Introduction to Quantum Logic
Chris Heunen
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Overview
I Boolean algebra
I Superposition
I Quantum logic
I Entanglement
I Quantum computation
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Boolean algebra
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Boolean algebra
A Boolean algebra is a set (of “logical propositions”) with
I special elements 0, 1 (“false” and “true”)
I binary operations ∨,∧ (“or” and “and”)
I a unary operation ¬ (“not”)
that satisfy laws:
associativity x ∨ (y ∨ z) = (x ∨ y) ∨ z x ∧ (y ∧ z) = (x ∧ y) ∧ zcommutativity x ∨ y = y ∨ x x ∧ y = y ∧ xidentity x ∨ 0 = x x ∧ 1 = xannihilation x ∨ 1 = 1 x ∧ 0 = 0idempotence x ∨ x = x x ∧ x = xabsorption x ∧ (x ∨ y) = x x ∨ (x ∧ y) = xcomplementation x ∧ ¬x = 0 x ∨ ¬x = 1de Morgan ¬(x ∨ y) = ¬x ∧ ¬y ¬(x ∧ y) = ¬x ∨ ¬ydouble negation ¬(¬x) = xdistributivity x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)
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Boolean algebra
A Boolean algebra is a set (of “logical propositions”) with
I special elements 0, 1 (“false” and “true”)
I binary operations ∨,∧ (“or” and “and”)
I a unary operation ¬ (“not”)
that satisfy laws:
associativity x ∨ (y ∨ z) = (x ∨ y) ∨ z x ∧ (y ∧ z) = (x ∧ y) ∧ zcommutativity x ∨ y = y ∨ x x ∧ y = y ∧ xidentity x ∨ 0 = x x ∧ 1 = xannihilation x ∨ 1 = 1 x ∧ 0 = 0idempotence x ∨ x = x x ∧ x = xabsorption x ∧ (x ∨ y) = x x ∨ (x ∧ y) = xcomplementation x ∧ ¬x = 0 x ∨ ¬x = 1de Morgan ¬(x ∨ y) = ¬x ∧ ¬y ¬(x ∧ y) = ¬x ∨ ¬ydouble negation ¬(¬x) = xdistributivity x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)
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Venn diagram
A
B
C
“logical proposition” ∼ subset“and” ∼ intersection
“or” ∼ union“not” ∼ complement
“true” ∼ whole set“false” ∼ empty set
Think of a “logical proposition” as the set of states in which it is trueThe larger the subset, the “more true” the proposition
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Venn diagram
A
B
C
“logical proposition” ∼ subset“and” ∼ intersection
“or” ∼ union“not” ∼ complement
“true” ∼ whole set“false” ∼ empty set
Think of a “logical proposition” as the set of states in which it is trueThe larger the subset, the “more true” the proposition
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Hasse diagramEvery Boolean algebra has a partial order
x ≤ y ⇐⇒ x ∧ y = x
with greatest lower bounds, least upper bounds, and complements.Conversely, every such partial order gives a Boolean algebra.
{}
{, ,
}
{ } { } { }
{,
} {,
} {,}
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Hasse diagramEvery Boolean algebra has a partial order
x ≤ y ⇐⇒ x ∧ y = x
with greatest lower bounds, least upper bounds, and complements.
Conversely, every such partial order gives a Boolean algebra.
{}
{, ,
}
{ } { } { }
{,
} {,
} {,}
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Hasse diagramEvery Boolean algebra has a partial order
x ≤ y ⇐⇒ x ∧ y = x
with greatest lower bounds, least upper bounds, and complements.Conversely, every such partial order gives a Boolean algebra.
{}
{, ,
}
{ } { } { }
{,
} {,
} {,}
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Hasse diagramEvery Boolean algebra has a partial order
x ≤ y ⇐⇒ x ∧ y = x
with greatest lower bounds, least upper bounds, and complements.Conversely, every such partial order gives a Boolean algebra.
