quantum logic and structure - joshua sack · 2 there is a q 2l such that there are at least two...

Quantum Logic and Structure

Joshua Sack

Department of Mathematics and StatisticsCSU Long Beach

Chapman University, 2015 November 4

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Two perspectives on quantum logic

Express testable properties

Testable properties may be ranges of values of a particle’sposition, momentum, or velocity.

Organize testable properties as a Hilbert lattice: the lattice ofclosed-linear subspaces of a Hilbert spaces

Express quantum dynamics

Dynamics resulting from two kinds of actions: unitaryevolutions and quantum tests

Model the effects of actions using a labelled transition system

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Original Quantum Logic: focus on testable properties

The original quantum logic helps us reason about relationshipsamong testable properties:

1 p, q denote testable properties (closed linear subspaces)

2 p ∧ q the intersection of two closed subspaces

3 p ∨ q the closure of the span of two closed subspaces

4 ¬p the orthocomplement of a closed subspace

The testable properties form a lattice with a complement.

p ≤ q p is at least as strong a property as q

Founding paper on quantum logic:

Birkhoff and von Neumann. The Logic of Quantum Mechanics. TheAnnals of Mathematics, 1936.

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Basic results

Dunn, Hagge, Moss, Wang (2005)

Quantum logic over finite dimensional Hilbert spaces is decidable

Decision procedures rely on computations involving real numbers.

Hermann & Ziegler (2011)

1 The 1 and 2 dimensional satisfiability problems areNP-complete(same as classical Boolean logic)

2 The n dimensional satisfiability problem is complete forNon-deterministic Blum-Shub-Smale computation for n ≥ 3(random access to registers that contain real values)

How well can we reason about Hilbert lattices without involvingreal numbers?

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An essential property of a Hilbert lattice

Hilbert lattices are bounded and satisfy

1 p⊥⊥ = p;

2 p ≤ q implies q⊥ ≤ p⊥;

3 p ∧ p⊥ = O and p ∨ p⊥ = I .

A lattice with these properties is called an ortholattice, and itslogic orthologic.

The negation of orthologic is “classical” as opposed to intuitionistic

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Distributivity and Weak Modularity

Hilbert lattices are not distributive: consider x , y , z non-equalone-dimensional subspaces of R2.

1 x ∨ (y ∧ z) 6= (x ∨ y) ∧ (x ∨ z)

2 x ∧ (y ∨ z) 6= (x ∧ y) ∨ (x ∧ z).

But Hilbert lattices do satisfy weak modularity (akaorthomodularity):

q ≤ p ⇒ p[q] = q,

where p[q] := p ∧ (p⊥ ∨ q).

Weak modularity adds some distributivity to the lattice.

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Projection and lattice dynamics

Projection is an action arising from a quantum test.

Sasaki Projection p[q] := p ∧ (p⊥ ∨ q)The result of projecting q onto p.

Weak modularity ensures that p[−] is idempotent.

Sasaki Hook [p]q := p⊥ ∨ (p ∧ q)The precondition of a projection onto p resulting in q.

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Additional significance of Weak Modularity

Proposition (Coecke and Smets 2004)

In an ortholattice L = (L,≤,−⊥) satisfies weak modularity if andonly if p[−] is a left adjoint of [p]−, that is

p[−] a [p]−

In the context of Hilbert lattices, this is the equivalence of

1 The projection of x is contained in y

2 x is contained the precondition of projecting onto y .

Coecke and Smets. The Sasaki hook is not a [static] implicativeconnective but induces a backward [in time] dynamic one thatassigns causes. International Journal of Theoretical Physics, 2004.

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Completeness and Atomicity

A Hilbert lattice is complete:(Closed linear subspaces are closed under arbitrary intersections)

A Hilbert lattice is atomic:(Every positive-dimensional subspace contains a one-dimensionalsubspace)

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Projections of atoms

Covering Law

If a is an atom and a ∧ p = O, then a ∨ p covers p.

In an orthomodular lattice, this is equivalent to

If a is at atom and a 6≤ p⊥, then p[a] is an atom.

One dimensional subspaces project onto one-dimensionalsubspaces.

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Propositional System

A propositional system is an orthoattice with all the propertiesmentioned(weak modularity, completeness, atomicity, covering law)

Hilbert geometries are projective geometries with an orthogonalityoperator

A categorical equivalence has been established betweenpropositional systems and Hilbert geometries.

