Linear algebraic proof of Wigner theorem and its consequences
-
Jáchym Barvínek
Abstract
We present new proof of non-bijective Wigner theorem on symmetries of quantum systems using only basic linear algebra. It is based on showing that any non-zero Jordan ∗-homomorphism between matrix algebras preserving rank-one projections is implemented by either a unitary or an anitiunitary map. As a new application we extend hitherto known results on preservers of quantum relative entropy to infinite quantum systems.
Acknowledgement
The authors would like to thank to the referee for many valuable suggestions, especially for pointing out to gaps in the arguments. It enabled to improve earlier version of the paper significantly.
This work was supported by the “Grant Agency of the Czech Republic” grant number P201/12/0290, “Topological and geometrical properties of Banach spaces and operator algebras”.
References
[1] Barvínek, J.: Quantum Entropy and its Preservation, Bachelor thesis, CVUT, 2013 (In Czech).Suche in Google Scholar
[2] Cassinelli, G.—De Vito, E.—Levrero, A.—Lahti, P.: Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry transformations, Rev. Math. Phys. 8 (1997), 921–941.10.1142/S0129055X97000324Suche in Google Scholar
[3] Emch, G.: Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Dover Publishing, 1972.Suche in Google Scholar
[4] Gehér, P.: An elementary proof for the non-bijective version of Wigner’s theorem, Phys. Lett. A 387 (2014), 2054–2057.10.1016/j.physleta.2014.05.039Suche in Google Scholar
[5] Hamhalter, J.: Quantum Measure Theory, Kluwer Publishers, Dordrecht, Boston, London, 2003.10.1007/978-94-017-0119-8Suche in Google Scholar
[6] Hanche-Olsen, H.—Størmer, E.: Jordan Operator Algebras, Pitman, 1984.Suche in Google Scholar
[7] Herstein, I. N.: Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331–341.10.1090/S0002-9947-1956-0076751-6Suche in Google Scholar
[8] Hou, J.: Rank-preserving linear maps on B(K), Sci. China Ser. A 32 (1989), 929–940.Suche in Google Scholar
[9] Jacobson, N.—Rickart, C.: Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502.10.1007/978-1-4612-3694-8_7Suche in Google Scholar
[10] Kadison, R. V.: Isometries of operator algebras, Ann. of Math. 54 (1951), 325–338.10.2307/1969534Suche in Google Scholar
[11] Kadison, R. V.—Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras American Mathematical Society, Vol. I, II, III, IV, 1994.Suche in Google Scholar
[12] Molnar, L.: Wigner’s unitary-antiunitary theorem via Herstein theorem on Jordan homomorphisms, J. Nat. Geom. 10 (1996), 137–148.Suche in Google Scholar
[13] Molnar, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Springer, 2007.Suche in Google Scholar
[14] Molnar, L.: Maps on states preserving the relative entropy. J. Math. Phys. 49 (2008), 0321114.10.1063/1.2898693Suche in Google Scholar
[15] Molnar, L.—Szokol, P.: Maps on states preserving the relative entropy II., Linear Algebra Appl. 432 (2010), 3343–3350.10.1016/j.laa.2010.01.025Suche in Google Scholar
[16] Nielsen, M. A.—Chuang, I. J.: Quantum Computation and Quantum Information, Cambridge University Press, 2001.Suche in Google Scholar
[17] Palmer, T. W.: Banch algebras and the general theory of *-algebras, I, II, University Press, Cambridge, 1994.10.1017/CBO9781107325777Suche in Google Scholar
[18] Ohya, M.—Petz, D.: Quantum Entropy and Its Use. Texts and Monographs in Physics, Springer Verlag, 1993.10.1007/978-3-642-57997-4Suche in Google Scholar
[19] Petz, D.: Quantum Information Theory and Quantum Statistics, Springer Verlag, Berlin, Heidelberg, 2008.Suche in Google Scholar
[20] Størmer, E.: On the Jordan structure of C*-algebras, Trans. Amer. Math. Soc. 120 (1965), 438–447.10.2307/1994536Suche in Google Scholar
[21] Størmer, E.: Positive Linear Maps of Operator Algebras, Springer, 2013.10.1007/978-3-642-34369-8Suche in Google Scholar
[22] Simon, R.—Mukunda, N.—Chaturvedi, S.—Srinavasan, V.: Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics, Phys. Lett. A 372 (2008), 6847–6852.10.1016/j.physleta.2008.09.052Suche in Google Scholar
