''Encoding''': A map <math>\mathcal{E}: \{0, 1\}^n \to \mathcal{D}(\mathcal{H}_{2^m})</math> that assigns a density matrix <math>\rho_x</math> to every input string <math>x</math>.		# '''Encoding''': A map <math>\mathcal{E}: \{0, 1\}^n \to \mathcal{D}(\mathcal{H}_{2^m})</math> that assigns a density matrix <math>\rho_x</math> to every input string <math>x</math>.
	# '''Decoding''': A set of ~~[[POVM\|~~Positive Operator-Valued Measures]] (POVMs) <math>\{M_i\}_{i=1}^n</math>. Each <math>M_i = \{E_i^0, E_i^1\}</math> is a two-outcome measurement designed to guess the value of the <math>i</math>-th bit.		# '''Decoding''': A set of Positive Operator-Valued Measures (POVMs) <math>\{M_i\}_{i=1}^n</math>. Each <math>M_i = \{E_i^0, E_i^1\}</math> is a two-outcome measurement designed to guess the value of the <math>i</math>-th bit.

	The probability of correctly guessing the <math>i</math>-th bit of <math>x</math> is given by:		The probability of correctly guessing the <math>i</math>-th bit of <math>x</math> is given by:
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	== Comparison with Classical RACs ==		== Comparison with Classical RACs ==
	A classical random access code (RAC) is defined similarly, but the encoding maps the <math>n</math> bits into <math>m</math> ''classical'' bits. A fundamental result in information theory, derived from [[Holevo's theorem]], states that for a classical or quantum <math>(n, m, p)</math>-RAC to exist with <math>p > 1/2</math>, we must have <math>m \ge n(1 - H(p))</math>, where <math>H(p)</math> is the ~~[[Binary entropy function\|~~binary entropy function]]. Effectively, to recover ''any'' bit with high probability, the encoded size must be close to the original size.<ref name="Nayak1999">{{cite book \|last1=Nayak \|first1=A. \|chapter=Optimal lower bounds for quantum automata and random access codes \|title=40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039) \|year=1999 \|pages=369–376 \|doi=10.1109/SFFCS.1999.814608 \|arxiv=quant-ph/9904093 \|isbn=0-7695-0409-4 }}</ref>		A classical random access code (RAC) is defined similarly, but the encoding maps the <math>n</math> bits into <math>m</math> ''classical'' bits. A fundamental result in information theory, derived from Holevo's theorem, states that for a classical or quantum <math>(n, m, p)</math>-RAC to exist with <math>p > 1/2</math>, we must have <math>m \ge n(1 - H(p))</math>, where <math>H(p)</math> is the binary entropy function. Effectively, to recover ''any'' bit with high probability, the encoded size must be close to the original size.<ref name="Nayak1999">{{cite book \|last1=Nayak \|first1=A. \|chapter=Optimal lower bounds for quantum automata and random access codes \|title=40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039) \|year=1999 \|pages=369–376 \|doi=10.1109/SFFCS.1999.814608 \|arxiv=quant-ph/9904093 \|isbn=0-7695-0409-4 }}</ref>

