# Building Multiple Access Channels with a Single Particle

Yujie Zhang1, Xinan Chen2, and Eric Chitambar2

1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
2Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

### Abstract

A multiple access channel describes a situation in which multiple senders are trying to forward messages to a single receiver using some physical medium. In this paper we consider scenarios in which this medium consists of just a single classical or quantum particle. In the quantum case, the particle can be prepared in a superposition state thereby allowing for a richer family of encoding strategies. To make the comparison between quantum and classical channels precise, we introduce an operational framework in which all possible encoding strategies consume no more than a single particle. We apply this framework to an $N$-port interferometer experiment in which each party controls a path the particle can traverse. When used for the purpose of communication, this setup embodies a multiple access channel (MAC) built with a single particle.
We provide a full characterization of the $N$-party classical MACs that can be built from a single particle, and we show that every non-classical particle can generate a MAC outside the classical set. To further distinguish the capabilities of a single classical and quantum particle, we relax the locality constraint and allow for joint encodings by subsets of ${1\lt K\le N}$ parties. This generates a richer family of classical MACs whose polytope dimension we compute. We identify a "generalized fingerprinting inequality'' as a valid facet for this polytope, and we verify that a quantum particle distributed among $N$ separated parties can violate this inequality even when ${K=N-1}$. Connections are drawn between the single-particle framework and multi-level coherence theory. We show that every pure state with $K$-level coherence can be detected in a semi-device independent manner, with the only assumption being conservation of particle number.

A single quantum particle, unlike its classical counterpart, can be in different locations simultaneously. This property of quantum particles is known as superposition in quantum mechanics and has led to numerous genuine quantum phenomena. For example, in the celebrated double-slit experiment, if quantum particles being in superposition states are employed, non-classical interference can be observed. Here we consider different multiple access communication channels, in which, multiple specially separated senders are trying to forward messages to a single receiver

Our discussion begins with building the framework of multiple access channels with a single particle. We then fully characterize the structures of different multiple access channels built with a single classical particle. Different non-trivial equalities and inequalities that constrain the classical multiple access channel are found and analyzed, where the equalities are intimately related to the multi-slit experiment. We demonstrate the much more fruitful structure of quantum multiple access channels by showing violation of those equalities and inequalities found in classical multiple access communication sceneries. We also draw a connection between the multilevel coherent of a quantum particle and its power in different communication scenarios we framed.

The fundamental differences of single classical particle and quantum particle in the multiple access communication scenarios are beneficial for quantum-enhanced multi-party communication, and the connection of our results and multi-level coherence theory can also be applied to other quantum technologies including quantum sensing and quantum biology.

### ► References

[1] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, USA, 2006. ISBN 0471241954. 10.5555/​1146355. URL https:/​/​dl.acm.org/​doi/​10.5555/​1146355.
https:/​/​doi.org/​10.5555/​1146355

[2] David Tse and Pramod Viswanath. Fundamentals of Wireless Communication. Cambridge University Press, Cambridge, 2005. ISBN 9780521845274. 10.1017/​CBO9780511807213. URL https:/​/​doi.org/​10.1017/​CBO9780511807213.
https:/​/​doi.org/​10.1017/​CBO9780511807213

[3] Hlér Kristjánsson, Wenxu Mao, and Giulio Chiribella. Witnessing latent time correlations with a single quantum particle. Phys. Rev. Research, 3: 043147, Nov 2021. 10.1103/​PhysRevResearch.3.043147. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.3.043147.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.043147

[4] Daniel Ebler, Sina Salek, and Giulio Chiribella. Enhanced communication with the assistance of indefinite causal order. Phys. Rev. Lett., 120: 120502, Mar 2018. 10.1103/​PhysRevLett.120.120502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.120.120502.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.120502

[5] K. Goswami, Y. Cao, G. A. Paz-Silva, J. Romero, and A. G. White. Increasing communication capacity via superposition of order. Phys. Rev. Research, 2: 033292, Aug 2020. 10.1103/​PhysRevResearch.2.033292. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.2.033292.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.033292

[6] Sina Salek, Daniel Ebler, and Giulio Chiribella. Quantum communication in a superposition of causal orders, 2018. URL https:/​/​arxiv.org/​abs/​1809.06655.
arXiv:1809.06655

