On the distribution of the mean energy in the unitary orbit of quantum states

Raffaele Salvia1 and Vittorio Giovannetti2

1Scuola Normale Superiore and University of Pisa, I-56127 Pisa, Italy
2NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy

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Abstract

Given a closed quantum system, the states that can be reached with a cyclic process are those with the same spectrum as the initial state. Here we prove that, under a very general assumption on the Hamiltonian, the distribution of the mean extractable work is very close to a gaussian with respect to the Haar measure. We derive bounds for both the moments of the distribution of the mean energy of the state and for its characteristic function, showing that the discrepancy with the normal distribution is increasingly suppressed for large dimensions of the system Hilbert space.

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[1] A. Hayashi, T. Hashimoto, and M. Horibe. Reexamination of optimal quantum state estimation of pure states. Physical Review A, 72 (3), September 2005. 10.1103/​physreva.72.032325. URL https:/​/​doi.org/​10.1103/​physreva.72.032325.
https:/​/​doi.org/​10.1103/​physreva.72.032325

[2] A J Scott. Optimizing quantum process tomography with unitary2-designs. Journal of Physics A: Mathematical and Theoretical, 41 (5): 055308, January 2008. 10.1088/​1751-8113/​41/​5/​055308. URL https:/​/​doi.org/​10.1088/​1751-8113/​41/​5/​055308.
https:/​/​doi.org/​10.1088/​1751-8113/​41/​5/​055308

[3] Fernando G. S. L. Brandão and Michal Horodecki. Exponential quantum speed-ups are generic. Quantum Info. Comput., 13 (11–12): 901–924, November 2013. ISSN 1533-7146.

[4] Joseph Emerson, Robert Alicki, and Karol Życzkowski. Scalable noise estimation with random unitary operators. Journal of Optics B: Quantum and Semiclassical Optics, 7 (10): S347–S352, September 2005. 10.1088/​1464-4266/​7/​10/​021. URL https:/​/​doi.org/​10.1088/​1464-4266/​7/​10/​021.
https:/​/​doi.org/​10.1088/​1464-4266/​7/​10/​021

[5] David C. McKay, Sarah Sheldon, John A. Smolin, Jerry M. Chow, and Jay M. Gambetta. Three-qubit randomized benchmarking. Physical Review Letters, 122 (20), May 2019. 10.1103/​physrevlett.122.200502. URL https:/​/​doi.org/​10.1103/​physrevlett.122.200502.
https:/​/​doi.org/​10.1103/​physrevlett.122.200502

[6] Jonas Helsen, Xiao Xue, Lieven M. K. Vandersypen, and Stephanie Wehner. A new class of efficient randomized benchmarking protocols. npj Quantum Information, 5 (1), August 2019. 10.1038/​s41534-019-0182-7. URL https:/​/​doi.org/​10.1038/​s41534-019-0182-7.
https:/​/​doi.org/​10.1038/​s41534-019-0182-7

[7] D.P. DiVincenzo, D.W. Leung, and B.M. Terhal. Quantum data hiding. IEEE Transactions on Information Theory, 48 (3): 580–598, March 2002. 10.1109/​18.985948. URL https:/​/​doi.org/​10.1109/​18.985948.
https:/​/​doi.org/​10.1109/​18.985948

[8] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. Quantum entanglement growth under random unitary dynamics. Phys. Rev. X, 7: 031016, Jul 2017. 10.1103/​PhysRevX.7.031016. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.7.031016.
https:/​/​doi.org/​10.1103/​PhysRevX.7.031016

[9] Cheryne Jonay, David A. Huse, and Adam Nahum. Coarse-grained dynamics of operator and state entanglement. 2018.

