Entanglement capacity of fermionic Gaussian states. (English) Zbl 1533.81012
Summary: We study the capacity of entanglement as an alternative to entanglement entropies in estimating the degree of entanglement of quantum bipartite systems over fermionic Gaussian states. In particular, we derive the exact and asymptotic formulas of average capacity of two different cases – with and without particle number constraints. For the later case, the obtained formulas generalize some partial results of average capacity in the literature. The key ingredient in deriving the results is a set of new tools for simplifying finite summations developed very recently in the study of entanglement entropy of fermionic Gaussian states.
MSC:
| 81P40 | Quantum coherence, entanglement, quantum correlations |
| 81P42 | Entanglement measures, concurrencies, separability criteria |
| 81V74 | Fermionic systems in quantum theory |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 15B52 | Random matrices (algebraic aspects) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 62F30 | Parametric inference under constraints |
| 28D20 | Entropy and other invariants |
Keywords:
quantum entanglement; entanglement capacity; fermionic Gaussian states; random matrix theory; orthogonal polynomials; special functionsReferences:
| [1] | Page, D. N., Average entropy of a subsystem, Phys. Rev. Lett., 71, 1291-4 (1993) · Zbl 0972.81504 · doi:10.1103/PhysRevLett.71.1291 |
| [2] | Foong, S. K.; Kanno, S., Proof of Page’s conjecture on the average entropy of a subsystem, Phys. Rev. Lett., 72, 1148-51 (1994) · Zbl 0973.81502 · doi:10.1103/PhysRevLett.72.1148 |
| [3] | Sánchez-Ruiz, J., Simple proof of Page’s conjecture on the average entropy of a subsystem, Phys. Rev. E, 52, 5653-5 (1995) · doi:10.1103/PhysRevE.52.5653 |
| [4] | Hayden, P.; Leung, D.; Winter, A., Aspects of generic entanglement, Commun. Math. Phys., 265, 95-117 (2006) · Zbl 1107.81011 · doi:10.1007/s00220-006-1535-6 |
| [5] | Vivo, P.; Pato, M. P.; Oshanin, G., Random pure states: quantifying bipartite entanglement beyond the linear statistics, Phys. Rev. E, 93 (2016) · doi:10.1103/PhysRevE.93.052106 |
| [6] | Wei, L., Proof of Vivo-Pato-Oshanin’s conjecture on the fluctuation of von Neumann entropy, Phys. Rev. E, 96 (2017) · doi:10.1103/PhysRevE.96.022106 |
| [7] | Sarkar, A.; Kumar, S., Bures-Hall ensemble: spectral densities and average entropies, J. Phys. A: Math. Theor., 52 (2019) · Zbl 1509.60011 · doi:10.1088/1751-8121/ab2675 |
| [8] | Wei, L., Skewness of von Neumann entanglement entropy, J. Phys. A: Math. Theor., 53 (2020) · Zbl 1514.81032 · doi:10.1088/1751-8121/ab63a7 |
| [9] | Wei, L., Proof of Sarkar-Kumar conjectures on average entanglement entropies over the Bures-Hall ensemble, J. Phys. A: Math. Theor., 53 (2020) · Zbl 1514.81033 · doi:10.1088/1751-8121/ab8d07 |
| [10] | Wei, L., Exact variance of von Neumann entanglement entropy over the Bures-Hall measure, Phys. Rev. E, 102 (2020) · doi:10.1103/PhysRevE.102.062128 |
| [11] | Huang, Y.; Wei, L.; Collaku, B., Kurtosis of von Neumann entanglement entropy, J. Phys. A: Math. Theor., 54 (2021) · Zbl 1507.81043 · doi:10.1088/1751-8121/ac367c |
| [12] | Lubkin, E., Entropy of an \(n\)-system from its correlation with a \(k\)-reservoir, J. Math. Phys., 19, 1028-31 (1978) · Zbl 0389.94006 · doi:10.1063/1.523763 |
| [13] | Sommers, H-J; Życzkowski, K., Statistical properties of random density matrices, J. Phys. A: Math. Gen., 37, 35 (2004) · Zbl 1062.82005 · doi:10.1088/0305-4470/37/35/004 |
| [14] | Osipov, V.; Sommers, H-J; Życzkowski, K., Random Bures mixed states and the distribution of their purity, J. Phys. A: Math. Theor., 43 (2010) · Zbl 1186.81033 · doi:10.1088/1751-8113/43/5/055302 |
| [15] | Giraud, O., Distribution of bipartite entanglement for random pure states, J. Phys. A: Math. Theor., 40, 2793 (2007) · Zbl 1111.81030 · doi:10.1088/1751-8113/40/11/014 |
| [16] | Li, S-H; Wei, L., Moments of quantum purity and biorthogonal polynomial recurrence, J. Phys. A: Math. Theor., 54 (2021) · Zbl 1519.81099 · doi:10.1088/1751-8121/ac2a53 |
