Abstract
We characterize bistochastic Kadison–Schwarz operators acting on \(M_2(\mathbb C)\). The obtained characterization allows us to find positive operators that are not Kadison–Schwarz ones. Moreover, we provide several examples of Kadison–Schwarz operators which are not completely positive.
Similar content being viewed by others
References
I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge Univ. Press, Cambridge, 2006).
R. Bhatia, Positive Definite Matrices (Princeton Univ. Press, Princeton, NJ, 2009).
M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
D. Chruściński, “Quantum-correlation breaking channels, quantum conditional probability and Perron–Frobenius theory,” Phys. Lett. A 377 (8), 606–611 (2013).
D. Chruściński, “A class of symmetric Bell diagonal entanglement witnesses—a geometric perspective,” J. Phys. A: Math. Theor. 47 (42), 424033 (2014).
D. Chruściński, “On Kossakowski construction of positive maps on matrix algebras,” Open Syst. Inf. Dyn. 21 (3), 1450001 (2014).
D. Chruściński and A. Kossakowski, “Divisibility of dynamical maps with time independent invariant state,” Open Syst. Inf. Dyn. 26 (4), 1950019 (2019).
D. Chruściński and F. Mukhamedov, “Dissipative generators, divisible dynamical maps, and the Kadison–Schwarz inequality,” Phys. Rev. A 100 (5), 052120 (2019).
D. Chruściński, F. Mukhamedov, and M. A. Hajji, “On Kadison–Schwarz approximation to positive maps,” Open Syst. Inf. Dyn. 27 (3), 2050016 (2020).
D. Chruściński and G. Sarbicki, “Exposed positive maps in \(M_4(\mathbb C)\),” Open Syst. Inf. Dyn. 19 (3), 1250017 (2012).
D. Chruściński and G. Sarbicki, “Entanglement witnesses: Construction, analysis and classification,” J. Phys. A.: Math. Theor. 47 (48), 483001 (2014).
N. N. Ganikhodzhaev and F. M. Mukhamedov, “Ergodic properties of quantum quadratic stochastic processes defined on von Neumann algebras,” Russ. Math. Surv. 53 (6), 1350–1351 (1998) [transl. from Usp. Mat. Nauk 53 (6), 243–244 (1998)].
R. Ganikhodzhaev, F. Mukhamedov, and U. Rozikov, “Quadratic stochastic operators and processes: Results and open problems,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2), 279–335 (2011).
U. Groh, “Uniform ergodic theorems for identity preserving Schwarz maps on \(W^*\)-algebras,” J. Oper. Theory 11, 395–404 (1984).
K.-C. Ha, “Entangled states with strong positive partial transpose,” Phys. Rev. A 81 (6), 064101 (2010).
K.-C. Ha and S.-H. Kye, “Entanglement witnesses arising from exposed positive linear maps,” Open Syst. Inf. Dyn. 18 (4), 323–337 (2011).
M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: Necessary and sufficient conditions,” Phys. Lett. A 223 (1–2), 1–8 (1996).
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81 (2), 865–942 (2009).
C. King and M. B. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” IEEE Trans. Inf. Theory 47 (1), 192–209 (2001).
A. Kossakowski, “A class of linear positive maps in matrix algebras,” Open Syst. Inf. Dyn. 10 (3), 213–220 (2003).
W. A. Majewski, “On non-completely positive quantum dynamical maps on spin chains,” J. Phys. A: Math. Theor. 40 (38), 11539–11545 (2007).
W. A. Majewski, “On positive decomposable maps,” Rep. Math. Phys. 59 (3), 289–298 (2007).
W. A. Majewski, “On the structure of positive maps: Finite-dimensional case,” J. Math. Phys. 53 (2), 023515 (2012).
W. A. Majewski and M. Marciniak, “On a characterization of positive maps,” J. Phys. A: Math. Gen. 34 (29), 5863–5874 (2001).
F. Mukhamedov, “On pure quasi-quantum quadratic operators of \(\mathbb M_2(\mathbb C)\). II,” Open Syst. Inf. Dyn. 22 (4), 1550024 (2015).
