Quantum Causal Structures Christian Majenz University of Copenhagen - - PowerPoint PPT Presentation

Quantum Causal Structures

Christian Majenz

University of Copenhagen Joint work with Rafael Chaves and David Gross, University of Freiburg (arXiv:1407.3800)

04.12.2014

Motivation

Question

Question

Can we rule out certain causal relationships by just looking at statistical data?

Question

Can we rule out certain causal relationships by just looking at statistical data? Example:

Smoking Lung Cancer Gene

• bserved
• bserved

unobserved

Question

Can we rule out certain causal relationships by just looking at statistical data? Example:

Smoking Lung Cancer Gene

• bserved
• bserved

unobserved

Answer: Causal Inference

◮ Rigorous mathematical way to distinguish different causal

relations

Answer: Causal Inference

◮ Rigorous mathematical way to distinguish different causal

relations

◮ Introduced by Pearl in the 1980s

Answer: Causal Inference

◮ Rigorous mathematical way to distinguish different causal

relations

◮ Introduced by Pearl in the 1980s ◮ Necessary conditions for probability distributions to come from

more restrictive causal model

Answer: Causal Inference

◮ Rigorous mathematical way to distinguish different causal

relations

◮ Introduced by Pearl in the 1980s ◮ Necessary conditions for probability distributions to come from

more restrictive causal model

◮ Tools: Bayesian networks, entropies, convexity

Answer: Causal Inference

◮ Rigorous mathematical way to distinguish different causal

relations

◮ Introduced by Pearl in the 1980s ◮ Necessary conditions for probability distributions to come from

more restrictive causal model

◮ Tools: Bayesian networks, entropies, convexity

[?] What if the underlying processes are quantum?

Answer: Causal Inference

◮ Rigorous mathematical way to distinguish different causal

relations

◮ Introduced by Pearl in the 1980s ◮ Necessary conditions for probability distributions to come from

more restrictive causal model

◮ Tools: Bayesian networks, entropies, convexity

[?] What if the underlying processes are quantum?

◮ Bell nonlocality etc. special cases of this

Structure

Motivation Classical Bayesian Networks Entropic Description Quantum Quantum Causal Structure Entropic Description Application Information Causality Computational Techniques

Classical

DAGs

◮ Any directed acyclic graph (DAG) specifies a causal structure:

1 2 3 4 5 6

DAGs

◮ Any directed acyclic graph (DAG) specifies a causal structure:

1 2 3 4 5 6

◮ Directed graph: G = (V , E) with E ⊂ V × V

DAGs

◮ Any directed acyclic graph (DAG) specifies a causal structure:

1 2 3 4 5 6

◮ Directed graph: G = (V , E) with E ⊂ V × V ◮ Acyclic: no cycle (causality)

DAGs

◮ Any directed acyclic graph (DAG) specifies a causal structure:

1 2 3 4 5 6

◮ Directed graph: G = (V , E) with E ⊂ V × V ◮ Acyclic: no cycle (causality) ◮ parents of v: pa(v) = {w ∈ V |(w, v) ∈ E}

DAGs

◮ Any directed acyclic graph (DAG) specifies a causal structure:

1 2 3 4 5 6

◮ Directed graph: G = (V , E) with E ⊂ V × V ◮ Acyclic: no cycle (causality) ◮ parents of v: pa(v) = {w ∈ V |(w, v) ∈ E} ◮ children, ancestors, descendants, non-descendants etc.

Bayesian networks

Definition (Bayesian network)

Let G = (V , E) be a DAG, X = (Xv)v∈V a collection of random variables indexed by V with probability distribution p on X V . Then p is called Markov w.r.t. G if p(x1, ..., xn) =

• i∈V

p

• xi|xpa(i)
• ,

assuming WLOG V = {1, ..., n}. In that case G together with X is called a Bayesian network.

