Quantum Suprematism with Triadas ofMalevich's Squares identifying spin

(qubit) states; new entropic inequalities

Margarita A. Man'ko1, Vladimir I. Man'ko1,2

1 � 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, 2018

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 identi�ed with

three probability distributions 1 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.

1V.I. Man'ko, G. Marmo, F. Ventriglia, and Vitale, J. Phys A: Math Theor., 50,335302 (2017)

2V.N.Chernega, O.V.Man'ko, and V.I.Man'ko, J. Russ. Laser Res. 38, 141-149,234-333, 416-425 (2017)

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 3 4.

3A. Ibort, V.I. Man'ko, G. Marmo, A. Simoni, and F. Ventriglia, "An introductionto the tomographic picture of quantum mechanics." Phys. Scr. 79, 065013 (2009).

4M. Asorey, A. Ibort, G. Marmo, and F Ventriglia, "Quantum Tomography twentyyears later." Phys. Scr. 90, 074031 (2015).

Quantum suprematism andMalevich squares5

5A. Shatskikh, Black Square: Malevich and the Origin of Suprematism. YaleUniversity Press, New Haven (2012).

Probability distribution for three classical coins

Triada of Malevich's squaresdetermined by the triangle A1A2A3

6

y1 = (2+ 2p21 − 4p1 − 2p2 + 2p22 + 2p1p2)1/2

y2 = (2+ 2p22 − 4p2 − 2p3 + 2p23 + 2p2p3)1/2

y3 = (2+ 2p13 − 4p3 − 2p1 + 2p21 + 2p3p1)1/2

6V. N. Chernega, O. V. Man'ko, and V. I. Man'ko J. Russ. Laser Res. 38, 141(2017)

Density matrix for spin-1/2 state

Smin ≤ S ≤ Smax

S = 3/2 � for maximally mixed state

Triangle area in terms ofprobabilities p1, p2, p3

yk = (2+ 2p2k − 4pk − 2pk+1 + 2p2k+1 + 2pkpk+1)1/2

Str =√

3/2 for classical case

Str =√

3/8 for maximally mixed case

Qutrit as two ququarts

ρ =

ρ11 ρ12 ρ13ρ21 ρ22 ρ23ρ31 ρ32 ρ33

ρ(1) =

ρ11 ρ12 ρ13 0

ρ21 ρ22 ρ23 0

ρ31 ρ32 ρ33 0

0 0 0 0

, ρ(2) =

0 0 0 0

0 ρ11 ρ12 ρ130 ρ21 ρ22 ρ230 ρ31 ρ32 ρ33

R(1) =

(ρ11 + ρ22 ρ13

ρ31 ρ33

), R(2) =

(ρ11 + ρ33 ρ12

ρ21 ρ22

),

R(3) =

(ρ11 ρ13ρ31 ρ33 + ρ22

), R(4) =

(ρ22 ρ23ρ32 ρ11 + ρ33

).

R(k) =

(p(k)3 (p

(k)1 −

12 )− i(p

(k)2 −

12 )

(p(k)1 −

12 ) + i(p

(k)2 −

12 ) 1− p

(k)3

),

k = 1, 2, 3, 4

ρ33 = 1− p(1)3 ,

ρ22 = 1− p(2)3 = p

(4)3 ,

ρ11 = p(1)3 + p

(2)3 − 1,

ρ21 = (p(2)1 −

1

2) + i(p

(2)2 −

1

2),

ρ31 = (p(1)1 −

1

2) + i(p

(1)2 −

1

2),

ρ32 = (p(4)1 −

1

2) + i(p

(4)2 −

1

2).

Coin probabilities for qutrit

p(1)1 = p

(3)1 , p

(1)2 = p

(3)2 , p

(4)3 = 1− p

(2)3 , p

(1)3 = p

(3)3 + p

(4)3

ρ =

p(1)3

+ p(2)3− 1 (p

(2)1− 1

2)− i(p

(2)2− 1

2) (p

(1)1− 1

2)− i(p

(1)2− 1

2)

(p(2)1− 1

2) + i(p

(2)2− 1

2) 1− p

(2)3

(p(4)1− 1

2)− i(p

(4)2− 1

2)

(p(1)1− 1

2) + i(p

(1)2− 1

2) (p

(4)1− 1

2) + i(p

(4)2− 1

2) 1− p

(1)3

Probabilities ρ

(k)j , j = 1, 2, 3, k = 1, 2, 3, 4 satisfy the inequalities

∑3

j=1

(p(k)j − 1

2

)2≤ 1

4, k = 1, 2, 3, 4.

p(1)3 = p

(33)3 , p

(2)3 = p

(22)3 , p

(1)1 = p

(31)1 , p

(1)2 = p

(31)2 ,

p(2)1 = p

(21)1 , p

(2)2 = p

(21)2 , p

(4)1 = p

(32)1 , p

(4)2 = p

(32)2 .

ρjk = (p(jk)1 − 1

2) + i(p

(jk)2 − 1

2), j > k

ρjj = 1− p(j j)3 , j ≥ 2,

ρ11 = 1−3

∑j=2

ρjj .

Density matrix of qutrit in terms ofcoin probabilities

ρ =

(p(33)3

+ p(22)3− 1 (p

(21)1− 1

2)− i(p

(21)2− 1

2) (p

(31)1− 1

2)− i(p

(31)2− 1

2)

(p(21)1− 1

2) + i(p

(21)2− 1

2) 1− p

(22)3

(p(32)1− 1

2)− i(p

(32)2− 1

2)

(p(31)1− 1

2) + i(p

(31)2− 1

2) (p

(32)1− 1

2) + i(p

(32)2− 1

2) 1− p

(33)3

)

ρ(1) =

(ρ 0

0 0

), ρ(2) =

(0 0

0 ρ

).