Stubbe and van Steirteghem. Propositional systems, Hilbertlattices, and generalized Hilbert spaces. Handbook of QuantumLogic and Quantum Structures, 2007.

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Superposition Principle

Superposition Principle

For distinct atoms a and b, there is an atom c distinct from theothers, such that a ∨ c = b ∨ c = a ∨ b.

a ∨ b is a two-dimensional space. Each atom is spanned by theother two.

Proposition

For a propositional system, the following are equivalent:

the Superposition Principle holds;

The lattice is irreducible.

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Piron lattice

A Piron lattice is a propositional system satisfying theSuperposition Principle.

A generalized Hilbert space (aka orthomodular vector space)extends the notion of Hilbert space to modules over division rings.

A Piron lattice with at least 4 mutually orthogonal atoms can berealized by a the bi-orthogonally closed subspaces of a generalizedHilbert space.

Piron. Foundations of Quantum Physics. W.A. Benjamin Inc, 1976.

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Ortholattice characterization of Hilbert spaces

A Piron lattice L = (L,≤,−⊥) is said to satisfy Mayet’s conditionif there is an automorphism k : L→ L such that

1 there is a p ∈ L such that k(p) < p, and

2 there is a q ∈ L such that there are at least two distinct atomsbelow q and k(r) = r for all r ≤ q.

A Piron lattices satisfying Mayet’s condition characterize thelattices of closed subspaces of infinite-dimensional Hilbert spacesover the reals, complex numbers, or quaternions.

Mayet. Some Characterizations of the Underlying Division Ring of aHilbert Lattice by Automorphisms. International Journal ofTheoretical Physics, 1998.

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Lattice theoretic characterization of Hilbert space

Summary

Infinite dimensional Hilbert spaces over complex numbers, realnumbers, or quaternions can be characterized lattice theoreticallyby the properties given earlier.

Finite dimensional Hilbert spaces are harder to characterize, sincethere are more possible division rings underlying finite-dimensionalorthomodular vector spaces.

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Labelled transition systems and dynamics

A basic labelled transition system consists of

1 A set S of states

2 A set A of actions

3 Relationsa−→⊆ S × S for each a ∈ A

sa−→ t means that action a can transform state s into state t.

Logics, such as Hennesey Milner logic and propositional dynamiclogic, that are interpreted on labelled transition systems are havebeen widely used for reasoning about classical computation.

For every action a there is a model operator [a] in the language s.t.for any formula φ

s |= [a]φ⇔ t |= φ whenever sa−→ t

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Quantum computation

1 States are one dimensional subspaces(atoms of the Hilbert lattice)

2 Actions include

Projections onto testable propertiesUnitary operations

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Probabilistic Logic for Quantum Actions

We use the labelled transition system setting to reason aboutquantum computation.

Probabilistic Quantum Dynamic Logic

A decidable probabilistic quantum dynamic logic that can expressquantum algorithms, such as Grover’s Search algorithm.

Baltag, Bergfeld, Kishida, Sack, Smets, Zhong. PLQP & Company:Decidable Logics for Quantum Algorithms. International Journal ofTheoretical Physics, 2014.

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Probabilistic Language for Quantum Actions

Let

N be a fixed finite subset of the natural numbers.

U be a set of symbols for unitary operators.

The language is two-sorted (where I ⊆ N, r ∈ Q, u ∈ U):

ϕ ::= p | ϕ ∧ ϕ | ¬ϕ | [π]ϕ | KIϕ | P≥rϕπ ::= u | ϕ? | π;π | π ∪ π

Basic propositional language

Propositional dynamic logic language

Probabilistic Language for Quantum Actions

KIϕ - a “knowledge” operator capturing properties local to aquantum subsystem

P≥rϕ - probability that a test of ϕ succeeds is at least r .

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Probabilistic Hilbert Semantics

Fix a finite set N of numbers. Let H =⊗

i Hi be the tensorproduct of finite Hilbert spaces.

States the one-dimensional subspaces of H.

Actions all possible terms π ::= u | ϕ? | π;π | π ∪ πProjection: For each formula φ,

T = [[φ]] is a set of states (explained more on future slide),T is the smallest closed subspace containing it, and[[ϕ?]] = ProjT is the projection operator of vectors in H ontoT , acting on states.