{}
{, ,
}
{ } { } { }
{,
} {,
} {,}
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Implication
Can axiomatise Boolean algebra in terms of ∧,∨, or in terms of ≤,or in terms of →:
x ∧ y ≤ z ⇐⇒ x ≤ y → z
where (y → z) = (¬y ∨ z)
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Quantum information
I Boolean logic governs propositions and states.
I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)
I Quantum information is weird:I superpositionI entanglement
I Quantum computers use this weirdness in a positive wayto achieve more than classical computers
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Quantum information
I Boolean logic governs propositions and states.
I Computers manipulate information(information stored on physical system)
Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)
I Quantum information is weird:I superpositionI entanglement
I Quantum computers use this weirdness in a positive wayto achieve more than classical computers
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Quantum information
I Boolean logic governs propositions and states.
I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)
I Quantum information is weird:I superpositionI entanglement
I Quantum computers use this weirdness in a positive wayto achieve more than classical computers
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Quantum information
I Boolean logic governs propositions and states.
I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)
I Quantum information is weird:I superpositionI entanglement
I Quantum computers use this weirdness in a positive wayto achieve more than classical computers
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Quantum information
I Boolean logic governs propositions and states.
I Computers manipulate information(information stored on physical system)Quantum computers manipulate quantum information(information stored on quantum-mechanical systems)
I Quantum information is weird:I superpositionI entanglement
I Quantum computers use this weirdness in a positive wayto achieve more than classical computers
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States and propositions
Physical system has set of statesProposition about physical system is subset
Quantum system has space of statesProposition about quantum system is subspace
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States and propositions
Physical system has set of statesProposition about physical system is subset
Quantum system has space of statesProposition about quantum system is subspace
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Quantum weirdness: superposition
Classical bits
(what went in comes out)
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Superposition
Quantum bits
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Superposition
Quantum bits
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Superposition
Quantum bits
(if you open different door than you closed, random colour comesout)
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Qubits
Quantum bit has state space R2
Could be
(10
), could be
(01
), or could be “in between”
(ab
).
You can ask for the value of a quantum bit in many ways,
using any angle θ. Say
(cos θ − sin θsin θ cos θ
)(ab
)=
(cd
).
Get answer 0 with probability c2, answer 1 with probability d2.
So propositions are the subspaces
{(t cos θt sin θ
): t ∈ R
},
{(00
)}, R2.
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Qubits
Quantum bit has state space R2
Could be
(10
), could be
(01
), or could be “in between”
(ab
).
You can ask for the value of a quantum bit in many ways,
using any angle θ. Say
(cos θ − sin θsin θ cos θ
)(ab
)=
(cd
).
Get answer 0 with probability c2, answer 1 with probability d2.
So propositions are the subspaces
{(t cos θt sin θ
): t ∈ R
},
{(00
)}, R2.
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Qubits
Quantum bit has state space R2
Could be
(10
), could be
(01
), or could be “in between”
(ab
).
You can ask for the value of a quantum bit in many ways,
using any angle θ. Say
(cos θ − sin θsin θ cos θ
)(ab
)=
(cd
).
Get answer 0 with probability c2, answer 1 with probability d2.
So propositions are the subspaces
{(t cos θt sin θ
): t ∈ R
},
{(00
)}, R2.
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Distributivity
(or
)and
6=(and
)or(
and)
tea
coffeebiscuit
nothing
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Distributivity
(or
)and
6=(and
)or(
and)
tea
coffeebiscuit
nothing
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Orthomodularity
I There is still order: ≤ is set inclusion.
I There are still least upper bounds, greatest lower bounds.
I There is still negation: ¬R2 =
{(00
)}, and
¬{(
t cos θt sin θ
): t ∈ R
}=
{(t cos(θ + π/2)t sin(θ + π/2)
): t ∈ R
}I The orthomodular law still holds:
x ≤ y =⇒ x ∨ (¬x ∧ y) = y
(distributivity x ∨ (z ∧ y) = (x ∨ z) ∧ (x ∨ y) for z = ¬x, x ≤ y)
I Quantum logic is study of partial orders with 0,1, least upperbounds, greatest lower bounds, complements, satisfyingorthomodular law.
