A 2-categorical Analysis of
Complementary Families and Quantum
Key Distribution
Krzysztof Bar
1 Department
1
Jamie Vicary
2
of Computer Science, University of Oxford
2 Centre
for Quantum Technologies, National University of Singapore
and Department of Computer Science, University of Oxford
QPL Kyoto, 6th June 2014
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
A
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
1-cells Lines
Quantum systems
B
S
A
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
1-cells Lines
Quantum systems
B
S
A
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
1-cells Lines
Quantum systems
2-cells Vertices Quantum dynamics
S0
B
α
S
A
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
1-cells Lines
Quantum systems
2-cells Vertices Quantum dynamics
S0
B
α
S
A
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
1-cells Lines
Quantum systems
2-cells Vertices Quantum dynamics
S0
Horizontal composition
α
B
A T C
S
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
1-cells Lines
Quantum systems
2-cells Vertices Quantum dynamics
S0
Horizontal composition
α
B
A T C
S
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
S 00
γ
1-cells Lines
Quantum systems
2-cells Vertices Quantum dynamics
S0
Horizontal composition
α
B
A T C
S
Vertical composition
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
S 00
γ
1-cells Lines
Quantum systems
2-cells Vertices Quantum dynamics
S0
Horizontal composition
α
B
A T C
S
Vertical composition
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
S 00
γ
U0
U 00
1-cells Lines
Quantum systems
D
2-cells Vertices Quantum dynamics
S0
δ
Horizontal composition
α
B
A T C
S
Vertical composition
E
F
U
Tensor product
Introduction
2-categories allow us to reason about quantum measurements in a more
elegant way and with fewer equations
Graphical calculus:
0-cells Regions Classical information
S 00
γ
U0
U 00
1-cells Lines
Quantum systems
D
2-cells Vertices Quantum dynamics
S0
δ
Horizontal composition
α
B
A T C
S
Vertical composition
E
F
U
Tensor product
Semantics given by the symmetric monoidal 2-category 2Hilb that has:
0-cells given by natural numbers
1-cells - matrices whose entries are finite-dimensional Hilbert spaces
2-cells given by matrices whose entries are linear maps
Quantum key distribution via E91
alice
bob
.
Quantum key distribution via E91
Creation of entangled state
alice
bob
.
Quantum key distribution via E91
bob: choose a random basis
alice: choose a random basis
Creation of entangled state
alice
bob
.
Quantum key distribution via E91
bob: controlled measurement
bob: choose a random basis
alice: controlled measurement
alice: choose a random basis
Creation of entangled state
alice
bob
basis information
key information
.
Quantum key distribution via E91
alice, bob: compare bases
bob: controlled measurement
bob: choose a random basis
alice: controlled measurement
alice: choose a random basis
Creation of entangled state
alice
bob
basis information
key information
.
Quantum key distribution via E91
Vertical 2-cell composition corresponds to temporal composition.
Horizontal 2-cell composition corresponds to spatial composition
alice, bob: compare bases
bob: controlled measurement
bob: choose a random basis
time
alice: controlled measurement
alice: choose a random basis
Creation of entangled state
alice
bob
By choosing a 2-category these diagrams are interpreted in, we choose the
theory of Physics to work in. Quantum theory is modelled by 2Hilb.
Quantum key distribution via E91
alice, bob: compare bases
bob: controlled measurement
bob: choose a random basis
alice: controlled measurement
alice: choose a random basis
Creation of entangled state
alice
bob
basis information
key information
.
Quantum key distribution via E91
alice, bob: compare bases
bob: controlled measurement
bob: choose a random basis
eve: prepare fake system
eve: copy measurement result
eve: intercept and measure
alice: controlled measurement
alice: choose a random basis
eve: choose a random basis
Creation of entangled state
alice
bob
eve
basis information
key information
.
Quantum key distribution via E91
alice, bob: compare bases
bob: controlled measurement
bob: choose a random basis
eve: prepare fake system
eve: copy measurement result
eve: intercept and measure
alice: controlled measurement
alice: choose a random basis
eve: choose a random basis
Creation of entangled state
alice
bob
eve
basis information
key information
.
Quantum key distribution via E91
alice, bob: compare bases
eve: prepare counterfeit system
bob: choose a random basis
eve: copy measurement result
eve: intercept system
eve: choose a random basis
alice: controlled preparation
alice: choose a random basis
alice: copy the bit
alice: choose random bit
alice
bob
eve
basis information
key information
.
Quantum key distribution via E91
alice, bob: compare bases
bob: controlled measurement
bob: choose a random basis
eve: prepare fake system
eve: copy measurement result
eve: intercept and measure
alice: controlled measurement
alice: choose a random basis
eve: choose a random basis
Creation of entangled state
alice
bob
eve
basis information
key information
.
Quantum key distribution via E91
=
Ps
alice
bob
eve
basis information
key information
.
Quantum key distribution via E91
Pd
=
+ Ps
alice
bob
eve
basis information
key information
.
Quantum key distribution via E91
Pd
ψ
=
+ Ps
alice
bob
eve
.
This is the QKD equation
Complementarity of families of controlled
operations
Pd
Complementarity of families of controlled
operations
R measurement
L measurement
Results copied
L measurement
Pd
L
R
Complementarity of families of controlled
operations
R measurement
L measurement
Results copied
L measurement
Pd
L
R
= Pnd
L
R
L measurement
Random data
Complementarity of families of controlled
operations
R measurement
L measurement
Results copied
L measurement
Pd
L
R
= Pnd
φ
L
R
Controlled phase φ
L measurement
Random data
Complementarity of families of controlled
operations
Controlled complementarity A family of controlled operations is
complementary, if there exists a phase ψ such that:
R measurement
L measurement
Results copied
L measurement
Pd
L
R
= Pnd
φ
L
R
Controlled phase φ
L measurement
Random data
Theorem
Solutions to the controlled complementarity equation in 2Hilb
correspond to families of mutually unbiased bases
Complementarity of families of controlled
operations
Controlled complementarity A family of controlled operations is
complementary, if there exists a phase ψ such that:
R measurement
L measurement
Results copied
L measurement
Pd
L
R
= Pnd
φ
L
R
Controlled phase φ
L measurement
Random data
Theorem
Solutions to the controlled complementarity equation in 2Hilb
correspond to families of mutually unbiased bases
Theorem
The QKD equation and the Controlled complementarity equation are
topologically equivalent
Summary
What is the significance of this work?
Summary
What is the significance of this work?
Completely syntactic proof of QKD ⇔ MUB equivalence that uses
only the logical structure
Summary
What is the significance of this work?
Completely syntactic proof of QKD ⇔ MUB equivalence that uses
only the logical structure
Also in the paper:
Logical correctness proof of Klappenecker and Roettler’s
construction of a solution to the Mean King’s problem from a family
of mutually unbiased bases
Summary
What is the significance of this work?
Completely syntactic proof of QKD ⇔ MUB equivalence that uses
only the logical structure
Also in the paper:
Logical correctness proof of Klappenecker and Roettler’s
construction of a solution to the Mean King’s problem from a family
of mutually unbiased bases
Future directions:
Application to other quantum protocols
Nonstandard, ’classical’ models
Summary
What is the significance of this work?
Completely syntactic proof of QKD ⇔ MUB equivalence that uses
only the logical structure
Also in the paper:
Logical correctness proof of Klappenecker and Roettler’s
construction of a solution to the Mean King’s problem from a family
of mutually unbiased bases
Future directions:
Application to other quantum protocols
Nonstandard, ’classical’ models
Thank you!