Unitary operators: For each u ∈ U fix a unitary operator [[u]]on H acting on states.Sequential composition: [[π1;π2]] = [[π1]] ◦ [[π2]] is relationcompositionChoice: [[π1 ∪ π2]] = [[π1]] ∪ [[π2]]

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Interpretation of the formulas

For each formula ϕ, we define the set [[ϕ]] of states

[[p]] is a closed subspace of H.

[[¬ϕ]] = S \ [[ϕ]] (set complement, not orthocomplement)

[[π]]ϕ is the largest set of states T , such that [[π]][T ] ⊆ [[ϕ]]

[[P≥rϕ]] denotes the set of states whose non-zero vectors vare such that 〈v | Proj[[ϕ]] | v〉 ≥ r .

[[KIϕ]] denotes the set of states whose image under everyI -remote unitary, IdI ⊗ UN\I , is in [[ϕ]].

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Abbreviations

⊥ def= ϕ ∧ ¬ϕ

> def= ¬⊥

∼ϕ def= [ϕ?]⊥

ϕ t ψ def= ∼(∼ϕ ∧ ∼ψ)

ϕ ∨ ψ def= ¬(¬ϕ ∧ ¬ψ)

ϕ→ ψdef= ¬(ϕ ∧ ¬ψ)

〈π〉ϕ def= ¬[π]¬ϕ

♦ϕdef= 〈ϕ?〉>

�ϕdef= ¬♦¬ϕ

Eϕdef= ♦♦ϕ

Aϕdef= ¬E¬ϕ

ϕIdef= ¬KI¬ϕ

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Quantum Search Example

(N + 1)-qubit Quantum Search Problem

Let |f0〉 be a classical state of type N:

f0 ∈N{0, 1} and |f0〉 = ⊗i∈N |f0(i)〉i

Let O be an action (“oracle”), such that

O(|f 〉 ⊗ |b〉N) =

{|f 〉 ⊗ |1− b〉N , if f = f0,|f 〉 ⊗ |b〉N , if f ∈ N{0, 1} \ {f0}

Goal: determine |f0〉 from O.

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Quantum Search Example

Quantum Search Algorithm

1 Input: s = ⊗Ni=0 |ai 〉i , where ai = 0 for 0 ≤ i ≤ N − 1 and

aN = 1.

2 Apply Hadamard gate to each qubit3 Repeat the following K times (K largest integer less than

π√

2N/4):1 Apply Oracle O to all qubits2 Apply Hadamard gate to all but last qubit3 Apply Conditional Phase Shift gate to all but last qubit.4 Apply Hadamard gate to all but last qubit

4 Measure (classical basis): let |g〉 be the resulting state of thefirst N qubits

Correctness Criterion

The algorithm is correct if |g〉 = |f0〉 with probability p > 0.5.

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Classical state

An N-bit classical state is a function f : N → 2, where 2 = {0, 1}.Let

fdef=∧i∈N

f (i)i

CState(p)def= Ep ∧ A

(p →

∨f :N→2

f)

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Oracle

Ora(O) :=∨

f :N→2

A[(f ∧ 0n → [O](f ∧ 1n)

)∧(f ∧ 1n → [O](f ∧ 0n)

)∧

∧g :N→2,g 6=f

((g ∧ 0n → [O](g ∧ 0n)

)∧(g ∧ 1n → [O](g ∧ 1n)

))].

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Quantum Search Algorithm Expressed

QSA := Ora(O) ∧ CState(p)

∧A(p ∧ 0n → [O]1n

)∧ A(p ∧ 1n → [O]0n

)∧ 0 ∧ 1n

→ [H0; · · · ;Hn][O;H0; · · · ;Hn−1;P;H0; · · · ;Hn−1]kP>0.5p.

O is an oracle and p is a classical statef is the state selected by the OracleAfter performing the operations and measurement, the result willmore likely than not to be correct.

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Returning to simplicity

Can we relate the quantum labelled transition systems, used forthe probabilistic logic for quantum actions, to Hilbert lattices?

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Quantum labelled transition systems

A dynamic frame is a tuple (Σ,L, { P?−→}P∈L) such that

Σ is a set (states of the system);

L ⊆ P(Σ) (testable properties);P?−→ ⊆ Σ× Σ for each P ∈ L (dynamics of tests);

Non-Orthogonality and Orthogonality

s → t ⇐⇒ there is some P ∈ L such that sP?→ t.

s 6→ t ⇐⇒ there is no P ∈ L such that sP?→ t.