[7] Lorenzo M. Procopio, Francisco Delgado, Marco Enríquez, Nadia Belabas, and Juan Ariel Levenson. Communication enhancement through quantum coherent control of n channels in an indefinite causal-order scenario. Entropy, 21 (10), 2019. ISSN 1099-4300. 10.3390/​e21101012. URL https:/​/​www.mdpi.com/​1099-4300/​21/​10/​1012.
https:/​/​doi.org/​10.3390/​e21101012
https:/​/​www.mdpi.com/​1099-4300/​21/​10/​1012

[8] Lorenzo M. Procopio, Francisco Delgado, Marco Enríquez, Nadia Belabas, and Juan Ariel Levenson. Sending classical information via three noisy channels in superposition of causal orders. Phys. Rev. A, 101: 012346, Jan 2020. 10.1103/​PhysRevA.101.012346. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.101.012346.
https:/​/​doi.org/​10.1103/​PhysRevA.101.012346

[9] Giulio Chiribella, Matt Wilson, and H. F. Chau. Quantum and classical data transmission through completely depolarizing channels in a superposition of cyclic orders. Phys. Rev. Lett., 127: 190502, Nov 2021. 10.1103/​PhysRevLett.127.190502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.127.190502.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.190502

[10] Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, and Cyril Branciard. Communication through coherent control of quantum channels. Quantum, 4: 333, September 2020. ISSN 2521-327X. 10.22331/​q-2020-09-24-333. URL https:/​/​doi.org/​10.22331/​q-2020-09-24-333.
https:/​/​doi.org/​10.22331/​q-2020-09-24-333

[11] N. Gisin, N. Linden, S. Massar, and S. Popescu. Error filtration and entanglement purification for quantum communication. Phys. Rev. A, 72: 012338, Jul 2005. 10.1103/​PhysRevA.72.012338. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.72.012338.
https:/​/​doi.org/​10.1103/​PhysRevA.72.012338

[12] Giulio Chiribella and Hlér Kristjánsson. Quantum shannon theory with superpositions of trajectories. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475 (2225): 20180903, 2019. 10.1098/​rspa.2018.0903. URL https:/​/​doi.org/​10.1098/​rspa.2018.0903.
https:/​/​doi.org/​10.1098/​rspa.2018.0903

[13] Hlér Kristjánsson, Giulio Chiribella, Sina Salek, Daniel Ebler, and Matthew Wilson. Resource theories of communication. New Journal of Physics, 22 (7): 073014, jul 2020. 10.1088/​1367-2630/​ab8ef7.
https:/​/​doi.org/​10.1088/​1367-2630/​ab8ef7

[14] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Phys. Rev. Lett., 87: 167902, Sep 2001. 10.1103/​PhysRevLett.87.167902. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.87.167902.
https:/​/​doi.org/​10.1103/​PhysRevLett.87.167902

[15] Rolf T. Horn, A. J. Scott, Jonathan Walgate, Richard Cleve, A. I. Lvovsky, and Barry C. Sanders. Classical and quantum fingerprinting with shared randomness and one-sided error. Quantum Inf. Comput., 5, 2005a. URL https:/​/​dl.acm.org/​doi/​10.5555/​2011637.2011643.
https:/​/​dl.acm.org/​doi/​10.5555/​2011637.2011643

[16] S. Massar. Quantum fingerprinting with a single particle. Phys. Rev. A, 71: 012310, Jan 2005. 10.1103/​PhysRevA.71.012310. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.71.012310.
https:/​/​doi.org/​10.1103/​PhysRevA.71.012310

[17] Rolf T. Horn, S. A. Babichev, Karl-Peter Marzlin, A. I. Lvovsky, and Barry C. Sanders. Single-qubit optical quantum fingerprinting. Phys. Rev. Lett., 95: 150502, Oct 2005b. 10.1103/​PhysRevLett.95.150502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.95.150502.
https:/​/​doi.org/​10.1103/​PhysRevLett.95.150502

[18] Flavio Del Santo and Borivoje Dakić. Two-way communication with a single quantum particle. Phys. Rev. Lett., 120: 060503, Feb 2018. 10.1103/​PhysRevLett.120.060503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.120.060503.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.060503

[19] Li-Yi Hsu, Ching-Yi Lai, You-Chia Chang, Chien-Ming Wu, and Ray-Kuang Lee. Carrying an arbitrarily large amount of information using a single quantum particle. Phys. Rev. A, 102: 022620, Aug 2020. 10.1103/​PhysRevA.102.022620. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.102.022620.
https:/​/​doi.org/​10.1103/​PhysRevA.102.022620