| [17] | Malacarne, L. C.; Mendes, R. S.; Lenzi, E. K., Average entropy of a subsystem from its average Tsallis entropy, Phys. Rev. E, 65 (2002) · Zbl 1244.82005 · doi:10.1103/PhysRevE.65.046131 |
| [18] | Wei, L., On the exact variance of Tsallis entanglement entropy in a random pure state, Entropy, 21, 539 (2019) · doi:10.3390/e21050539 |
| [19] | Borot, G.; Nadal, C., Purity distribution for generalized random Bures mixed states, J. Phys. A: Math. Theor., 45 (2012) · Zbl 1239.81022 · doi:10.1088/1751-8113/45/7/075209 |
| [20] | Bianchi, E.; Hackl, L.; Kieburg, M., The Page curve for fermionic Gaussian states, Phys. Rev. B, 103 (2021) · doi:10.1103/PhysRevB.103.L241118 |
| [21] | Huang, Y.; Wei, L., Second-order statistics of fermionic Gaussian states, J. Phys. A: Math. Theor., 55 (2022) · Zbl 1505.81017 · doi:10.1088/1751-8121/ac4e20 |
| [22] | Bianchi, E.; Hackl, L.; Kieburg, M.; Rigol, M.; Vidmar, L., Volume-law entanglement entropy of typical pure quantum states, PRX Quantum, 3 (2022) · doi:10.1103/PRXQuantum.3.030201 |
| [23] | Huang, Y.; Wei, L., Entropy fluctuation formulas of fermionic Gaussian states, Ann. Henri Poincaré (2023) · Zbl 1534.81014 · doi:10.1007/s00023-023-01342-w |
| [24] | Yao, H.; Qi, X-L, Entanglement entropy and entanglement spectrum of the Kitaev model, Phys. Rev. Lett., 105 (2010) · doi:10.1103/PhysRevLett.105.080501 |
| [25] | Nandy, P., Capacity of entanglement in local operators, J. High Energy Phys., JHEP07(2021)019 (2021) · Zbl 1468.81019 · doi:10.1007/JHEP07(2021)019 |
| [26] | Arias, R.; Di Giulio, G.; Keski-Vakkuri, E.; Tonni, E., Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement, J. High Energy Phys., JHEP03(2023)175 (2023) · Zbl 07690739 · doi:10.1007/JHEP03(2023)175 |
| [27] | de Boer, J.; Järvelä, J.; Keski-Vakkuri, E., Aspects of capacity of entanglement, Phys. Rev. D, 99 (2019) · doi:10.1103/PhysRevD.99.066012 |
| [28] | Okuyama, K., Capacity of entanglement in random pure state, Phys. Lett. B, 820 (2021) · Zbl 07414591 · doi:10.1016/j.physletb.2021.136600 |
| [29] | Wei, L., Average capacity of quantum entanglement, J. Phys. A: Math. Theor., 56 (2023) · Zbl 1519.81091 · doi:10.1088/1751-8121/acb114 |
| [30] | Bhattacharjee, B.; Nandy, P.; Pathak, T., Eigenstate capacity and Page curve in fermionic Gaussian states, Phys. Rev. B, 104 (2021) · doi:10.1103/PhysRevB.104.214306 |
| [31] | Preskill, J., Quantum computing in the NISQ era and beyond, Quantum, 2, 79 (2018) · doi:10.22331/q-2018-08-06-79 |
| [32] | Surace, J.; Tagliacozzo, L., Fermionic Gaussian states: an introduction to numerical approaches (2021) |
| [33] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (1972), Dover · Zbl 0543.33001 |
| [34] | Bengtsson, I.; Życzkowski, K., Geometry of Quantum States: An Introduction to Quantum Entanglement (2017), Cambridge University Press · Zbl 1392.81005 |
| [35] | Kieburg, M.; Forrester, P.; Ipsen, J. R., Multiplicative convolution of real asymmetric and real anti-symmetric matrices, Adv. Pure Appl. Math., 10, 467 (2019) · Zbl 1475.15043 · doi:10.1515/apam-2018-0037 |
| [36] | Lydżba, P.; Rigol, M.; Vidmar, L., Eigenstate entanglement entropy in random quadratic Hamiltonians, Phys. Rev. Lett., 125 (2020) · doi:10.1103/PhysRevLett.125.180604 |
| [37] | Lydżba, P.; Rigol, M.; Vidmar, L., Entanglement in many-body eigenstates of quantum-chaotic quadratic Hamiltonians, Phys. Rev. B, 103 (2021) · doi:10.1103/PhysRevB.103.104206 |
| [38] | Mehta, M. L., Random Matrices (2004), Elsevier · Zbl 1107.15019 |
| [39] | Forrester, P., Log-Gases and Random Matrices (2010), Princeton University Press · Zbl 1217.82003 |
| [40] | Bernard, D.; Piroli, L., Entanglement distribution in the quantum symmetric simple exclusion process, Phys. Rev. E, 104 (2021) · doi:10.1103/PhysRevE.104.014146 |
| [41] | Szegő, G., Orthogonal Polynomials (1975), American Mathematical Society · JFM 61.0386.03 |
| [42] | Luke, Y. L., The Special Functions and Their Approximations, vol 1 (1969), Academic · Zbl 0193.01701 |
| [43] | Brychkov, Y. A., Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas (2008), CRC Press · Zbl 1158.33001 |
| [44] | Milgram, M., On some sums of digamma and polygamma functions (2017) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