[10] Daniel Gottesman. Theory of fault-tolerant quantum computation. Physical Review A, 57 (1): 127–137, January 1998. 10.1103/​physreva.57.127. URL https:/​/​doi.org/​10.1103/​physreva.57.127.
https:/​/​doi.org/​10.1103/​physreva.57.127

[11] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The resource theory of stabilizer quantum computation. New Journal of Physics, 16 (1): 013009, January 2014. 10.1088/​1367-2630/​16/​1/​013009. URL https:/​/​doi.org/​10.1088/​1367-2630/​16/​1/​013009.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​1/​013009

[12] D. Gross, K. Audenaert, and J. Eisert. Evenly distributed unitaries: On the structure of unitary designs. Journal of Mathematical Physics, 48 (5): 052104, May 2007. 10.1063/​1.2716992. URL https:/​/​doi.org/​10.1063/​1.2716992.
https:/​/​doi.org/​10.1063/​1.2716992

[13] Adam Sawicki and Katarzyna Karnas. Universality of single-qudit gates. Annales Henri Poincaré, 18 (11): 3515–3552, August 2017. 10.1007/​s00023-017-0604-z. URL https:/​/​doi.org/​10.1007/​s00023-017-0604-z.
https:/​/​doi.org/​10.1007/​s00023-017-0604-z

[14] Eiichi Bannai, Gabriel Navarro, Noelia Rizo, and Pham Huu Tiep. Unitary t-groups. arXiv [math.RT], Oct 2018. URL https:/​/​arxiv.org/​abs/​1810.02507.
arXiv:1810.02507

[15] Jonas Haferkamp, Felipe Montealegre-Mora, Markus Heinrich, Jens Eisert, David Gross, and Ingo Roth. Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-clifford gates. arXiv [quant-ph], Feb 2020. URL https:/​/​arxiv.org/​abs/​2002.09524.
arXiv:2002.09524

[16] Massimiliano Esposito, Upendra Harbola, and Shaul Mukamel. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Reviews of Modern Physics, 81 (4): 1665–1702, December 2009. 10.1103/​revmodphys.81.1665. URL https:/​/​doi.org/​10.1103/​revmodphys.81.1665.
https:/​/​doi.org/​10.1103/​revmodphys.81.1665

[17] P. Talkner M. Campisi, P. Hänggi. Colloquium. quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys., 83: 771–791, 2011. 10.1103/​RevModPhys.83.771.
https:/​/​doi.org/​10.1103/​RevModPhys.83.771

[18] Thomas Speck and Udo Seifert. Distribution of work in isothermal nonequilibrium processes. Physical Review E, 70 (6), December 2004. 10.1103/​physreve.70.066112. URL https:/​/​doi.org/​10.1103/​physreve.70.066112.
https:/​/​doi.org/​10.1103/​physreve.70.066112

[19] Evžen Šubrt and Petr Chvosta. Exact analysis of work fluctuations in two-level systems. Journal of Statistical Mechanics: Theory and Experiment, 2007 (09): P09019–P09019, September 2007. 10.1088/​1742-5468/​2007/​09/​p09019. URL https:/​/​doi.org/​10.1088/​1742-5468/​2007/​09/​p09019.
https:/​/​doi.org/​10.1088/​1742-5468/​2007/​09/​p09019

[20] Thomas Speck. Work distribution for the driven harmonic oscillator with time-dependent strength: exact solution and slow driving. Journal of Physics A: Mathematical and Theoretical, 44 (30): 305001, June 2011. 10.1088/​1751-8113/​44/​30/​305001. URL https:/​/​doi.org/​10.1088/​1751-8113/​44/​30/​305001.
https:/​/​doi.org/​10.1088/​1751-8113/​44/​30/​305001

[21] Harry J. D. Miller, Matteo Scandi, Janet Anders, and Martí Perarnau-Llobet. Work fluctuations in slow processes: Quantum signatures and optimal control. Physical Review Letters, 123 (23), December 2019. 10.1103/​physrevlett.123.230603. URL https:/​/​doi.org/​10.1103/​physrevlett.123.230603.
https:/​/​doi.org/​10.1103/​physrevlett.123.230603

[22] Matteo Scandi, Harry J. D. Miller, Janet Anders, and Martí Perarnau-Llobet. Quantum work statistics close to equilibrium. Phys. Rev. Research, 2: 023377, Jun 2020. 10.1103/​PhysRevResearch.2.023377. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.2.023377.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.023377