[20] Flavio Del Santo and Borivoje Dakić. Coherence equality and communication in a quantum superposition. Phys. Rev. Lett., 124: 190501, May 2020. 10.1103/​PhysRevLett.124.190501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.124.190501.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.190501

[21] Sebastian Horvat and Borivoje Dakić. Quantum enhancement to information acquisition speed. New Journal of Physics, 23 (3): 033008, mar 2021a. 10.1088/​1367-2630/​abe9d4. URL https:/​/​doi.org/​10.1088/​1367-2630/​abe9d4.
https:/​/​doi.org/​10.1088/​1367-2630/​abe9d4

[22] Juan Carlos Garcia-Escartin and Pedro Chamorro-Posada. swap test and hong-ou-mandel effect are equivalent. Phys. Rev. A, 87: 052330, May 2013. 10.1103/​PhysRevA.87.052330. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.87.052330.
https:/​/​doi.org/​10.1103/​PhysRevA.87.052330

[23] Juan Miguel Arrazola and Norbert Lütkenhaus. Quantum communication with coherent states and linear optics. Phys. Rev. A, 90: 042335, Oct 2014. 10.1103/​PhysRevA.90.042335. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.042335.
https:/​/​doi.org/​10.1103/​PhysRevA.90.042335

[24] Michał Horodecki and Jonathan Oppenheim. (quantumness in the context of) resource theories. International Journal of Modern Physics B, 27 (01n03): 1345019, 2013. 10.1142/​S0217979213450197. URL https:/​/​doi.org/​10.1142/​S0217979213450197.
https:/​/​doi.org/​10.1142/​S0217979213450197

[25] Bob Coecke, Tobias Fritz, and Robert W. Spekkens. A mathematical theory of resources. Information and Computation, 250: 59 – 86, 2016. ISSN 0890-5401. https:/​/​doi.org/​10.1016/​j.ic.2016.02.008. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0890540116000353.
https:/​/​doi.org/​10.1016/​j.ic.2016.02.008
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0890540116000353

[26] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91: 025001, Apr 2019. 10.1103/​RevModPhys.91.025001. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.91.025001.
https:/​/​doi.org/​10.1103/​RevModPhys.91.025001

[27] A. Winter. The capacity of the quantum multiple-access channel. IEEE Transactions on Information Theory, 47 (7): 3059–3065, 2001. 10.1109/​18.959287. URL https:/​/​doi.org/​10.1109/​18.959287.
https:/​/​doi.org/​10.1109/​18.959287

[28] M. Hsieh, I. Devetak, and A. Winter. Entanglement-assisted capacity of quantum multiple-access channels. IEEE Transactions on Information Theory, 54 (7): 3078–3090, 2008. 10.1109/​TIT.2008.924726. URL https:/​/​doi.org/​10.1109/​TIT.2008.924726.
https:/​/​doi.org/​10.1109/​TIT.2008.924726

[29] Felix Leditzky, Mohammad A. Alhejji, Joshua Levin, and Graeme Smith. Playing games with multiple access channels. Nature Communications, 11 (1): 1497, Mar 2020. ISSN 2041-1723. 10.1038/​s41467-020-15240-w. URL https:/​/​doi.org/​10.1038/​s41467-020-15240-w.
https:/​/​doi.org/​10.1038/​s41467-020-15240-w

[30] Lee A. Rozema, Zhao Zhuo, Tomasz Paterek, and Borivoje Dakić. Higher-order interference between multiple quantum particles interacting nonlinearly. Phys. Rev. A, 103: 052204, May 2021. 10.1103/​PhysRevA.103.052204. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.103.052204.
https:/​/​doi.org/​10.1103/​PhysRevA.103.052204

[31] P Grangier, G Roger, and A Aspect. Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences. Europhysics Letters (EPL), 1 (4): 173–179, feb 1986. 10.1209/​0295-5075/​1/​4/​004. URL https:/​/​doi.org/​10.1209.
https:/​/​doi.org/​10.1209/​0295-5075/​1/​4/​004

[32] V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch. Single-photon wavefront-splitting interference. The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics, 35 (3): 561–565, Sep 2005. ISSN 1434-6079. 10.1140/​epjd/​e2005-00201-y. URL https:/​/​doi.org/​10.1140/​epjd/​e2005-00201-y.
https:/​/​doi.org/​10.1140/​epjd/​e2005-00201-y

[33] E. Merzbacher. Quantum Mechanics. Wiley, 1998. ISBN 9780471887027. URL https:/​/​www.worldcat.org/​title/​quantum-mechanics/​oclc/​246969310.
https:/​/​www.worldcat.org/​title/​quantum-mechanics/​oclc/​246969310

