Noncommutative disintegration Arthur J. Parzygnat ∗ & Benjamin P. Russo † ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberFest 2018 The City College of New York (CUNY) October 28, 2018 Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 1 / 29
Outline Deterministic and nondeterministic processes 1 Stochastic matrices 2 Standard definitions The category of stochastic maps Classical disintegrations 3 Classical disintegrations: intuition Diagrammatic disintegrations Classical disintegrations exist and are unique a.e. Quantum disintegrations 4 Completely positive maps and ∗ -homomorphisms Non-commutative disintegrations Existence and uniqueness Examples Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 2 / 29
� � � � � � � � � Deterministic and nondeterministic processes Category theory as a theory of processes Processes can be deterministic or non-deterministic a f b d i c j e g k h Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 3 / 29
Stochastic matrices Standard definitions Stochastic maps � X assigns a Let X and Y be finite sets. A stochastic map r : Y probability measure on X to every point in Y . It is a function whose value at a point “spreads out” over the codomain. Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 4 / 29
Stochastic matrices Standard definitions Stochastic maps � X assigns a Let X and Y be finite sets. A stochastic map r : Y probability measure on X to every point in Y . It is a function whose value at a point “spreads out” over the codomain. X Y r y • y Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 4 / 29
Stochastic matrices Standard definitions Stochastic maps � X assigns a Let X and Y be finite sets. A stochastic map r : Y probability measure on X to every point in Y . It is a function whose value at a point “spreads out” over the codomain. X Y r y • y The value r y ( x ) of r y at x is denoted by r xy . Since r y is a probability measure, r xy ≥ 0 for all x and y . Also, � x ∈ X r xy = 1 for all y . Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 4 / 29
Stochastic matrices Standard definitions Stochastic maps from functions � Y via A function f : X → Y induces a stochastic map f : X f yx := δ yf ( x ) X Y f x • • x f ( x ) where δ yy ′ is the Kronecker delta and equals 1 if and only if y = y ′ and is zero otherwise. Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 5 / 29
Stochastic matrices Standard definitions Composing stochastic maps � Z of µ : X � Y followed by ν : Y � Z The composition ν ◦ µ : X is defined by matrix multiplication � ( ν ◦ µ ) zx := ν zy µ yx . y ∈ Y Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 6 / 29
Stochastic matrices Standard definitions Composing stochastic maps � Z of µ : X � Y followed by ν : Y � Z The composition ν ◦ µ : X is defined by matrix multiplication � ( ν ◦ µ ) zx := ν zy µ yx . y ∈ Y This is completely intuitive! If we start at x and end at z , we have the possibility of passing through any intermediate step y . These “paths” have associated probabilities, which must be added. Z Y X y • • • • • • x • • z • • Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 6 / 29
Stochastic matrices Standard definitions Composing stochastic maps � Z of µ : X � Y followed by ν : Y � Z The composition ν ◦ µ : X is defined by matrix multiplication � ( ν ◦ µ ) zx := ν zy µ yx . y ∈ Y This is completely intuitive! If we start at x and end at z , we have the possibility of passing through any intermediate step y . These “paths” have associated probabilities, which must be added. Z Y X y • µ yx • • ν zy • • • x • • z • • Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 7 / 29
Stochastic matrices Standard definitions Composing stochastic maps � Z of µ : X � Y followed by ν : Y � Z The composition ν ◦ µ : X is defined by matrix multiplication � ( ν ◦ µ ) zx := ν zy µ yx . y ∈ Y This is completely intuitive! If we start at x and end at z , we have the possibility of passing through any intermediate step y . These “paths” have associated probabilities, which must be added. Z Y X y • µ yx • • ν zy • • • x • • z • • Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 8 / 29
Stochastic matrices Standard definitions Composing stochastic maps � Z of µ : X � Y followed by ν : Y � Z The composition ν ◦ µ : X is defined by matrix multiplication � ( ν ◦ µ ) zx := ν zy µ yx . y ∈ Y This is completely intuitive! If we start at x and end at z , we have the possibility of passing through any intermediate step y . These “paths” have associated probabilities, which must be added. Z Y X y • µ yx • • ν zy • • • x • • z • • Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 9 / 29
Stochastic matrices Standard definitions Composing stochastic maps � Z of µ : X � Y followed by ν : Y � Z The composition ν ◦ µ : X is defined by matrix multiplication � ( ν ◦ µ ) zx := ν zy µ yx . y ∈ Y This is completely intuitive! If we start at x and end at z , we have the possibility of passing through any intermediate step y . These “paths” have associated probabilities, which must be added. Z Y X y • µ yx • • ν zy • • • x • • z • • Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 10 / 29
Stochastic matrices Standard definitions Composing stochastic maps � Z of µ : X � Y followed by ν : Y � Z The composition ν ◦ µ : X is defined by matrix multiplication � ( ν ◦ µ ) zx := ν zy µ yx . y ∈ Y This is completely intuitive! If we start at x and end at z , we have the possibility of passing through any intermediate step y . These “paths” have associated probabilities, which must be added. Z Y X y • µ yx • • ν zy • • • x • • z • • Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 11 / 29
Stochastic matrices Standard definitions Special case: probability measures A probability measure µ on X can be viewed as a stochastic map � X from a single element set. µ : {•} Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 12 / 29
Stochastic matrices Standard definitions Special case: probability measures A probability measure µ on X can be viewed as a stochastic map � X from a single element set. µ : {•} � Y is the If f : X → Y is a function, the composition f ◦ µ : {•} pushforward of µ along f . Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory OctoberF Noncommutative disintegration October 28, 2018 12 / 29
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