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Information-Theoretic Implications of Classical and Quantum Causal Structures Rafael Chaves QIP 2015 RC, L. Luft, T. Maciel, D. Gross, D. Janzing, B. Schlkopf (arXiv:1407.2256) RC, C. Majenz, D. Gross (arXiv:1407.3800) RC, C. Majenz &

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Information-Theoretic Implications of Classical and Quantum Causal Structures

Rafael Chaves QIP 2015

RC, L. Luft, T. Maciel, D. Gross, D. Janzing, B. Schölkopf (arXiv:1407.2256) RC, C. Majenz, D. Gross (arXiv:1407.3800)

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Dominik Janzing Bernhard Schölkopf

Thiago Maciel Christian Majenz

A joint work with

Lukas Luft David Gross

RC, C. Majenz & D. Gross, Nature Communications 6, 5766 (2015) RC, L. Luft, T. Maciel, D. Gross, D. Janzing, B. Schölkopf, Proceedings of Uncertainty in Artificial Intelligence 2014

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Given some empirically observable variables, which correlations between them are compatible with a presumed causal structure?

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  • Distinguishing direct influence from common cause… Is obesity contagious?
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  • Distinguishing direct influence from common cause… Is obesity contagious?
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  • Distinguishing direct influence from common cause… Is obesity contagious?
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  • Bell’s Theorem: Quantum correlations are incompatible with “local realism”.
  • Distinguishing direct influence from common cause… Is obesity contagious?
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  • Classical causal structures
  • The information-theoretic approach to classical causal inference
  • The generalization to quantum causal structures
  • Where to go from here?

Outline

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  • Classical causal structures
  • The information-theoretic approach to classical causal inference
  • The generalization to quantum causal structures
  • Where to go from here?
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  • For n variables X1, ... ,Xn, the causal relationships are encoded in a causal structure,

represented by a directed acyclic graph (DAG)

  • ith variable being a deterministic function

xi=fi(pai,ui)

  • f its parents pai and „local randomness“ ui

[See J. Pearl, Causality]

Cl Class assical ical Caus Causal S al Str truc uctur tures es

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  • For n variables X1, ... ,Xn, the causal relationships are encoded in a causal structure,

represented by a directed acyclic graph (DAG)

  • ith variable being a deterministic function

xi=fi(pai,ui)

  • f its parents pai and „local randomness“ ui
  • Causal relationships are encoded in the conditional

independencies (CIs) implied by the DAG [See J. Pearl, Causality]

Cl Class assical ical Caus Causal S al Str truc uctur tures es ...

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Is a given probability distribution compatible with a presumed causal structure?

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Is a given probability distribution compatible with a presumed causal structure?

Iff the given probability distribution fullfils all the CIs implied by the DAG

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Is a given probability distribution compatible with a presumed causal structure?

Example: Is a given compatible with

Iff the given probability distribution fullfils all the CIs implied by the DAG

?

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Is a given probability distribution compatible with a presumed causal structure?

Example: Is a given compatible with

Iff the given probability distribution fullfils all the CIs implied by the DAG

? ...

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Is a given probability distribution compatible with a presumed causal structure?

Example: Is a given compatible with

Iff the given probability distribution fullfils all the CIs implied by the DAG

... ?

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Is a given probability distribution compatible with a presumed causal structure?

Example: Is a given compatible with

  • If the full probability distribution (of all nodes in a DAG) is available, CIs hold

all information required to solve the compatibility problem

However…

Iff the given probability distribution fullfils all the CIs implied by the DAG

? ...

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Mar Margina ginal S l Sce cena narios rios

  • Usually and for a variety of reasons not all variables in a DAG are observable, i.e., not

all CIs are available from empirical data

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Mar Margina ginal S l Sce cena narios rios

  • Usually and for a variety of reasons not all variables in a DAG are observable, i.e., not

all CIs are available from empirical data

...

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Mar Margina ginal S l Sce cena narios rios

  • Usually and for a variety of reasons not all variables in a DAG are observable, i.e., not

all CIs are available from empirical data

...