[34] Sebastian Horvat. Quantum superposition as a resource for quantum communication. Master's thesis, University of Zagreb, Croatia, 2019. URL https:/​/​repozitorij.pmf.unizg.hr/​islandora/​object/​pmf:7648.
https:/​/​repozitorij.pmf.unizg.hr/​islandora/​object/​pmf:7648

[35] Sebastian Horvat and Borivoje Dakić. Interference as an information-theoretic game. Quantum, 5: 404, March 2021b. ISSN 2521-327X. 10.22331/​q-2021-03-08-404. URL https:/​/​doi.org/​10.22331/​q-2021-03-08-404.
https:/​/​doi.org/​10.22331/​q-2021-03-08-404

[36] Rafael D. Sorkin. Quantum mechanics as quantum measure theory. Modern Physics Letters A, 09 (33): 3119–3127, 1994. 10.1142/​S021773239400294X. URL https:/​/​doi.org/​10.1142/​S021773239400294X.
https:/​/​doi.org/​10.1142/​S021773239400294X

[37] Urbasi Sinha, Christophe Couteau, Thomas Jennewein, Raymond Laflamme, and Gregor Weihs. Ruling out multi-order interference in quantum mechanics. Science, 329 (5990): 418–421, 2010. ISSN 0036-8075. 10.1126/​science.1190545. URL https:/​/​science.sciencemag.org/​content/​329/​5990/​418.
https:/​/​doi.org/​10.1126/​science.1190545
https:/​/​science.sciencemag.org/​content/​329/​5990/​418

[38] Cozmin Ududec, Howard Barnum, and Joseph Emerson. Three slit experiments and the structure of quantum theory. Foundations of Physics, 41 (3): 396–405, Mar 2011. ISSN 1572-9516. 10.1007/​s10701-010-9429-z. URL https:/​/​doi.org/​10.1007/​s10701-010-9429-z.
https:/​/​doi.org/​10.1007/​s10701-010-9429-z

[39] Ciarán M. Lee and John H. Selby. Higher-order interference in extensions of quantum theory. Foundations of Physics, 47 (1): 89–112, Jan 2017. ISSN 1572-9516. 10.1007/​s10701-016-0045-4. URL https:/​/​doi.org/​10.1007/​s10701-016-0045-4.
https:/​/​doi.org/​10.1007/​s10701-016-0045-4

[40] B Dakić, T Paterek, and Č Brukner. Density cubes and higher-order interference theories. New Journal of Physics, 16 (2): 023028, feb 2014. 10.1088/​1367-2630/​16/​2/​023028. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​2/​023028

[41] Gilad Gour and Robert W Spekkens. The resource theory of quantum reference frames: manipulations and monotones. New Journal of Physics, 10 (3): 033023, mar 2008. 10.1088/​1367-2630/​10/​3/​033023. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​10/​3/​033023

[42] R. Cleve, P. Hoyer, B. Toner, and J. Watrous. Consequences and limits of nonlocal strategies. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004., pages 236–249, 2004. 10.1109/​CCC.2004.1313847. URL https:/​/​doi.org/​10.1109/​CCC.2004.1313847.
https:/​/​doi.org/​10.1109/​CCC.2004.1313847

[43] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/​RevModPhys.86.419. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.86.419.
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[44] A. Barvinok. A Course in Convexity. Graduate studies in mathematics. American Mathematical Society, 2002. URL https:/​/​bookstore.ams.org/​gsm-54/​.
https:/​/​bookstore.ams.org/​gsm-54/​

[45] Ll Masanes. Tight bell inequality for d-outcome measurements correlations. Quantum Info. Comput., 3 (4): 345–358, July 2003. ISSN 1533-7146. 10.5555/​2011528.2011532. URL https:/​/​dl.acm.org/​doi/​10.5555/​2011528.2011532.
https:/​/​doi.org/​10.5555/​2011528.2011532

[46] Stefano Pironio. Lifting bell inequalities. Journal of Mathematical Physics, 46 (6): 062112, 2005. 10.1063/​1.1928727. URL https:/​/​doi.org/​10.1063/​1.1928727.
https:/​/​doi.org/​10.1063/​1.1928727

[47] T. Christof and A. Löbel. porta, URL http:/​/​porta.zib.de, 1997. URL http:/​/​porta.zib.de/​.
http:/​/​porta.zib.de/​