Conditional independence

◮ equivalent characterization: v⊥nd(v)|pa(v)

Conditional independence

◮ equivalent characterization: v⊥nd(v)|pa(v) ◮ example:

X → Y → Z = ⇒ Z⊥X|Y ⇔ p(x, y, z)p(y) = p(x, y)p(y, z)

Conditional independence

◮ equivalent characterization: v⊥nd(v)|pa(v) ◮ example:

X → Y → Z = ⇒ Z⊥X|Y ⇔ p(x, y, z)p(y) = p(x, y)p(y, z)

◮ formalism requires access to joint probability distribution

Conditional independence

◮ equivalent characterization: v⊥nd(v)|pa(v) ◮ example:

X → Y → Z = ⇒ Z⊥X|Y ⇔ p(x, y, z)p(y) = p(x, y)p(y, z)

◮ formalism requires access to joint probability distribution

= ⇒ obstacle for quantum generalization

Conditional independence

◮ equivalent characterization: v⊥nd(v)|pa(v) ◮ example:

X → Y → Z = ⇒ Z⊥X|Y ⇔ p(x, y, z)p(y) = p(x, y)p(y, z)

◮ formalism requires access to joint probability distribution

= ⇒ obstacle for quantum generalization

◮ hard algebraic equations

Entropies

Entropies

X random variable (RV) on a finite alphabet X probability distribution p

Entropies

X random variable (RV) on a finite alphabet X probability distribution p Shannon entropy: H(X) = −

• x∈X

p(x) log p(x)

Entropies

X random variable (RV) on a finite alphabet X probability distribution p Shannon entropy: H(X) = −

• x∈X

p(x) log p(x) ρ ∈ S(H) mixed state on a finite-dimensional Hilbert space H = Cd

Entropies

X random variable (RV) on a finite alphabet X probability distribution p Shannon entropy: H(X) = −

• x∈X

p(x) log p(x) ρ ∈ S(H) mixed state on a finite-dimensional Hilbert space H = Cd Von Neumann entropy S(ρ) = −trρ log ρ

Multiparticle Entropies

Classical:

◮ Collection of RV’s X = (X1, ..., Xn)

Multiparticle Entropies

Classical:

◮ Collection of RV’s X = (X1, ..., Xn)

XI, I ⊂ {1, ..., n}

Multiparticle Entropies

Classical:

◮ Collection of RV’s X = (X1, ..., Xn)

XI, I ⊂ {1, ..., n}

◮ Shannon entropy vector: h(X) = (H(XI))I⊂{1,...,n} ∈ R2n

Multiparticle Entropies

Classical:

◮ Collection of RV’s X = (X1, ..., Xn)

XI, I ⊂ {1, ..., n}

◮ Shannon entropy vector: h(X) = (H(XI))I⊂{1,...,n} ∈ R2n

Quantum:

◮ Multipartite state ρ ∈ S(H) with H = H1 ⊗ ... ⊗ Hn

Multiparticle Entropies

Classical:

◮ Collection of RV’s X = (X1, ..., Xn)

XI, I ⊂ {1, ..., n}

◮ Shannon entropy vector: h(X) = (H(XI))I⊂{1,...,n} ∈ R2n

Quantum:

◮ Multipartite state ρ ∈ S(H) with H = H1 ⊗ ... ⊗ Hn

ρI = trIc ρ

Multiparticle Entropies

Classical:

◮ Collection of RV’s X = (X1, ..., Xn)

XI, I ⊂ {1, ..., n}

◮ Shannon entropy vector: h(X) = (H(XI))I⊂{1,...,n} ∈ R2n

Quantum:

◮ Multipartite state ρ ∈ S(H) with H = H1 ⊗ ... ⊗ Hn

ρI = trIc ρ

◮ Von Neumann entropy vector: s(ρ) = (S(ρI))I⊂{1,...,n} ∈ R2n

Convex cones

Convex cones

What is a convex cone?

Convex cones

What is a convex cone?

Convex cones

What is a convex cone?