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Mar Margina ginal S l Sce cena narios rios

  • CIs impose non-trivial constraints on the level of the
  • bservable variables, for example, Bell inequalities.

Pic from [Rev. Mod. Phys. 86, 419 (2014)]

  • Usually and for a variety of reasons not all variables in a DAG are observable, i.e., not

all CIs are available from empirical data

...

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Challeng Challenge

  • Describe marginals compatible with DAGs
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Challeng Challenge

  • Describe marginals compatible with DAGs
  • The observable probability contains the full causal information empirically available...
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Challeng Challenge

  • Describe marginals compatible with DAGs
  • The observable probability contains the full causal information empirically available...
  • ..very difficult, non-convex sets (algebraic geometry methods required, see for

instance [Geiger & Meek, UAI 1999])

Picture from [Steeg & Galstyan, UAI 2011]

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Challeng Challenge

  • Describe marginals compatible with DAGs
  • The observable probability contains the full causal information empirically available...
  • ..very difficult, non-convex sets (algebraic geometry methods required, see for

instance [Geiger & Meek, UAI 1999])

Picture from [Steeg & Galstyan, UAI 2011]

Our idea Rely on entropic information!

  • Concise characterization as a convex set
  • Naturally encodes the causal constraints
  • Quantitative and stable tool
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  • Causal structures
  • The information-theoretic approach to classical causal inference
  • The generalization to quantum causal structures
  • Where to go from here?

[T. Fritz and RC, IEEE Trans. Inf. Th. 59, 803 (2013)] [RC, L. Luft, D. Gross, NJP 16, 043001 (2014)] [RC, L. Luft, T. Maciel, D. Gross, D. Janzing, B. Schölkopf, UAI 2014]

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Step 1/3: Unconstrained, global object

Caus Causal E al Entr ntrop

  • pic

ic co cone ne

  • Entropic vector :each entry is the Shannon

entropy H(XS) indexed by subset Example: 2 vars →

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  • Defines a convex cone. Structure not fully understood, but...

Step 1/3: Unconstrained, global object

  • Entropic vector :each entry is the Shannon

entropy H(XS) indexed by subset Example: 2 vars →

Caus Causal E al Entr ntrop

  • pic

ic co cone ne

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  • Defines a convex cone. Structure not fully understood, but...

Step 1/3: Unconstrained, global object

  • Entropic vector :each entry is the Shannon

entropy H(XS) indexed by subset Example: 2 vars →

  • ...contained in Shannon Cone , defined by strong subadditivity and monotonicity

Caus Causal E al Entr ntrop

  • pic

ic co cone ne

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  • Piece of cake! Conditional independences are

naturally embedded in mutual informations

  • We can even relax (stable!)
  • C: set of constraints

Step 2/3: Choose candidate structure and add causal constraints

→

  • New global cone of entropies subject to causal structure

Caus Causal E al Entr ntrop

  • pic

ic co cone ne

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  • Geometrically trivial:

just restrict to observable coordinates

  • Algorithmically costly: represented in

terms of inequalities (use, eg, Fourier-Motzkin elimination) Step 3/3: Marginalize to

  • : set of joint observables

Final result: description of marginal, causal entropic cone in terms of „entropic Bell inequalities“

[T. Fritz and RC, IEEE Trans. Inf. Th. 59, 803 (2013)] [RC, L. Luft, D. Gross, NJP 16, 043001 (2014)] [RC, L. Luft, T. Maciel, D. Gross, D. Janzing, B. Schölkopf, UAI 2014]

Caus Causal E al Entr ntrop

  • pic

ic co cone ne

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App ppli lica cations tions

  • Entropic Bell inequalities [Braunstein & Caves PRL 61, 662 (1988)]
  • Common ancestors problem: Can the correlations between n observable variables be

explained by independent common ancestors connecting at most M of them? [Steudel

& Ay, arXiv:1010.5720]

  • Quantifying Causal Influences [D. Janzing et al, Ann. of Stat. 41, 2324 (2013)]
  • Witnessing direction of causation from pairwise information
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  • Causal structures
  • The information-theoretic approach to (classical) causal inference
  • The generalization to quantum causal structures
  • Where to go from here?