[23] Simon, R.—Mukunda, N.—Chaturvedi, S.—Srinavasan, V.—Hamhalter, J.: Comment on: Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics [Phys. Lett. A 372 (2008), 6847], Phys. Lett. A 378 (2014), 2332–2335.10.1016/j.physleta.2014.03.058Suche in Google Scholar
[24] Weinberger, S.: The Quantum Theory of Fields, Cambridge, USA, 1995.10.1017/CBO9781139644167Suche in Google Scholar
[25] Wigner, E. P.: Group Theory and Its Applications to the Quantum Theory of Atomic Spectra, Academic Press Inc., New York, 1959.Suche in Google Scholar
© 2017 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- 10.1515/ms-2015-0200
- Zero-divisor graphs of lower dismantlable lattices I
- Some results on the intersection graph of submodules of a module
- Class number parities of compositum of quadratic function fields
- Examples of beurling prime systems
-
Connection between multiplication theorem for Bernoulli polynomials and first factor
- On permutational invariance of the metric discrepancy results
- Evaluation of sums containing triple aerated generalized Fibonomial coefficients
- Linear algebraic proof of Wigner theorem and its consequences
- A note on groups with finite conjugacy classes of subnormal subgroups
- Groups with the same complex group algebras as some extensions of psl(2, pn)
- Klee-Phelps convex groupoids
- On analytic functions with generalized bounded Mocanu variation in conic domain with complex order
- Weak interpolation for the lipschitz class
- Generalized Padé approximants for plane condenser and distribution of points
- Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials
- Positive solutions of nonlocal integral BVPS for the nonlinear coupled system involving high-order fractional differential
- Existence of positive solutions for a nonlinear nth-order m-point p-Laplacian impulsive boundary value problem
- Dirichlet boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses
- On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term
- Homoclinic solutions for ordinary (q, p)-Laplacian systems with a coercive potential
- Semi-equivelar maps on the torus and the Klein bottle with few vertices
- A problem considered by Friedlander & Iwaniec and the discrete Hardy-Littlewood method
Artikel in diesem Heft
- 10.1515/ms-2015-0200
- Zero-divisor graphs of lower dismantlable lattices I
- Some results on the intersection graph of submodules of a module
- Class number parities of compositum of quadratic function fields
- Examples of beurling prime systems
-
Connection between multiplication theorem for Bernoulli polynomials and first factor
- On permutational invariance of the metric discrepancy results
- Evaluation of sums containing triple aerated generalized Fibonomial coefficients
- Linear algebraic proof of Wigner theorem and its consequences
- A note on groups with finite conjugacy classes of subnormal subgroups
- Groups with the same complex group algebras as some extensions of psl(2, pn)
- Klee-Phelps convex groupoids
- On analytic functions with generalized bounded Mocanu variation in conic domain with complex order
- Weak interpolation for the lipschitz class
- Generalized Padé approximants for plane condenser and distribution of points
- Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials
- Positive solutions of nonlocal integral BVPS for the nonlinear coupled system involving high-order fractional differential
- Existence of positive solutions for a nonlinear nth-order m-point p-Laplacian impulsive boundary value problem
- Dirichlet boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses
- On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term
- Homoclinic solutions for ordinary (q, p)-Laplacian systems with a coercive potential
- Semi-equivelar maps on the torus and the Klein bottle with few vertices
- A problem considered by Friedlander & Iwaniec and the discrete Hardy-Littlewood method