	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 <math>p \approx 0.85</math>, and 3 bits into 1 qubit with <math>p \approx 0.79</math>. This demonstrates a quantum advantage in terms of encoding efficiency.<ref name="Ambainis2002"/>		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 <math>p \approx 0.85</math>, and 3 bits into 1 qubit with <math>p \approx 0.79</math>. This demonstrates a quantum advantage in terms of encoding efficiency.<ref name="Ambainis2002"/>
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	The <math>(2, 1, p)</math>-QRAC encodes <math>n=2</math> classical bits (<math>x_1, x_2</math>) into <math>m=1</math> qubit. The optimal strategy yields a success probability of <math>p = \frac{1}{2} + \frac{1}{2\sqrt{2}} \approx 0.854</math>.<ref name="Hayashi2006">{{Cite book \|last1=Hayashi \|first1=Masahito \|last2=Iwama \|first2=Kazuo \|last3=Nishimura \|first3=Harumichi \|last4=Raymond \|first4=Rudy \|last5=Yamashita \|first5=Shigeru \|date=2007 \|editor-last=Thomas \|editor-first=Wolfgang \|chapter=Quantum Network Coding \|series=Lecture Notes in Computer Science \|volume=4393 \|editor2-last=Weil \|editor2-first=Pascal \|title=Stacs 2007 \|language=en \|location=Berlin, Heidelberg \|publisher=Springer \|pages=610–621 \|doi=10.1007/978-3-540-70918-3_52 \|arxiv=quant-ph/0601088 \|isbn=978-3-540-70918-3}}</ref>		The <math>(2, 1, p)</math>-QRAC encodes <math>n=2</math> classical bits (<math>x_1, x_2</math>) into <math>m=1</math> qubit. The optimal strategy yields a success probability of <math>p = \frac{1}{2} + \frac{1}{2\sqrt{2}} \approx 0.854</math>.<ref name="Hayashi2006">{{Cite book \|last1=Hayashi \|first1=Masahito \|last2=Iwama \|first2=Kazuo \|last3=Nishimura \|first3=Harumichi \|last4=Raymond \|first4=Rudy \|last5=Yamashita \|first5=Shigeru \|date=2007 \|editor-last=Thomas \|editor-first=Wolfgang \|chapter=Quantum Network Coding \|series=Lecture Notes in Computer Science \|volume=4393 \|editor2-last=Weil \|editor2-first=Pascal \|title=Stacs 2007 \|language=en \|location=Berlin, Heidelberg \|publisher=Springer \|pages=610–621 \|doi=10.1007/978-3-540-70918-3_52 \|arxiv=quant-ph/0601088 \|isbn=978-3-540-70918-3}}</ref>

	The encoding maps the four possible strings <math>00, 01, 10, 11</math> to four qubit states located on the equator of the ~~[[Physics:~~Bloch sphere~~\|Bloch sphere]]~~:		The encoding maps the four possible strings <math>00, 01, 10, 11</math> to four qubit states located on the equator of the Bloch sphere:
	* <math>00 \to \|+\rangle</math>		* <math>00 \to \|+\rangle</math>
	* <math>01 \to \|+i\rangle</math>		* <math>01 \to \|+i\rangle</math>
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	== Entanglement-Assisted QRACs ==		== Entanglement-Assisted QRACs ==
	The concept can be extended to scenarios where the sender and receiver share ~~[[Quantum~~ entanglement~~\|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 <math>x</math> 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.<ref name="Pawlowski2010">{{cite journal \|last1=Pawłowski \|first1=M. \|last2=Żukowski \|first2=M. \|title=Entanglement-assisted random access codes \|journal=Physical Review A \|volume=81 \|issue=4 \|article-number=042326 \|year=2010 \|doi=10.1103/PhysRevA.81.042326 \|arxiv=0912.3524 \|bibcode=2010PhRvA..81d2326P }}</ref>		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 <math>x</math> 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.<ref name="Pawlowski2010">{{cite journal \|last1=Pawłowski \|first1=M. \|last2=Żukowski \|first2=M. \|title=Entanglement-assisted random access codes \|journal=Physical Review A \|volume=81 \|issue=4 \|article-number=042326 \|year=2010 \|doi=10.1103/PhysRevA.81.042326 \|arxiv=0912.3524 \|bibcode=2010PhRvA..81d2326P }}</ref>

	== Applications ==		== Applications ==

	=== Quantum Finite Automata ===		=== Quantum Finite Automata ===
	QRACs were originally developed to prove lower bounds on the size of ~~[[Quantum finite automaton\|~~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.<ref name="Ambainis2002"/>		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.<ref name="Ambainis2002"/>

	=== Foundations of Quantum Mechanics ===		=== Foundations of Quantum Mechanics ===