Entropy Cones

Sets of all entropy vectors:

Entropy Cones

Sets of all entropy vectors: Classical Σn = {h(X)|X n-party random variable}

Entropy Cones

Sets of all entropy vectors: Classical Σn = {h(X)|X n-party random variable} Quantum Γn = {s(ρ)|ρ n-party quantum state}

Entropy Cones

Sets of all entropy vectors: Classical Σn = {h(X)|X n-party random variable} Quantum Γn = {s(ρ)|ρ n-party quantum state} Σn as well as Γn are convex cones (Zhang, Yeung, 1997 and Pippenger, 2003)

Information Inequalities

Describe convex cone via linear inequalities

Information Inequalities

Describe convex cone via linear inequalities

Information Inequalities

Describe convex cone via linear inequalities Inequalities for Σn: information inequalities

Information Inequalities

Describe convex cone via linear inequalities Inequalities for Σn: information inequalities Example: monotonicity H(X1X2) − H(X2) ≥ 0

Information Inequalities

Describe convex cone via linear inequalities Inequalities for Σn: information inequalities Example: monotonicity H(X1X2) − H(X2) ≥ 0 Inequalities for Γn: quantum information inequalities

Information Inequalities

Describe convex cone via linear inequalities Inequalities for Σn: information inequalities Example: monotonicity H(X1X2) − H(X2) ≥ 0 Inequalities for Γn: quantum information inequalities Example: weak monotonicity S(ρ12) + S(ρ13) − S(ρ2) − S(ρ3) ≥ 0

Information Inequalities

Describe convex cone via linear inequalities Inequalities for Σn: information inequalities Example: monotonicity H(X1X2) − H(X2) ≥ 0 Inequalities for Γn: quantum information inequalities Example: weak monotonicity S(ρ12) + S(ρ13) − S(ρ2) − S(ρ3) ≥ 0 known (quantum) information inequalities provide outer approximation of entropy cones

Entropy and Bayesian networks

◮ Consider entropy cone of RVs from Bayesian network

Entropy and Bayesian networks

◮ Consider entropy cone of RVs from Bayesian network ◮ Conditonal independence: Z⊥X|Y ⇔ I(X : Z|Y ) = 0

Entropy and Bayesian networks

◮ Consider entropy cone of RVs from Bayesian network ◮ Conditonal independence: Z⊥X|Y ⇔ I(X : Z|Y ) = 0 ◮ Conditional Mutual information

I(X : Z|Y ) = H(XY ) + H(YZ) − H(XZY ) − H(Y )

Entropy and Bayesian networks

◮ Consider entropy cone of RVs from Bayesian network ◮ Conditonal independence: Z⊥X|Y ⇔ I(X : Z|Y ) = 0 ◮ Conditional Mutual information

I(X : Z|Y ) = H(XY ) + H(YZ) − H(XZY ) − H(Y )

◮ Linear equation

Entropy and Bayesian networks

◮ Consider entropy cone of RVs from Bayesian network ◮ Conditonal independence: Z⊥X|Y ⇔ I(X : Z|Y ) = 0 ◮ Conditional Mutual information

I(X : Z|Y ) = H(XY ) + H(YZ) − H(XZY ) − H(Y )

◮ Linear equation ◮ intersection with entropy cone is convex cone

Marginal scenario I

◮ Only some RVs are observed

Marginal scenario I

◮ Only some RVs are observed ◮ Example:

Marginal scenario I

◮ Only some RVs are observed ◮ Example:

→ Marginal scenario: {A, B, C}

Marginal scenario I

◮ Only some RVs are observed ◮ Example:

→ Marginal scenario: {A, B, C}

◮ Plan: Remove unobserved variables from inequality

description of entropy cone

Marginal scenario II

Marginal scenario II

Definition (Marginal Scenario)

Let n ∈ N. A subset M ⊂ 2{1,...,n} such that for I ∈ M and J ⊂ I also J ∈ M is called marginal scenario.

Marginal scenario II

Definition (Marginal Scenario)

Let n ∈ N. A subset M ⊂ 2{1,...,n} such that for I ∈ M and J ⊂ I also J ∈ M is called marginal scenario. Projection of entropy cone to a marginal scenario = ⇒ observable entropy cone.