[23] C. Jarzynski. Nonequilibrium equality for free energy differences. Physical Review Letters, 78 (14): 2690–2693, April 1997. 10.1103/​physrevlett.78.2690. URL https:/​/​doi.org/​10.1103/​physrevlett.78.2690.
https:/​/​doi.org/​10.1103/​physrevlett.78.2690

[24] W. Pusz and S. L. Woronowicz. Passive states and KMS states for general quantum systems. Communications in Mathematical Physics, 58 (3): 273–290, October 1978. 10.1007/​bf01614224. URL https:/​/​doi.org/​10.1007/​bf01614224.
https:/​/​doi.org/​10.1007/​bf01614224

[25] A. Lenard. Thermodynamical proof of the gibbs formula for elementary quantum systems. Journal of Statistical Physics, 19 (6): 575–586, December 1978. 10.1007/​bf01011769. URL https:/​/​doi.org/​10.1007/​bf01011769.
https:/​/​doi.org/​10.1007/​bf01011769

[26] A. E. Allahverdyan, R Balian, and Th. M. Nieuwenhuizen. Maximal work extraction from finite quantum systems. Europhysics Letters (EPL), 67 (4): 565–571, aug 2004. 10.1209/​epl/​i2004-10101-2. URL https:/​/​doi.org/​10.1209.
https:/​/​doi.org/​10.1209/​epl/​i2004-10101-2

[27] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy. Quantum batteries. In Fundamental Theories of Physics, pages 207–225. Springer International Publishing, 2018. 10.1007/​978-3-319-99046-0_8. URL https:/​/​doi.org/​10.1007/​978-3-319-99046-0_8.
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_8

[28] R. Alicki and M. Fannes. Entanglement boost for extractable work from ensembles of quantum batteries. Phys. Rev. E, 87: 042123, 2013.

[29] Karen V. Hovhannisyan, Martí Perarnau-Llobet, Marcus Huber, and Antonio Acín. Entanglement generation is not necessary for optimal work extraction. Physical Review Letters, 111 (24), dec 2013. 10.1103/​physrevlett.111.240401. URL https:/​/​doi.org/​10.1103/​physrevlett.111.240401.
https:/​/​doi.org/​10.1103/​physrevlett.111.240401

[30] Felix C Binder, Sai Vinjanampathy, Kavan Modi, and John Goold. Quantacell: powerful charging of quantum batteries. New Journal of Physics, 17 (7): 075015, July 2015. 10.1088/​1367-2630/​17/​7/​075015. URL https:/​/​doi.org/​10.1088/​1367-2630/​17/​7/​075015.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​7/​075015

[31] Sergi Julià-Farré, Tymoteusz Salamon, Arnau Riera, Manabendra N. Bera, and Maciej Lewenstein. Bounds on the capacity and power of quantum batteries. Physical Review Research, 2 (2), May 2020. 10.1103/​physrevresearch.2.023113. URL https:/​/​doi.org/​10.1103/​physrevresearch.2.023113.
https:/​/​doi.org/​10.1103/​physrevresearch.2.023113

[32] a. Given $f(E)$ a generic function of the random variable (8), its mean value with respect to $P(E| \hat{\rho}; \hat{H})$, i.e. the quantity $ \langle f(E) \rangle := \int dE P(E| \hat{\rho}; \hat{H}) f(E) \nonumber = \int d\mu(\hat{U}) f(E(\hat{U} \hat{\rho} \hat{U}^\dagger; \hat{H})),$ with $d\mu(\hat{U})$ representing the Harr measure on $\mathbf{U}(d)$.

[33] b. Notice that the quantity (20) can be equivalently be expressed as $ \langle E^p\rangle = Tr[\mathcal{T}^{(p)}(\hat{\rho}^{\otimes p}) \hat{H}^{\otimes p} ] $ where $ \mathcal{T}^{(p)}(\cdots ) := \int d\mu(\hat{U}) \hat{U}^{\otimes p} \cdots \hat{U}^{\dagger\otimes p} $ are Completely Positive, trace preserving map known as Twirling channels which have a number of applications in quantum information theory [63-68].