[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

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Quan Quantum C tum Cau ausal Str sal Struc uctur tures es

  • Different formulations have been proposed. For an incomplete list see:

[R. Tucci, arXiv:quant-ph/0701201 (2007)] [M. S. Leifer & R. W. Spekkens, Phys. Rev. A 88, 052130 (2013)] [T. Fritz, arXiv:1404.4812] [J.Henson, R. Lal, M. F. Pusey New J. Phys. 16, 113043 (2014)] [J. Pienaar & C. Brukner, arXiv:1406.0430]

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Quan Quantum C tum Cau ausal Str sal Struc uctur tures es

  • Different formulations have been proposed. For an incomplete list see:

[R. Tucci, arXiv:quant-ph/0701201 (2007)] [M. S. Leifer & R. W. Spekkens, Phys. Rev. A 88, 052130 (2013)] [T. Fritz, arXiv:1404.4812] [J.Henson, R. Lal, M. F. Pusey New J. Phys. 16, 113043 (2014)] [J. Pienaar & C. Brukner, arXiv:1406.0430]

  • We propose a graphical representation that allow us to generalize the information-

theoretic approach

  • Informally, a quantum causal structure specifies the functional dependency between a

collection of quantum states and classical variables.

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Building Blocks

  • Classical variable
  • Quantum State
  • Quantum Operation (CPTP map)

Quan Quantum C tum Cau ausal Str sal Struc uctur tures es

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[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

Entr Entrop

  • pic desc

ic description ription of

  • f Qua

Quant ntum Cau um Causal sal Str Struc uctu tures es

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  • Replace Shannon entropy with von Neumann entropy

(For purely classical variables both coincide)

[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

Entr Entrop

  • pic desc

ic description ription of

  • f Qua

Quant ntum Cau um Causal sal Str Struc uctu tures es

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  • Replace Shannon entropy with von Neumann entropy

(For purely classical variables both coincide)

  • von Neumann entropy respect strong subadditivity but

not monotonicity (replaced by weak monotonicity)

[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

Entr Entrop

  • pic desc

ic description ription of

  • f Qua

Quant ntum Cau um Causal sal Str Struc uctu tures es

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  • Measurements disturb/destroy the quantum system
  • Replace Shannon entropy with von Neumann entropy

(For purely classical variables both coincide)

  • von Neumann entropy respect strong subadditivity but

not monotonicity (replaced by weak monotonicity)

[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

Entr Entrop

  • pic desc

ic description ription of

  • f Qua

Quant ntum Cau um Causal sal Str Struc uctu tures es

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  • Measurements disturb/destroy the quantum system

Some CIs that are classically valid cannot be defined in the quantum case

  • Replace Shannon entropy with von Neumann entropy

(For purely classical variables both coincide)

  • von Neumann entropy respect strong subadditivity but

not monotonicity (replaced by weak monotonicity)

[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

Entr Entrop

  • pic desc

ic description ription of

  • f Qua

Quant ntum Cau um Causal sal Str Struc uctu tures es

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  • Measurements disturb/destroy the quantum system

Some CIs that are classically valid cannot be defined in the quantum case

  • Replace Shannon entropy with von Neumann entropy

(For purely classical variables both coincide)

  • von Neumann entropy respect strong subadditivity but

not monotonicity (replaced by weak monotonicity) Independecies still hold

[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

Entr Entrop

  • pic desc

ic description ription of

  • f Qua

Quant ntum Cau um Causal sal Str Struc uctu tures es

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  • Measurements disturb/destroy the quantum system
  • We need a rule mapping the quantum states to classical variables (data processing)

Some CIs that are classically valid cannot be defined in the quantum case

  • Replace Shannon entropy with von Neumann entropy

(For purely classical variables both coincide)

