Evolution of negative probability distributions Pawel Kurzynski and - - PowerPoint PPT Presentation

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Evolution of negative probability distributions Pawel Kurzynski and - - PowerPoint PPT Presentation

Jun 01, 2023 422 likes •620 views

Evolution of negative probability distributions Pawel Kurzynski and Marcin Karczewski Why negative probabilities? NP emerge in QT when we try to use classical description Wigner function Contextuality QT NP ? Can QT emerge from

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SLIDE 1

Pawel Kurzynski and Marcin Karczewski Evolution of negative probability distributions

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SLIDE 2

Why negative probabilities?

  • NP emerge in QT when we try to use classical description

– Wigner function – Contextuality

  • Can QT emerge from NP theories?

– Complementarity (Feynman, ...) – No-signaling (Abramsky and Brandenburger)

  • To answer the above we need to keep in mind that QT describes not only

states, but also their dynamics – QT from dynamical constrains?

QT NP ?

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SLIDE 3

Piponi's example

http://blog.sigfpe.com/2008/04/negative-probabilities.html A B prob 1/2 1 1/2 1 1/2 1 1

  • 1/2

p(A=0) = 1/2 + 1/2 = 1 p(A=1) = 1/2 - 1/2 = 0 p(B=0) = 1/2 + 1/2 = 1 p(A=0) = 1/2 + 1/2 = 1 p(B=1) = 1/2 - 1/2 = 0 p(A=B) = p(C=0) = 1/2 - 1/2 = 0 p(A≠B) = p(C=1) = 1/2 + 1/2 = 1

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SLIDE 4

General probability distributions

a b c d p = A0 = ( 1 1 0 0 ) A1 = ( 0 0 1 1 ) B0 = ( 1 0 1 0 ) B1 = ( 0 1 0 1 ) C0 = ( 1 0 0 1 ) C1 = ( 0 1 1 0 ) Id = ( 1 1 1 1 ) This set is tomographically complete p(A=0) = A0 p = a + b . All measurable probabilities need to be non-negative:

  • Sum of any two entries must be non-negative
  • At most single entry can be negative
  • Say, a ≥ b ≥ c ≥ d
  • Only d can be negative and c ≥ |d|
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SLIDE 5

Extreme points

1/2 1/2 1/2

  • 1/2

p001 = 1/2 1/2

  • 1/2

1/2 p010 = 1/2

  • 1/2

1/2 1/2 p100 =

  • 1/2

1/2 1/2 1/2 p111 = 1 p110 = 1 p101 = 1 p011 = 1 p000 = Negative probabilities allow to somehow store 3 bits in 2 – Random Access Codes Any allowed state can be represented as a convex combination of the above eight

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SLIDE 6
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SLIDE 7

Random walk with Piponi's coins

1/2 1/2 1/2

  • 1/2
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SLIDE 8

Random walk with Piponi's coins

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SLIDE 9

Random walk with Piponi's coins

Position X Position Y X-Y

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SLIDE 10

Standard transformations

  • Reversible (permutations)
  • Irreversible:

– Stochastic matrices – Bi-stochastic matrices

(uniform stationary state, do not decrease entropy)

1 1 1 1 1/3 1/2 1/2 1/3 1/3 1/3 1/2 1/3 1/3 1/2 1/3 1/3 1/3 1/3 1/6 1/2 2/3 1/6 1/6 1/3 1/3 1/3 1/2 1/2

  • 1/2

1/2 = 1/2

  • 1/2

1/2 1/2

  • 1/12
  • 1/12

7/12 7/12 = 1/2

  • 1/2

1/2 1/2 1/6 1/6 1/2 1/6 = 1/2

  • 1/2

1/2 1/2

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SLIDE 11

Quasi-stochastic transformations

  • Quasi-stochastic matrix – negative but columns sum to 1
  • Which transformations change proper states into proper states?
  • Enough to consider transformation of extreme points

p000 p111 p001 p100 p110 p010 p011 p101

  • D. Chruscinski et. al. Phys. Scr. 90, 115202 (2015)
  • 1/2

1/2 1/2 1/2 1/2

  • 1/2

1/2 1/2 1/2 1/2

  • 1/2

1/2 1/2 1/2 1/2

  • 1/2
  • Columns need to be proper states
  • Sum of any three columns minus the

fourth one needs to be a proper state

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SLIDE 12

Quantum subset?

Model qubit (spin 1/2) using Piponi's system (similar to Spekkens model)

a b c d p = X = A0 – A1 = a + b – c – d Y = B0 – B1 = a + c – b – d Z = A0 – A1 = a + d – b – c 1/2 1/2 +x = 1/2 1/2 +y = 1/2 1/2 +z = 1/2 1/2

  • x =

1/2 1/2

  • y =

1/2 1/2

  • z =

p000 p111 p001 p100 p110 p010 p011 p101

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SLIDE 13

Is quantum set negative? - Yes

p000 p111 p001 p100 p110 p010 p011 p101 1/4 1/4 1/4 1/4 + (1-p) 1/2

  • 1/2

1/2 1/2 p 0.394

  • 0.183

0.394 0.394 = 1/4 1/4 1/4 1/4 + (1-p) 1 p 0.105 0.683 0.105 0.105 =

  • Transformations of the quantum subset are quasi-stochastic
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SLIDE 14

p000

Some example transformations

  • 1/2

1/2 1/2 1/2 1/2

  • 1/2

1/2 1/2 1/2 1/2

  • 1/2

1/2 1/2 1/2 1/2

  • 1/2

1/2 1/2 1/2

  • 1/2
  • 1/2

1/2 1/2 1/2 1/2

  • 1/2

1/2 1/2 1/2 1/2

  • 1/2
  • 1/2

p111 p001 p100 p110 p010 p011 p101 Pi/2 rotation about Y p000 p111 p001 p100 p110 p010 p011 p101 Universal NOT

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SLIDE 15

General rotations – are they allowed?

p000 p111 p001 p100 p110 p010 p011 p101 p000 p111 p001 p100 p110 p010 p011 p101

  • Either we limit the set of available transformations or we limit

the set of allowed states

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SLIDE 16

Composite systems

a b c d p = a' b' c' d' x

But in general there can be 16-dim states that are not convex sums of such products

Q = ( 0 , 1/2, 0, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2 )^T

PR – box:

<CHSH> = < X X' > + < X Y' > + < X' Y > - < Y Y' > = 4

QT from dynamical constrains: How alowed local transformations affect the above state?

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SLIDE 17

Permutations Pi x Pj

p Q + (1-p) C <CHSH> = 4 p Q = ( 0 , 1/2, 0, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2 )^T C = 1/16 ( 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 )^T

  • Do observable probabilities become negative?
  • Does negativity vanish for some value of p > ½?
  • Negativity is observable even for p < 1/2
  • Many distributions give rise to PR-box

Q = {0.1875, 0.1875, -0.0625, -0.0625, 0.1875, -0.0625, 0.1875, -0.0625,

  • 0.0625, 0.1875, -0.0625, 0.1875, -0.0625, -0.0625, 0.1875, 0.1875}^T
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SLIDE 18

Global permutations - CNOT

  • At the moment we know that CNOT leads to

measurable negativities, even for states in the „Quantum” regime

  • ...