[34] Don Weingarten. Asymptotic behavior of group integrals in the limit of infinite rank. Journal of Mathematical Physics, 19 (5): 999–1001, May 1978. 10.1063/​1.523807. URL https:/​/​doi.org/​10.1063/​1.523807.
https:/​/​doi.org/​10.1063/​1.523807

[35] Michael Creutz. On invariant integration over SU(n). Journal of Mathematical Physics, 19 (10): 2043, 1978. 10.1063/​1.523581. URL https:/​/​doi.org/​10.1063/​1.523581.
https:/​/​doi.org/​10.1063/​1.523581

[36] I. Bars. U(n) integral for the generating functional in lattice gauge theory. Journal of Mathematical Physics, 21 (11): 2678–2681, November 1980. 10.1063/​1.524368. URL https:/​/​doi.org/​10.1063/​1.524368.
https:/​/​doi.org/​10.1063/​1.524368

[37] Benoı̂t Collins. Moments and cumulants of polynomial random variables on unitary groups, the itzykson-zuber integral and free probability. International Mathematics Research Notices, 2003 (17): 953, 2003. 10.1155/​s107379280320917x. URL https:/​/​doi.org/​10.1155/​s107379280320917x.
https:/​/​doi.org/​10.1155/​s107379280320917x

[38] A. Young. On quantitative substitutional analysis. Proceedings of the London Mathematical Society, s1-33 (1): 97–145, November 1900. 10.1112/​plms/​s1-33.1.97. URL https:/​/​doi.org/​10.1112/​plms/​s1-33.1.97.
https:/​/​doi.org/​10.1112/​plms/​s1-33.1.97

[39] Wilhelm Specht. Zur theorie der matrizen. ii. Jahresbericht der Deutschen Mathematiker-Vereinigung, 50: 19–23, 1940. URL http:/​/​eudml.org/​doc/​146243.
http:/​/​eudml.org/​doc/​146243

[40] Jean-Bernard Zuber. Revisiting SU(n) integrals. Journal of Physics A: Mathematical and Theoretical, 50 (1): 015203, November 2016. 10.1088/​1751-8113/​50/​1/​015203. URL https:/​/​doi.org/​10.1088/​1751-8113/​50/​1/​015203.
https:/​/​doi.org/​10.1088/​1751-8113/​50/​1/​015203

[41] William Fulton and Joe Harris. Representation Theory. Springer New York, 2004. 10.1007/​978-1-4612-0979-9. URL https:/​/​doi.org/​10.1007/​978-1-4612-0979-9.
https:/​/​doi.org/​10.1007/​978-1-4612-0979-9

[42] Marcel Noaves. Elementary derivation of weingarten functions of classical lie groups. arXiv [math-ph], Jan 2014. URL http:/​/​arxiv.org/​abs/​1406.2182.
arXiv:1406.2182

[43] c. Notice that the requirement $c[\sigma]\leq d$ is trivial as long as $p\leq d$; if we extend the sum in (43) to all the irreducible representations of $S_p$, we obtain the Weingarten function for the special unitary group $S\mathbf{U}(d)$. Therefore, when $c[\sigma]<d$; is equivalent to do the integral (21) on the unitary group $\mathbf{U}(d)$ or on the special unitary group $S\mathbf{U}(d)$.

[44] Benoı̂t Collins and Sho Matsumoto. Weingarten calculus via orthogonality relations: new applications. Latin American Journal of Probability and Mathematical Statistics, 14 (1): 631, 2017. 10.30757/​alea.v14-31. URL https:/​/​doi.org/​10.30757/​alea.v14-31.
https:/​/​doi.org/​10.30757/​alea.v14-31

[45] Fuad Kittaneh. Inequalities for the schatten p-norm. Glasgow Mathematical Journal, 26 (2): 141–143, July 1985. 10.1017/​s0017089500005905. URL https:/​/​doi.org/​10.1017/​s0017089500005905.
https:/​/​doi.org/​10.1017/​s0017089500005905

[46] d. The fact that $\eta_{\hat{H}}\geq \frac{1}{d^6}$ follows from the inequality $ \| \hat{\Theta}\|_{\infty} \leq \| \hat{\Theta}\|_3 \leq \| \hat{\Theta}\|_2\leq \sqrt{d} \| \hat{\Theta}\|_{\infty} , $ that holds true for all operators $\hat{\Theta}$.