Quantum

Quantum Causal Structures I

◮ DAG G = (V , E)

Quantum Causal Structures I

◮ DAG G = (V , E) ◮ Sink nodes s ∈ V : Hilbert space Hs

Quantum Causal Structures I

◮ DAG G = (V , E) ◮ Sink nodes s ∈ V : Hilbert space Hs ◮ Nodes v ∈ V with children: Hilbert space

Hv =

• w∈V

(v,w)∈E

Hv,w

Quantum Causal Structures I

◮ DAG G = (V , E) ◮ Sink nodes s ∈ V : Hilbert space Hs ◮ Nodes v ∈ V with children: Hilbert space

Hv =

• w∈V

(v,w)∈E

Hv,w

◮

H =

• v∈V

Hv

Quantum Causal Structures I

◮ DAG G = (V , E) ◮ Sink nodes s ∈ V : Hilbert space Hs ◮ Nodes v ∈ V with children: Hilbert space

Hv =

• w∈V

(v,w)∈E

Hv,w

◮

H =

• v∈V

Hv

◮ Parent Hilbert space

Hpa(v) =

• w∈V

(w,v)∈E

Hw,v

Quantum Causal Structures II

◮ Initial product state ρ0 = q ρq on Hilbert spaces of source

nodes q

Quantum Causal Structures II

◮ Initial product state ρ0 = q ρq on Hilbert spaces of source

nodes q

◮ CPTP maps Φv for each non-source node:

Φv : L

• Hpa(v)
• → L(Hv)

Quantum Causal Structures II

◮ Initial product state ρ0 = q ρq on Hilbert spaces of source

nodes q

◮ CPTP maps Φv for each non-source node:

Φv : L

• Hpa(v)
• → L(Hv)

◮

ρv = Φvρpa(v)

Quantum Causal Structures II

◮ Initial product state ρ0 = q ρq on Hilbert spaces of source

nodes q

◮ CPTP maps Φv for each non-source node:

Φv : L

• Hpa(v)
• → L(Hv)

◮

ρv = Φvρpa(v) [!] no global state

Quantum Causal Structures II

◮ Initial product state ρ0 = q ρq on Hilbert spaces of source

nodes q

◮ CPTP maps Φv for each non-source node:

Φv : L

• Hpa(v)
• → L(Hv)

◮

ρv = Φvρpa(v) [!] no global state

◮ want classical nodes: pick the right Φv

Example

2 3 1

• bserved
• bserved

unobserved

◮ H = H1,2 ⊗ H1,3 ⊗ H2 ⊗ H3

Example

2 3 1

• bserved
• bserved

unobserved

◮ H = H1,2 ⊗ H1,3 ⊗ H2 ⊗ H3 ◮ States on the coexisting subsets of systems:

ρ0 = ρ(1,2),(1,3) ρ(1,3),2 = (Φ2 ⊗ 1) ρ(1,2),(1,3) ρ(1,2),3 = (1 ⊗ Φ3) ρ(1,2),(1,3) ρ2,3 = (Φ2 ⊗ Φ3) ρ(1,2),(1,3)

The entropy cone

◮ Look at entropy cone of states constructed like this

The entropy cone

◮ Look at entropy cone of states constructed like this ◮ H is a tensor product of n = |E| + |Vs| hilbert spaces, Vs: Set

• f sinks

The entropy cone

◮ Look at entropy cone of states constructed like this ◮ H is a tensor product of n = |E| + |Vs| hilbert spaces, Vs: Set

• f sinks

◮ Formally: Entropy vector v ∈ R2n

The entropy cone

◮ Look at entropy cone of states constructed like this ◮ H is a tensor product of n = |E| + |Vs| hilbert spaces, Vs: Set

• f sinks

◮ Formally: Entropy vector v ∈ R2n ◮ vI = S(ρI) if the state ρI exists

The entropy cone

◮ Look at entropy cone of states constructed like this ◮ H is a tensor product of n = |E| + |Vs| hilbert spaces, Vs: Set

• f sinks

◮ Formally: Entropy vector v ∈ R2n ◮ vI = S(ρI) if the state ρI exists ◮ vI arbitrary if ρI doesn’t exist

The entropy cone

◮ Look at entropy cone of states constructed like this ◮ H is a tensor product of n = |E| + |Vs| hilbert spaces, Vs: Set