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Implication
There is no good notion of quantum implication. Best we can do is
x& y ≤ z ⇐⇒ x ≤ y → z
where (x& y) = (x ∨ ¬y) ∧ yand (y → z) = ¬y ∨ (y ∧ z).
Here (x& y) = (x ∧ y) when x ≤ ¬y.
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Quantum computation
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Entanglement
2 quantum bits
random random
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Entanglement
2 quantum bits
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Entanglement
2 quantum bits
(same door, same colour!)
I information stored entirely in correlations, not locally!
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Entanglement
classical correlations
quantum correlations
I But: only one way to look at socks,but two ways to look in box!
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Entanglement
classical correlations quantum correlations
I But: only one way to look at socks,but two ways to look in box!
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Entanglement
classical correlations quantum correlations
I But: only one way to look at socks,but two ways to look in box!
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Tensor products
I State space of n bits is product of state spaces of individual bits.
I Product of n qubits R2 × · · · × R2 ' R2n has dimension 2n.
I Instead, use tensor product R2 ⊗ · · · ⊗ R2, with dimension 2n.
I Has many entangled states not in the product.
(ab
)×(cd
)=
abcd
but
(ab
)⊗(cd
)=
acadbcbd
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Quantum computation speed-up
10 classical bits: only 210 = 1024 possibilities
need 10 numbers to describe one possibility:
(all independent)
10 quantum bits:
need ∼ 1000 numbers to describe a single possibility!(many correlations)
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Quantum computation speed-up
10 classical bits: only 210 = 1024 possibilities
need 10 numbers to describe one possibility:
(all independent)
10 quantum bits:
need ∼ 1000 numbers to describe a single possibility!(many correlations)
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Quantum computation speed-up
10 classical bits: only 210 = 1024 possibilities
need 10 numbers to describe one possibility:
(all independent)
10 quantum bits:
need ∼ 1000 numbers to describe a single possibility!(many correlations)
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Deutsch–Josza
I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,
or f always gives the same output.Question: find out which.
I Classical algorithm requires n+ 1 calls to f .
I Quantum algorithm can do it in 1 step!(Caveat: “oracle” f needs to be quantum to start with)
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Deutsch–Josza
I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,
or f always gives the same output.Question: find out which.
I Classical algorithm requires n+ 1 calls to f .
I Quantum algorithm can do it in 1 step!(Caveat: “oracle” f needs to be quantum to start with)
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Deutsch–Josza
I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,
or f always gives the same output.Question: find out which.
I Classical algorithm requires n+ 1 calls to f .
I Quantum algorithm can do it in 1 step!
(Caveat: “oracle” f needs to be quantum to start with)
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Deutsch–Josza
I Given: algorithm f that inputs 2n bits and outputs 1 bit.Promised: either f outputs 0 on n and 1 on other half,
or f always gives the same output.Question: find out which.
I Classical algorithm requires n+ 1 calls to f .
I Quantum algorithm can do it in 1 step!(Caveat: “oracle” f needs to be quantum to start with)
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Deutsch–Josza
I Start with
(10
)⊗ · · · ⊗
(10
)⊗(
01
).
I Apply H ⊗ · · · ⊗H ⊗H, where H =
(cosπ/2 − sinπ/2sinπ/2 cosπ/2
)I Apply f
I Apply H ⊗ · · · ⊗HI Measure with angle 0
I Answer 1 with certainty if f was constant, 0 if f was balanced
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Deutsch–Josza
Correctness proof in vector space notation:
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Summary
I Quantum weirdness: superposition, entanglement
I Quantum computation can use weirdness
I Quantum logic has to deal with weirdness
Take-home message:
I Information is physical
I Logic is physical
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