[47] Donald E. Knuth, Ronald L. Graham, and Oren Patashnik. Concrete Mathematics. Addison-Wesley Longman Publishing Co., Inc., USA, 2nd edition, 1994. ISBN 0201558025.

[48] W. Allen Whitworth. The Messenger of Mathematics, Vol. VIII. Forgotten Books, 1879. ISBN 9780282986629.

[49] Renzo Sprugnoli. On the approximation of catalan numbers and other quantities. European Journal of Combinatorics, 11 (1): 65–72, January 1990. 10.1016/​s0195-6698(13)80057-1. URL https:/​/​doi.org/​10.1016/​s0195-6698(13)80057-1.
https:/​/​doi.org/​10.1016/​s0195-6698(13)80057-1

[50] Richard Stanley. Enumerative combinatorics. Wadsworth & Brooks/​Cole Advanced Books & Software Cambridge University Press, Monterey, Calif. Cambridge, U.K, 1986. ISBN 9780521789875.

[51] e. We observe incidentally that an alternative way to estimate $\left\langle e^{-itE} \right\rangle$ is via the direct evaluation of the associated integral, i.e. $\int d\mu(\hat{U}) e^{-itE} = \int d\mu(\hat{U}) e^{-it Tr[\hat{U} \hat{\rho} \hat{U}^\dagger \hat{H}]}$. The solution of this integral is known [69, 70] and it is given by $ \left\langle e^{-itE} \right\rangle = \left( \prod_{i=1}^{d-1} i! \right) \frac{ \det \exp \left[ t \lambda_i \epsilon_j \right] }{t^{\frac{d^2-d}{2}} \Delta(\rho) \Delta(H) } , $ where $\Delta(\cdot)$ denotes the Vandermonde determinants $\Delta(\rho) := \prod_{1 \leq i < j \leq d} \left( \lambda_j - \lambda_i \right)$ and $\Delta(H) := \prod_{1 \leq i < j \leq d} \left( \epsilon_j - \epsilon_i \right)$. We prefer however to estimate $\mathcal{G}(t)$ with the definition (\refmoment_genarating_function), as we lack of an appropriate expansion for the term $\det \exp \left[ t \lambda_i \epsilon_j \right]$.

[52] Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, USA, 1 edition, 2009. ISBN 0521898064.

[53] Freeman J. Dyson. The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics. Journal of Mathematical Physics, 3 (6): 1199–1215, November 1962. 10.1063/​1.1703863. URL https:/​/​doi.org/​10.1063/​1.1703863.
https:/​/​doi.org/​10.1063/​1.1703863

[54] Bertrand Eynard, Taro Kimura, and Sylvain Ribault. Random matrices, 2015.

[55] Eugene P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. The Annals of Mathematics, 62 (3): 548, November 1955. 10.2307/​1970079. URL https:/​/​doi.org/​10.2307/​1970079.
https:/​/​doi.org/​10.2307/​1970079

[56] Yan V. Fyodorov. Introduction to the random matrix theory: Gaussian unitary ensemble and beyond. London Mathematical Society Lecture Note Series, 322: 31–78, 2004.

[57] Hrant Gharibyan, Masanori Hanada, Stephen H. Shenker, and Masaki Tezuka. Onset of random matrix behavior in scrambling systems. Journal of High Energy Physics, 2018 (7), July 2018. 10.1007/​jhep07(2018)124. URL https:/​/​doi.org/​10.1007/​jhep07(2018)124.
https:/​/​doi.org/​10.1007/​jhep07(2018)124

[58] M. L. Mehta. Random matrices. Academic Press, Amsterdam San Diego, CA, 2004. ISBN 9780120884094.