• f sinks

◮ Formally: Entropy vector v ∈ R2n ◮ vI = S(ρI) if the state ρI exists ◮ vI arbitrary if ρI doesn’t exist ◮ for each J ⊂ E ∪ Vs of coexisting systems: quantum entropy

cone

The entropy cone

◮ Look at entropy cone of states constructed like this ◮ H is a tensor product of n = |E| + |Vs| hilbert spaces, Vs: Set

• f sinks

◮ Formally: Entropy vector v ∈ R2n ◮ vI = S(ρI) if the state ρI exists ◮ vI arbitrary if ρI doesn’t exist ◮ for each J ⊂ E ∪ Vs of coexisting systems: quantum entropy

cone

◮ extra monotonicities for classical systems

Data processing inequality

◮ replace conditional independence relations?

Data processing inequality

◮ replace conditional independence relations?

→ Data processing inequality: For a CPTP map Φ : HA → HB I(A : C|D) ≥ I(B : C|D) for any other systems C and D.

Data processing inequality

◮ replace conditional independence relations?

→ Data processing inequality: For a CPTP map Φ : HA → HB I(A : C|D) ≥ I(B : C|D) for any other systems C and D.

◮ relates entropies of noncoexisting systems

Data processing inequality

◮ replace conditional independence relations?

→ Data processing inequality: For a CPTP map Φ : HA → HB I(A : C|D) ≥ I(B : C|D) for any other systems C and D.

◮ relates entropies of noncoexisting systems ◮ replaces conditional independence

Marginal Scenario

◮ Again, only some systems are observed

Marginal Scenario

◮ Again, only some systems are observed ◮ Example:

Marginal Scenario

◮ Again, only some systems are observed ◮ Example:

Marginal Scenario

◮ Again, only some systems are observed ◮ Example:

Marginal Scenario

◮ Again, only some systems are observed ◮ Example: ◮ most interesting marginal scenario: {A, B, C} & assume these

nodes classical

Application

Information Causality

The IC principle is defined by a game:

Information Causality

The IC principle is defined by a game:

Information Causality

The IC principle is defined by a game:

Information Causality

The IC principle is defined by a game:

ΡAB

Information Causality

The IC principle is defined by a game:

X1,X2

ΡAB

Information Causality

The IC principle is defined by a game:

M

X1,X2

ΡAB

Information Causality

The IC principle is defined by a game:

M

X1,X2

i

ΡAB

Information Causality

The IC principle is defined by a game:

M

X1,X2

i

ΡAB Yi

Information Causality

The IC principle is defined by a game:

M

X1,X2

i

ΡAB Yi

Information Causality

◮ Corresponding DAG: X1 S X2 Y M AB

Information Causality

◮ Corresponding DAG: X1 S X2 Y M AB ◮ Nodes Xi, Y , M, S are classical, only the AB node is quantum

Information Causality

◮ Corresponding DAG: X1 S X2 Y M AB ◮ Nodes Xi, Y , M, S are classical, only the AB node is quantum ◮ Counterfactual reformulation: Yi = (Y |S = i)

Information Causality

◮ Marginal scenario {X1, Y1}, {X2, Y2}, {M} yields original

information causality inequality I(X1 : Y1) + I(X2 : Y2) ≤ H(M)

Information Causality

◮ Marginal scenario {X1, Y1}, {X2, Y2}, {M} yields original

information causality inequality I(X1 : Y1) + I(X2 : Y2) ≤ H(M) → More general marginal scenario {X1, X2, Y1, M}, {X1, X2, Y2, M}:

Information Causality

◮ Marginal scenario {X1, Y1}, {X2, Y2}, {M} yields original

information causality inequality I(X1 : Y1) + I(X2 : Y2) ≤ H(M) → More general marginal scenario {X1, X2, Y1, M}, {X1, X2, Y2, M}:

X1 S X2 Y M AB

Information Causality

◮ Marginal scenario {X1, Y1}, {X2, Y2}, {M} yields original

information causality inequality I(X1 : Y1) + I(X2 : Y2) ≤ H(M) → More general marginal scenario {X1, X2, Y1, M}, {X1, X2, Y2, M}: Strengthened information causality inequality I(X1 : Y1, M)+I(X2 : Y2, M)+I(X1 : X2|M) ≤ H(M)+I(X1 : X2)

Information Causality

Triangle scenario

1

Triangle scenario

◮ Classical: e.g.