  • von Neumann entropy respect strong subadditivity but

not monotonicity (replaced by weak monotonicity) Independecies still hold

[RC, C. Majenz & D. Gross, Nat. Comm. 6, 5766 (2015)]

Entr Entrop

  • pic desc

ic description ription of

  • f Qua

Quant ntum Cau um Causal sal Str Struc uctu tures es

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Inf Infor

  • rma

mation tion ca causa usali lity ty

[Pawlowski et al, Nature 461, 1101 (2009)]

Task: Bob must output a guess YS about the s-th bit XS of Alice

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Inf Infor

  • rma

mation tion ca causa usali lity ty

[Pawlowski et al, Nature 461, 1101 (2009)]

Task: Bob must output a guess YS about the s-th bit XS of Alice IC inequality

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Inf Infor

  • rma

mation tion ca causa usali lity ty

[Pawlowski et al, Nature 461, 1101 (2009)]

Task: Bob must output a guess YS about the s-th bit XS of Alice

  • Implicitly restricting to the marginal scenario: {Xi, Yi}, {M}, with i=1,...,n

IC inequality

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Inf Infor

  • rma

mation tion ca causa usali lity ty

[Pawlowski et al, Nature 461, 1101 (2009)]

Task: Bob must output a guess YS about the s-th bit XS of Alice

  • Implicitly restricting to the marginal scenario: {Xi, Yi}, {M}, with i=1,...,n
  • Most general marginal scenario: {X1,...,Xn,Ys,M} with s=1,...,n

IC inequality

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Inf Infor

  • rma

mation tion ca causa usali lity ty

[Pawlowski et al, Nature 461, 1101 (2009)]

Task: Bob must output a guess YS about the s-th bit XS of Alice

  • Implicitly restricting to the marginal scenario: {Xi, Yi}, {M}, with i=1,...,n

Tigher IC inequality

  • Most general marginal scenario: {X1,...,Xn,Ys,M} with s=1,...,n

IC inequality

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Inf Infor

  • rma

mation tion ca causa usali lity ty

[Pawlowski et al, Nature 461, 1101 (2009)]

Task: Bob must output a guess YS about the s-th bit XS of Alice

  • Implicitly restricting to the marginal scenario: {Xi, Yi}, {M}, with i=1,...,n

Tigher IC inequality

  • Most general marginal scenario: {X1,...,Xn,Ys,M} with s=1,...,n

IC inequality

  • The new inequality witness the non quantumness of distributions that could not be

detected by the original one.

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Inf Infor

  • rma

mation tion ca causa usali lity ty

[Pawlowski et al, Nature 461, 1101 (2009)]

Task: Bob must output a guess YS about the s-th bit XS of Alice

  • Implicitly restricting to the marginal scenario: {Xi, Yi}, {M}, with i=1,...,n

Tigher IC inequality

  • Most general marginal scenario: {X1,...,Xn,Ys,M} with s=1,...,n

IC inequality

  • Can be easily generalized to the case of a quantum message (Dense Coding)
  • The new inequality witness the non quantumness of distributions that are not

detected by the original one.

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  • Causal structures
  • The information-theoretic approach to (classical) causal inference
  • The generalization to quantum causal structures
  • Where to go from here?
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  • Entropies allow for a non-trivial, quantitative and operational discrimination between

causal relationships...

What we know...

... both in classical and quantum problems

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  • Entropies allow for a non-trivial, quantitative and operational discrimination between

causal relationships...

...and what we would like to know

  • Beyond Bell‘s theorem? Nonlocality in quantum networks... see [T. Fritz, NJP 14,

103001 (2012)]

  • New information-theoretical principles? Multipartite Information Causality?

What we know...

... both in classical and quantum problems

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Thanks!

  • RC, C. Majenz & D. Gross, ”Information-Theoretic Implications of Quantum Causal Structures”,

Nature Communications 6, 5766 (2015)

  • RC, L. Luft, T. Maciel, D. Gross, D. Janzing, B. Schölkopf, “Inferring latent structures via information

inequalities“, Proceedings of Uncertainty in Artificial Intelligence (2014)