A dynamic frame is a quantum dynamic frame if it satisfies sevenconditions in next slides.

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(1) Closure Condition

Closure Condition

L is closed under arbitrary intersection and orthocomplement,where the orthocomplement of A ⊆ Σ is

∼A := {s ∈ Σ | s → t ⇒ t 6∈ A,∀t ∈ Σ}.

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(2) Atomicity

Atomicity

For any s ∈ Σ, {s} ∈ L.

Every singleton is in L.

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(3) Adequacy

Adequacy

For any s ∈ Σ and P ∈ L, if s ∈ P, then sP?−→ s.

The restriction ofP?−→ to P is reflexive.

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(4) Repeatability

Repeatability

For any s, t ∈ Σ and P ∈ L, if sP?−→ t, then t ∈ P.

The image ofP?−→ is in P.

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(5) Self-Adjointness

Self-Adjointness

For any s, t, u ∈ Σ and P ∈ L, if sP?−→ t → u, then there is a

v ∈ Σ such that uP?−→ v → s.

sP? // t

��v

OO

uP?oo

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(6) Covering Property

Covering Property

Suppose sP?−→ t for s, t ∈ Σ and P ∈ L. Then, for any u ∈ P,

if u 6= t then u → v 6→ s for some v ∈ P.

sP? // t

�

P 3 v

�

OO

u ∈ Poo

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(7) Proper Superposition

Proper Superposition

For any s, t ∈ Σ there is a u ∈ Σ such that u → s and u → t.

→ composed with → is the total relation.

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Quantum Dynamic Frame (Summary)

1 Closure Condition: L is closed under arbitrary intersectionand orthocomplement.

2 Atomicity: For any s ∈ Σ, {s} ∈ L.

3 Adequacy: For any s ∈ Σ and P ∈ L, if s ∈ P, then sP?−→ s.

4 Repeatability: For any s, t ∈ Σ and P ∈ L, if sP?−→ t, then

t ∈ P.

5 Self-Adjointness: For any s, t, u ∈ Σ and P ∈ L, if

sP?−→ t → u, then there is a v ∈ Σ such that u

P?−→ v → s.

6 Covering Property: Suppose sP?−→ t for s, t ∈ Σ and P ∈ L.

Then, for any u ∈ P, if u 6= t then u → v 6→ s for somev ∈ P.

7 Proper Superposition: For any s, t ∈ Σ there is a u ∈ Σsuch that u → s and u → t.

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Mayet’s condition on frames

A quantum dynamic frame automorphism is a bijective mapg : Σ→ Σ that preserves both orthogonality andnon-orthogonality.

Definition (Mayet’s condition for Quantum Dynamic Frames)

A quantum dynamic frame F = (Σ,L, { P?−→}P∈L) is said to satisfyMayet’s condition if there is a strong automorphism g : Σ→ Σsuch that

1 there is a P ∈ L such that g−1[P] ⊂ P, and

2 there is a Q ∈ L that has at least two distinct elements andsuch that g(s) = s for all s ∈ Q.

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Relating Quantum Dynamic Frames with Piron Lattices

Quantum dynamic frames were introduced by Baltag and Smets.

Baltag-Smets conjecture

Any quantum dynamic frame satisfying Mayet’s condition can berealized by a Piron lattices with Mayet’s condition.

Baltag and Smets. Complete axiomatizations for quantum actions.International Journal of Theoretical Physics, 2005

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Our result

Theorem

There is a categorical duality between

1 Piron lattices and quantum dynamic frames

2 Piron lattices with Mayet’s condition and quantum dynamicframes with Mayet’s condition

Bergfeld, Kishida, Sack, and Zhong. Duality for the Logic ofQuantum Actions. Studia Logica, 2015.

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Concluding remarks

1 Richness of structure of testable properties

2 Connect quantum mechanical perspective (Hilbert spaces)with discrete (lattice) and computational (frame) perspectives

3 Exist quantum logics that describe probabilistic outcomes aswell as composite systems (quantum entanglement)

4 Future work: Dualities between algebraic and relationalquantum structures involving probabilities

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Thank you!

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quantum logic and structure - joshua sack · 2 there is a q 2l such that there are at least two...

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