Quantum Suprematism with Triadas of Malevich's Squares identifying spin (qubit) states; new entropic inequalities

Transcription

1 Quantum Suprematism with Triadas of Malevich's Squares identifying spin qubit states; new entropic inequalities Margarita A. Man'ko, Vladimir I. Man'ko,2 Lebedev Physical Institute 2 Moscow Institute of Physics and Technology 60 Years Alberto Ibort Fest Classical and Quantum Physics: Geometry, Dynamics and Control Madrid, 5-9 March, 208

2 Abstract For arbitrary N-level atom states, the density matrix elements are expressed in terms of a set of probability distributions describing the set of "classical coin" positions. For one qubit, the states are identied with three probability distributions and illustrated by three squares on the plane called "Triada of Malevich's Squares" 2. Using this approach, called the quantum suprematism picture, new entropic inequalities are obtained for density matrix elements of qudits and N-level atom states. Arbitrary quantum observables are bijectively mapped on the set of classical random variables, and formulas for quantum statistics of the observables are expressed in terms of classical-like statistics of random variables. V.I. Man'ko, G. Marmo, F. Ventriglia, and Vitale, J. Phys A: Math Theor., 50, V.N.Chernega, O.V.Man'ko, and V.I.Man'ko, J. Russ. Laser Res. 8, 4-49, 24-,

3 The aim of the talk is to discuss the possibility to describe the quantum states by the fair probability distributions and quantum observables by classiacal-like random variables. This aim is coherent with tomographic description of quantum states 4. A. Ibort, V.I. Man'ko, G. Marmo, A. Simoni, and F. Ventriglia, "An introduction to the tomographic picture of quantum mechanics." Phys. Scr. 79, M. Asorey, A. Ibort, G. Marmo, and F Ventriglia, "Quantum Tomography twenty years later." Phys. Scr. 90,

4 Quantum suprematism and Malevich squares 5 5 A. Shatskikh, Black Square: Malevich and the Origin of Suprematism. Yale University Press, New Haven 202.

5 Probability distribution for three classical coins

6 Triada of Malevich's squares determined by the triangle A A 2 A 6 y = 2 + 2p 2 4p 2p 2 + 2p p p 2 /2 y 2 = 2 + 2p 2 2 4p 2 2p + 2p 2 + 2p 2p /2 y = 2 + 2p 4p 2p + 2p 2 + 2p p /2 6 V. N. Chernega, O. V. Man'ko, and V. I. Man'ko J. Russ. Laser Res. 8, 4 207

7 Density matrix for spin-/2 state S min S S max S = /2 for maximally mixed state

8 Triangle area in terms of probabilities p, p 2, p y k = 2 + 2p 2 k 4p k 2p k+ + 2p 2 k+ + 2p kp k+ /2 S tr = /2 for classical case S tr = /8 for maximally mixed case

9 Qutrit as two ququarts ρ = ρ = ρ ρ 2 ρ 0 ρ 2 ρ 22 ρ 2 0 ρ ρ 2 ρ ρ ρ 2 ρ ρ 2 ρ 22 ρ 2 ρ ρ 2 ρ, ρ2 = ρ ρ 2 ρ 0 ρ 2 ρ 22 ρ 2 0 ρ ρ 2 ρ

10 ρ + ρ R = 22 ρ ρ ρ ρ + ρ, R2 = ρ 2 ρ 2 ρ 22, ρ ρ R = ρ ρ + ρ 22 ρ22 ρ, R4 = 2 ρ 2 ρ + ρ. Rk = p k p k 2 ipk 2 2 p k 2 + ipk 2 2 pk, k =, 2,, 4

11 ρ = p, ρ 22 = p 2 = p 4, ρ = p + p 2, ρ 2 = p ip2 2 2, ρ = p 2 + ip 2 2, ρ 2 = p ip4 2 2.

12 Coin probabilities for qutrit p = p, p 2 = p 2, p4 = p 2, p = p + p 4 ρ = p + p 2 p 2 2 ip2 2 2 p 2 ip 2 2 p ip2 2 2 p2 p 4 2 ip4 2 2 p 2 + ip 2 2 p4 2 + ip4 2 2 p Probabilities ρ k j, j =, 2,, k =, 2,, 4 satisfy the inequalities j= p k 2 j 2 4, k =, 2,, 4.

