Non-commutative disintegrations and regular conditional probabilities Arthur J. Parzygnat ∗ & Benjamin P. Russo † ∗ University of Connecticut † Farmingdale State College SUNY Category Theory 2019 Edinburgh, Scotland The University of Edinburgh July 9, 2019 Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 1 / 30
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 Applications and Examples Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 2 / 30
� � � � � � � � � 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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 3 / 30
� � � � � � � � � 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 The Kleisli category associated to a monad is one way to distinguish between two such kinds of morphisms. Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 3 / 30
Deterministic and nondeterministic processes Goal for non-commutative regular conditional probabilities Our goal will be to formulate concepts in probability theory categorically. This will enable us to abstract these concepts to contexts beyond their initial domain. We will focus our attention on quantum probability. categorical probability theory standard probability theory non-commutative probability theory Arthur J. Parzygnat ∗ & Benjamin P. Russo † ( ∗ University of Connecticut † Farmingdale State College SUNY Category Theory 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 4 / 30
Stochastic matrices Standard definitions Stochastic maps: “if y then x ” probabilistic statements � 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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 5 / 30
Stochastic matrices Standard definitions Stochastic maps: “if y then x ” probabilistic statements � 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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 5 / 30
Stochastic matrices Standard definitions Stochastic maps: “if y then x ” probabilistic statements � 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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 5 / 30
Stochastic matrices Standard definitions Stochastic maps from functions: “if x then y ” statements � 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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 6 / 30
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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 7 / 30
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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 7 / 30
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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 8 / 30
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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 9 / 30
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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 10 / 30
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 2019 Edinburgh, Non-commutative disintegrations and regular conditional probabilities July 9, 2019 11 / 30
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