Density matrix of ququart in termsof coin probabilities

ρ =

p(44)3

+ p(22)3

+ p(33)3

− 2 (p(21)1

− 1

2)− i(p

(21)2

− 1

2) (p

(31)1

− 1

2)− i(p

(31)2

− 1

2) (p

(41)1

− 1

2)− i(p

(41)2

− 1

2)

(p(21)1

− 1

2) + i(p

(21)2

− 1

2) 1− p

(22)3

(p(32)1

− 1

2)− i(p

(32)2

− 1

2) (p

(42)1

− 1

2)− i(p

(42)2

− 1

2)

(p(31)1

− 1

2) + i(p

(31)2

− 1

2) (p

(32)1

− 1

2) + i(p

(32)2

− 1

2) 1− p

(33)3

(p(43)1

− 1

2)− i(p

(43)2

− 1

2)

(p(41)1

− 1

2) + i(p

(41)2

− 1

2) (p

(42)1

− 1

2) + i(p

(42)2

− 1

2) (p

(43)1

− 1

2) + i(p

(43)2

− 1

2) 1− p

(44)3

New entropic inequalities

Re ρjk +1

2≥ 0, Im ρjk ≤

1

2

(1

2− Im ρjk

)ln

[ (1

2− Im ρjk

)(1

2− Im ρj ′k ′

) ]+(1

2+ Im ρjk

)ln

[ (1

2+ Im ρjk

)(1

2+ Im ρj ′k ′

) ] ≥ 0

ρjj ln

[ρjj(

1

2∓ Im ρj ′k

) ]+ (1− ρjj ) ln

[(1− ρjj )(1

2± Im ρj ′k

) ] ≥ 0.

ln 2 ≥ −(1

2∓ Imρjk

)ln

(1

2∓ Imρjk

)−(1

2± Imρjk

)ln

(1

2± Imρjk

)≥ 0

Quantum observables as set of classical random variables

H =

(H11 H12

H21 H22

), H† = H

H11 = z1, H22 = z2

H12 = x − iy , H21 = x + iy

Dichotomic random variables andcoin probabilities

~X =

(x−x

), ~Y =

(y−y

), ~Z =

(z1z2

)

~P1 =

(p1

1− p1

), ~P2 =

(p2

1− p2

), ~P3 =

(p3

1− p3

)

Dichotomic random variables and coin probabilities in quantum

mechanics

H =

(z1 x − iy

x + iy z2

)

ρ =

(p3 p1 − 1

2 − i(p212 )

p1 − 12 + i(p2

12 ) 1− p3

)

Quanum means in terms of classical means

〈H〉 = Tr(Hρ) =

= p1x +(1− p1)(−x)+ p2y +(1− p2)(−y)+ p3z1+(1− p3)z2 =

= 〈~X 〉+ 〈~Y 〉+ 〈~Z 〉

Superposition principle of two states

|ψ1〉 =( √

p3p1− 1

2√p3

+i(p2− 1

2)√

p3

)

|ψ2〉 =( √

P3P1− 1

2√P3

+i(P2− 1

2)√

P3

)

〈ψ1| ψ2〉 = 0

State vector of superposed state

|ψ〉 =√

Π3 |ψ1〉+√

1−Π3eiα |ψ2〉 =

( √π3

π1− 1

2√π3

+i(π2− 1

2)√

π3

)

cos(α) =Π1 − 1

2√Π3(1−Π3)

, sin(α) =Π2 − 1

2√Π3(1−Π3)

Classical coins �interference�

p1p2p3

⊕~Π

P1P2P3

=

π1

π2

π3

Born rule:

Trρ1ρ2 = 2+ 2[p3P3 + p1P1 + p2P2]− p1 −P1 − p2 −P2 − p3 −P3

π3 =1

T

{Π3p3 + (1−Π3)P3 + 2

√p3P3 (Π1 − 1/2)

}π1 − 1/2 =

1

T {Π3(p1 − 1/2) + (P1 − 1/2)(1−Π3)+

+ [(Π1 − 1/2)(p1 − 1/2) + (Π2 − 1/2)(p2 − 1/2)]

√P3p3

+

+ [(Π1 − 1/2)(P1 − 1/2)− (Π2 − 1/2)(P2 − 1/2)]√

p3P3

}π2 − 1/2 =

1

T {[(p2 − 1/2)Π3 + (P2 − 1/2)(1−Π3)] +

+

√P3p3

[(Π1 − 1/2)(p2 − 1/2)− (Π2 − 1/2)(p1 − 1/2)] +√p3P3

[(Π2 − 1/2)(P1 − 1/2) + (Π1 − 1/2)(P2 − 1/2)]}

T = 1+2√p3P3

{(Π1 − 1/2) [(p1 − 1/2)(P1 − 1/2) + (P2 − 1/2)(p2 − 1/2) + p3P3] +

+ (Π2 − 1/2) [(p2 − 1/2)(P1 − 1/2)− (p1 − 1/2)(P2 − 1/2)]}

Conclusion

1. States (density matrices) are interpreted as set of probability distributionsdescribing positions of coins.2. Observables � matrix elements, e.g. of Hamiltonians can be interpreted asclassical dichotomic random variables corresponding to playing coins.3. Quantum statistics � means of quantum observables, other moments can beexpressed in terms of the coin probabilities and dichotomic random variables.4. Superposition principle for quantum states is formulated as nonlinearaddition rule of coin probability distributions.5. Born rule also is formulated as another addition rule of coin probabilitieswhich provides the transition probability between the states.6.The qubit states can be mapped on triada of Malevich's squares and anyqudit states can be mapped onto set of such triadas.We illustrate obtained relations by the statement connected with the discussionof Bohr and Einstein "God does not play dice � God plays coins"

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