13 p = p, p 2 = p 22, p = p, p 2 = p 2, p 2 = p 2, p 2 2 = p 2 2, p 4 = p 2, p 4 2 = p 2 2. ρ jk = p jk 2 + ipjk 2 2, j > k j j ρ jj = p, j 2, ρ = j=2 ρ jj.

14 Density matrix of qutrit in terms of coin probabilities ρ = p + p 22 p 2 2 ip2 2 2 p 2 ip 2 2 p ip2 2 2 p22 p 2 2 ip2 2 2 p 2 + ip 2 2 p2 2 + ip2 2 2 p ρ = ρ 0 0 0, ρ2 = ρ.

15 Density matrix of ququart in terms of coin probabilities p 44 + p 22 + p 2 p 2 2 ip2 2 2 ρ = p ip2 2 2 p22 p 2 + ip 2 2 p2 2 + ip2 2 2 p ip4 2 2 p ip p 2 ip 2 2 p 2 2 ip2 2 2 p p4 2 + ip4 2 2 p4 2 ip4 2 2 p42 2 ip p 4 2 ip4 2 2 p44

16 New entropic inequalities Re ρ jk + 2 0, Im ρ jk 2 [ 2 Im ρ 2 jk ln Im ρ ] [ jk 2 Im ρ + j k 2 + Im ρ 2 jk ln + Im ρ ] jk 2 + Im ρ 0 j k [ ] [ ] ρ jj ρ jj ρ jj ln 2 Im ρ + ρ jj ln j k 2 ± Im ρ 0. j k ln 2 2 Imρ jk ln 2 Imρ jk 2 ± Imρ jk ln 2 ± Imρ jk 0

17 Quantum observables as set of classical random variables H H H = 2 H 2 H 22, H = H H = z, H 22 = z 2 H 2 = x iy, H 2 = x + iy

18 Dichotomic random variables and coin probabilities x X = x, Y = y y, Z = z z 2 P = p p p2, P 2 = p 2 p, P = p

19 Dichotomic random variables and coin probabilities in quantum mechanics ρ = z x iy H = x + iy z 2 p p 2 ip 2 2 p 2 + ip 2 2 p

20 Quanum means in terms of classical means H = TrHρ = = p x + p x + p 2 y + p 2 y + p z + p z 2 = = X + Y + Z

21 Superposition principle of two states ψ = ψ 2 = p p 2 p + ip 2 2 p P P 2 P + ip 2 2 P ψ ψ 2 = 0

22 State vector of superposed state ψ = Π ψ + Π e iα ψ 2 = π π 2 π + iπ 2 2 π cosα = Π 2 Π Π, sinα = Π 2 2 Π Π

23 Classical coins interference Born rule: p P p 2 P 2 = π π 2 Π p P π Trρ ρ 2 = 2 + 2[p P + p P + p 2 P 2 ] p P p 2 P 2 p P

24 π = T { Π p + Π P + } 2 p P Π /2 π /2 = T {Π p /2 + P /2 Π + + [Π /2p /2 + Π 2 /2p 2 /2] P + p p + [Π /2P /2 Π 2 /2P 2 /2] P π 2 /2 = T {[p 2 /2Π + P 2 /2 Π ] + P + [Π /2p 2 /2 Π 2 /2p /2] + p } p [Π 2 /2P /2 + Π /2P 2 /2] P T = + 2 p P {Π /2 [p /2P /2 + P 2 /2p 2 /2 + p P ] + + Π 2 /2 [p 2 /2P /2 p /2P 2 /2]} }

25 Conclusion. States density matrices are interpreted as set of probability distributions describing positions of coins. 2. Observables matrix elements, e.g. of Hamiltonians can be interpreted as classical dichotomic random variables corresponding to playing coins.. Quantum statistics means of quantum observables, other moments can be expressed in terms of the coin probabilities and dichotomic random variables. 4. Superposition principle for quantum states is formulated as nonlinear addition rule of coin probability distributions. 5. Born rule also is formulated as another addition rule of coin probabilities which provides the transition probability between the states. 6.The qubit states can be mapped on triada of Malevich's squares and any qudit states can be mapped onto set of such triadas. We illustrate obtained relations by the statement connected with the discussion of Bohr and Einstein "God does not play dice God plays coins"

26 Happy Birthday!