I(A : B) + I(A : C) ≤ H(A)

1

Triangle scenario

◮ Classical: e.g.

I(A : B) + I(A : C) ≤ H(A)

◮ New framework =

⇒ also holds for quantum

1

Triangle scenario

◮ Classical: e.g.

I(A : B) + I(A : C) ≤ H(A)

◮ New framework =

⇒ also holds for quantum

◮ independently proven by Henson et al., 2014 for non-signalling

resources1

1Henson, Joe, Raymond Lal, and Matthew F. Pusey. ”Theory-independent

limits on correlations from generalised Bayesian networks.” arXiv preprint arXiv:1405.2572 (2014).

Triangle scenario

◮ Classical: e.g.

I(A : B) + I(A : C) ≤ H(A)

◮ New framework =

⇒ also holds for quantum

◮ independently proven by Henson et al., 2014 for non-signalling

resources1

◮ using their techniques =

⇒ causes connecting at most m of n nodes yield entropic inequality

1Henson, Joe, Raymond Lal, and Matthew F. Pusey. ”Theory-independent

limits on correlations from generalised Bayesian networks.” arXiv preprint arXiv:1405.2572 (2014).

Computational Techniques

Fourier-Motzkin Elimination

◮ Fancy name for removing variables from inequalities

Fourier-Motzkin Elimination

◮ Fancy name for removing variables from inequalities ◮ Example: x ≤ 2y, z ≤ x + y, y ≤ x, remove y

Fourier-Motzkin Elimination

◮ Fancy name for removing variables from inequalities ◮ Example: x ≤ 2y, z ≤ x + y, y ≤ x, remove y

= ⇒

1 2x ≤ x, z − x ≤ x

Fourier-Motzkin Elimination

◮ Fancy name for removing variables from inequalities ◮ Example: x ≤ 2y, z ≤ x + y, y ≤ x, remove y

= ⇒

1 2x ≤ x, z − x ≤ x ◮ Problem: double exponential in number of variables...

Fourier-Motzkin Elimination

◮ Fancy name for removing variables from inequalities ◮ Example: x ≤ 2y, z ≤ x + y, y ≤ x, remove y

= ⇒

1 2x ≤ x, z − x ≤ x ◮ Problem: double exponential in number of variables...

... i.e. triple exponential in number of nodes

Fourier-Motzkin Elimination

◮ Fancy name for removing variables from inequalities ◮ Example: x ≤ 2y, z ≤ x + y, y ≤ x, remove y

= ⇒

1 2x ≤ x, z − x ≤ x ◮ Problem: double exponential in number of variables...

... i.e. triple exponential in number of nodes

◮ works for small instances (triangle, IC)

Linear Programming

◮ Easier: check candidate inequality

Linear Programming

◮ Easier: check candidate inequality ◮ formulation as LP:

minimize

• I⊂{1,...,n}

αIvI subject to SSA, WM, DPs

Linear Programming

◮ Easier: check candidate inequality ◮ formulation as LP:

minimize

• I⊂{1,...,n}

αIvI subject to SSA, WM, DPs

◮ minimum is 0 if the inequality holds

Linear Programming

◮ Easier: check candidate inequality ◮ formulation as LP:

minimize

• I⊂{1,...,n}

αIvI subject to SSA, WM, DPs

◮ minimum is 0 if the inequality holds ◮ −∞ else

Linear Programming

◮ Easier: check candidate inequality ◮ formulation as LP:

minimize

• I⊂{1,...,n}

αIvI subject to SSA, WM, DPs

◮ minimum is 0 if the inequality holds ◮ −∞ else

[!] still exponential in # of nodes

Summary

◮ Defined quantum causal structures

Summary

◮ Defined quantum causal structures ◮ Algorithm for characterization

Summary

◮ Defined quantum causal structures ◮ Algorithm for characterization ◮ Applications: strengthening of information causality, quantum

networks

Thank you for your attention!