F. Mukhamedov, “On circle preserving quadratic operators,” Bull. Malays. Math. Sci. Soc. 40 (2), 765–782 (2017).
F. Mukhamedov and A. Abduganiev, “On the description of bistochastic Kadison–Schwarz operators on \(\mathbb M_2(\mathbb C)\),” Open Syst. Inf. Dyn. 17 (3), 245–253 (2010).
F. Mukhamedov and A. Abduganiev, “On Kadison–Schwarz type quantum quadratic operators on \(\mathbb M_2(\mathbb C)\),” Abstr. Appl. Anal. 2013, 278606 (2013).
F. Mukhamedov and A. Abduganiev, “On pure quasi-quantum quadratic operators of \(\mathbb M_2(\mathbb C)\),” Open Syst. Inf. Dyn. 20 (4), 1350018 (2013).
F. Mukhamedov, H. Akın, S. Temir, and A. Abduganiev, “On quantum quadratic operators of \(\mathbb M_2(\mathbb C)\) and their dynamics,” J. Math. Anal. Appl. 376 (2), 641–655 (2011).
F. Mukhamedov and N. Ganikhodjaev, Quantum Quadratic Operators and Processes (Springer, Cham, 2015), Lect. Notes Math. 2133.
F. Mukhamedov, S. M. Syam, and S. A. Y. Almazrouei, “Few remarks on quasi quantum quadratic operators on \(\mathbb M_2(\mathbb C)\),” Open Syst. Inf. Dyn. 27 (2), 2050006 (2020).
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).
M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer, Berlin, 1993).
M. Ohya and I. Volovich, Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems (Springer, New York, 2011).
A. N. Pechen and N. B. Il’in, “On extrema of the objective functional for short-time generation of single-qubit quantum gates,” Izv. Math. 80 (6), 1200–1212 (2016) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 80 (6), 217–229 (2016)].
A. N. Pechen and D. J. Tannor, “Are there traps in quantum control landscapes?,” Phys. Rev. Lett. 106 (12), 120402 (2011).
A. G. Robertson, “Schwarz inequalities and the decomposition of positive maps on \(C^*\)-algebras,” Math. Proc. Cambridge Philos. Soc. 94, 291–296 (1983).
M. B. Ruskai, S. Szarek, and E. Werner, “An analysis of completely-positive trace-preserving maps on \(\mathcal M_2\),” Linear Algebra Appl. 347 (1–3), 159–187 (2002).
Acknowledgments
We are grateful to an anonymous referee, whose useful suggestions and comments allowed us to improve the presentation of this paper.
Funding
The work is supported by the UAEU UPAR grant (grant code G00003447).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 313, pp. 178–191 https://doi.org/10.4213/tm4179.
Appendix. Diagonalizability
Let \(\Phi\) be a bistochastic KS operator on \(M_2({\mathbb C})\). Then it can be represented by (3.2). Following [19], let us decompose the matrix \(T\) as \(T=RS\), where \(R\) is a rotation and \(S\) is a self-adjoint matrix (see [19]). Define a mapping \(\Phi_S\) as follows:
On the other hand, every self-adjoint operator \(S\) can be diagonalized by some unitary operator; i.e., there is a unitary \(V\in M_2({\mathbb C})\) such that \(S=VD_{\lambda_1,\lambda_2,\lambda_3}V^*\), where
Consequently, the mapping \(\Phi\) can be represented as
Using Theorem 2.1, one can characterize KS operators of the form \(\Phi_{(\lambda_1,\lambda_2,\lambda_3)}\).
Theorem A.1 [8].
Let \(|\lambda_k|\leq 1,\) \(k=1,2,3\). If one has
Rights and permissions
About this article
Cite this article
Mukhamedov, F., Akın, H. Characterization of Bistochastic Kadison–Schwarz Operators on \(M_2(\mathbb C)\). Proc. Steklov Inst. Math. 313, 165–177 (2021). https://doi.org/10.1134/S0081543821020164
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543821020164