[59] Sho Matsumoto. Weingarten calculus for matrix ensembles associated with compact symmetric spaces. Random Matrices: Theory and Applications, 02 (02): 1350001, April 2013. 10.1142/​s2010326313500019. URL https:/​/​doi.org/​10.1142/​s2010326313500019.
https:/​/​doi.org/​10.1142/​s2010326313500019

[60] John C. Oxtoby. Invariant measures in groups which are not locally compact. Transactions of the American Mathematical Society, 60: 215–215, 1946. 10.1090/​s0002-9947-1946-0018188-5. URL https:/​/​doi.org/​10.1090/​s0002-9947-1946-0018188-5.
https:/​/​doi.org/​10.1090/​s0002-9947-1946-0018188-5

[61] X. Wang, T. Hiroshima, A. Tomita, and M. Hayashi. Quantum information with gaussian states. Physics Reports, 448 (1-4): 1–111, August 2007. 10.1016/​j.physrep.2007.04.005. URL https:/​/​doi.org/​10.1016/​j.physrep.2007.04.005.
https:/​/​doi.org/​10.1016/​j.physrep.2007.04.005

[62] Alessio Serafini. Quantum continuous variables : a primer of theoretical methods. CRC Press, Taylor & Francis Group, Boca Raton, FL, 2017. ISBN 9781482246346.

[63] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54: 3824–3851, Nov 1996. 10.1103/​PhysRevA.54.3824. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.54.3824.
https:/​/​doi.org/​10.1103/​PhysRevA.54.3824

[64] Charles H. Bennett, David P. DiVincenzo, Tal Mor, Peter W. Shor, John A. Smolin, and Barbara M. Terhal. Unextendible product bases and bound entanglement. Phys. Rev. Lett., 82: 5385–5388, Jun 1999. 10.1103/​PhysRevLett.82.5385. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.82.5385.
https:/​/​doi.org/​10.1103/​PhysRevLett.82.5385

[65] K. G. H. Vollbrecht and R. F. Werner. Entanglement measures under symmetry. Phys. Rev. A, 64: 062307, Nov 2001. 10.1103/​PhysRevA.64.062307. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.64.062307.
https:/​/​doi.org/​10.1103/​PhysRevA.64.062307

[66] Soojoon Lee, Dong Pyo Chi, Sung Dahm Oh, and Jaewan Kim. Convex-roof extended negativity as an entanglement measure for bipartite quantum systems. Phys. Rev. A, 68: 062304, Dec 2003. 10.1103/​PhysRevA.68.062304. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.68.062304.
https:/​/​doi.org/​10.1103/​PhysRevA.68.062304

[67] Masahito Hayashi, Damian Markham, Mio Murao, Masaki Owari, and Shashank Virmani. Entanglement of multiparty-stabilizer, symmetric, and antisymmetric states. Physical Review A, 77 (1), January 2008. 10.1103/​physreva.77.012104. URL https:/​/​doi.org/​10.1103/​physreva.77.012104.
https:/​/​doi.org/​10.1103/​physreva.77.012104

[68] Yingkai Ouyang. Channel covariance, twirling, contraction, and some upper bounds on the quantum capacity. Quantum Information and Computation, 14: 0917–0936, Sep 2014. 10.26421/​QIC17.11-12. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.68.062304.
https:/​/​doi.org/​10.26421/​QIC17.11-12
https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.68.062304

[69] Harish-Chandra. Differential operators on a semisimple lie algebra. American Journal of Mathematics, 79 (1): 87, January 1957. 10.2307/​2372387. URL https:/​/​doi.org/​10.2307/​2372387.
https:/​/​doi.org/​10.2307/​2372387

[70] C. Itzykson and J.-B. Zuber. The planar approximation. II. Journal of Mathematical Physics, 21 (3): 411–421, March 1980. 10.1063/​1.524438. URL https:/​/​doi.org/​10.1063/​1.524438.
https:/​/​doi.org/​10.1063/​1.524438

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