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

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 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?

Piponi's example p(A=0) = 1/2 + 1/2 = 1 p(A=0) = 1/2 + 1/2 = 1 A B prob p(A=1) = 1/2 - 1/2 = 0 0 0 1/2 p(B=0) = 1/2 + 1/2 = 1 0 1 1/2 p(B=1) = 1/2 - 1/2 = 0 1 0 1/2 p(A=B) = p(C=0) = 1/2 - 1/2 = 0 p(A≠B) = p(C=1) = 1/2 + 1/2 = 1 1 1 -1/2 http://blog.sigfpe.com/2008/04/negative-probabilities.html

General probability distributions A0 = ( 1 1 0 0 ) a A1 = ( 0 0 1 1 ) b . p = p(A=0) = A0 p = a + b c B0 = ( 1 0 1 0 ) d B1 = ( 0 1 0 1 ) C0 = ( 1 0 0 1 ) All measurable probabilities need to be non-negative: C1 = ( 0 1 1 0 ) ● Sum of any two entries must be non-negative Id = ( 1 1 1 1 ) ● At most single entry can be negative ● Say, a ≥ b ≥ c ≥ d This set is tomographically complete ● Only d can be negative and c ≥ |d|

Extreme points 1/2 1/2 1/2 -1/2 1/2 1/2 -1/2 1/2 p001 = p010 = p100 = p111 = 1/2 -1/2 1/2 1/2 -1/2 1/2 1/2 1/2 0 0 0 1 0 0 1 0 p110 = p101 = p011 = p000 = 0 1 0 0 1 0 0 0 Any allowed state can be represented as a convex combination of the above eight Negative probabilities allow to somehow store 3 bits in 2 – Random Access Codes

Random walk with Piponi's coins -1/2 1/2 1/2 1/2

Random walk with Piponi's coins

Random walk with Piponi's coins Position X Position Y X-Y

Standard transformations ● Reversible (permutations) 1 0 0 0 1/2 1/2 0 0 1 0 -1/2 1/2 = 0 1 0 0 1/2 -1/2 0 0 0 1 1/2 1/2 ● Irreversible: 1/3 1/2 0 0 1/2 -1/12 – Stochastic matrices 0 1/2 1/3 0 -1/2 -1/12 = 1/3 0 1/3 1/2 1/2 7/12 1/3 0 1/3 1/2 1/2 7/12 1/3 1/3 1/3 0 1/2 1/6 – Bi-stochastic matrices 0 1/3 1/6 1/2 -1/2 1/6 = (uniform stationary state, 2/3 0 1/6 1/6 1/2 1/2 0 1/3 1/3 1/3 1/2 1/6 do not decrease entropy)

Quasi-stochastic transformations D. Chruscinski et. al. Phys. Scr. 90, 115202 (2015) ● Quasi-stochastic matrix – negative but columns sum to 1 ● Which transformations change proper states into proper states? ● Enough to consider transformation of extreme points ● Columns need to be proper states p001 p011 ● Sum of any three columns minus the p101 p111 fourth one needs to be a proper state -1/2 1/2 1/2 1/2 p000 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 p110

Quantum subset? Model qubit (spin 1/2) using Piponi's system (similar to Spekkens model) a X = A0 – A1 = a + b – c – d b Y = B0 – B1 = a + c – b – d p = c p001 Z = A0 – A1 = a + d – b – c p011 d p101 p111 1/2 1/2 1/2 1/2 0 0 +x = +y = +z = 0 1/2 0 p000 0 0 1/2 p010 p100 0 0 0 0 1/2 1/2 p110 -x = -y = -z = 1/2 0 1/2 1/2 1/2 0

Is quantum set negative? - Yes 1/2 1/4 0.394 -1/2 1/4 -0.183 p001 p + (1-p) = p011 1/2 1/4 0.394 p101 1/2 1/4 0.394 p111 0 1/4 0.105 p000 1 1/4 0.683 p010 p + (1-p) = 0 1/4 0.105 p100 0 1/4 0.105 p110 ● Transformations of the quantum subset are quasi-stochastic

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

General rotations – are they allowed? p001 p001 p011 p011 p101 p101 p111 p111 p000 p000 p010 p010 p100 p100 p110 p110 ● Either we limit the set of available transformations or we limit the set of allowed states

Composite systems a a' But in general there can be 16-dim b b' states that are not convex sums of p = x c c' such products d d' PR – box: Q = ( 0 , 1/2, 0, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2 )^T <CHSH> = < X X' > + < X Y' > + < X' Y > - < Y Y' > = 4 QT from dynamical constrains: How alowed local transformations affect the above state?

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

Global permutations - CNOT ● At the moment we know that CNOT leads to measurable negativities, even for states in the „Quantum” regime ● ...