[48] Max Born. Zur quantenmechanik der stoßvorgänge. Zeitschrift für Physik, 37 (12): 863–867, Dec 1926. ISSN 0044-3328. 10.1007/​BF01397477. URL https:/​/​doi.org/​10.1007/​BF01397477.
https:/​/​doi.org/​10.1007/​BF01397477

[49] Tanmoy Biswas, María García Díaz, and Andreas Winter. Interferometric visibility and coherence. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473 (2203): 20170170, 2017. 10.1098/​rspa.2017.0170. URL https:/​/​doi.org/​10.1098/​rspa.2017.0170.
https:/​/​doi.org/​10.1098/​rspa.2017.0170

[50] George Svetlichny. Distinguishing three-body from two-body nonseparability by a bell-type inequality. Phys. Rev. D, 35: 3066–3069, May 1987. 10.1103/​PhysRevD.35.3066. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevD.35.3066.
https:/​/​doi.org/​10.1103/​PhysRevD.35.3066

[51] Carl W. Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics, 1 (2): 231–252, Jun 1969. ISSN 1572-9613. 10.1007/​BF01007479. URL https:/​/​doi.org/​10.1007/​BF01007479.
https:/​/​doi.org/​10.1007/​BF01007479

[52] Ryuji Takagi and Bartosz Regula. General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks. Phys. Rev. X, 9: 031053, Sep 2019. 10.1103/​PhysRevX.9.031053. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.9.031053.
https:/​/​doi.org/​10.1103/​PhysRevX.9.031053

[53] Federico Levi and Florian Mintert. A quantitative theory of coherent delocalization. New Journal of Physics, 16 (3): 033007, mar 2014. 10.1088/​1367-2630/​16/​3/​033007. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​3/​033007

[54] Martin Ringbauer, Thomas R. Bromley, Marco Cianciaruso, Ludovico Lami, W. Y. Sarah Lau, Gerardo Adesso, Andrew G. White, Alessandro Fedrizzi, and Marco Piani. Certification and quantification of multilevel quantum coherence. Phys. Rev. X, 8: 041007, Oct 2018. 10.1103/​PhysRevX.8.041007. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.8.041007.
https:/​/​doi.org/​10.1103/​PhysRevX.8.041007

[55] Johan Äberg. Quantifying superposition, 2006.

[56] T. Baumgratz, M. Cramer, and M. B. Plenio. Quantifying coherence. Phys. Rev. Lett., 113: 140401, Sep 2014. 10.1103/​PhysRevLett.113.140401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.113.140401.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.140401

[57] Patrick J. Coles. Entropic framework for wave-particle duality in multipath interferometers. Phys. Rev. A, 93: 062111, Jun 2016. 10.1103/​PhysRevA.93.062111. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.93.062111.
https:/​/​doi.org/​10.1103/​PhysRevA.93.062111

[58] Carmine Napoli, Thomas R. Bromley, Marco Cianciaruso, Marco Piani, Nathaniel Johnston, and Gerardo Adesso. Robustness of coherence: An operational and observable measure of quantum coherence. Phys. Rev. Lett., 116: 150502, Apr 2016. 10.1103/​PhysRevLett.116.150502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.116.150502.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.150502

[59] Marco Piani, Marco Cianciaruso, Thomas R. Bromley, Carmine Napoli, Nathaniel Johnston, and Gerardo Adesso. Robustness of asymmetry and coherence of quantum states. Phys. Rev. A, 93: 042107, Apr 2016. 10.1103/​PhysRevA.93.042107. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.93.042107.
https:/​/​doi.org/​10.1103/​PhysRevA.93.042107

[60] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nature Photonics, 5 (4): 222–229, Apr 2011. ISSN 1749-4893. 10.1038/​nphoton.2011.35. URL https:/​/​doi.org/​10.1038/​nphoton.2011.35.
https:/​/​doi.org/​10.1038/​nphoton.2011.35

### Cited by

[1] Robert Czupryniak, Eric Chitambar, John Steinmetz, and Andrew N. Jordan, "Quantum telescopy clock games", Physical Review A 106 3, 032424 (2022).

[2] Ricardo Faleiro, Nikola Paunkovic, and Marko Vojinovic, "Operational interpretation of the vacuum and process matrices for identical particles", arXiv:2010.16042.

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-05 02:06:55) and SAO/NASA ADS (last updated successfully 2022-10-05 02:06:56). The list may be incomplete as not all publishers provide suitable and complete citation data.