1. Picture-proofs for quantum cryptography based on categorical quantum mechanics
Carl A. Miller
Fellow, Joint Center for Quantum Information and Computer Science (QuICS)
Mathematican, National Institute of Standards and Technology (NIST)
Co-authors: Spencer Breiner (NIST), Amir Kalev (QuICS),
and Neil J. Ross (Dalhousie University)
February 4, 2019

2. A starting point vs. Formalizing quantum information can be hard.
Picture-proofs can help.
vs.
From “Symmetry and Quantum Information,” lecture notes by Michael Walter
A formal statement!

3. Formal visual proofs for quantum crypto
This talk
Benefits:
- More trust in proofs (inc. computer verification)
- New results, and easy extensions of old results
- Unification with other fields
Formal visual proofs for quantum crypto
Categorical
quantum
mechanics

4. “Graphical Methods in Device- Independent Quantum Cryptography”
(Breiner, Miller, Ross)

5. Primer on quantum crypto
Security for q. crypto is based on physical assumptions (rather than computational hardness assumptions).
Quantum key distribution was invented by Bennett and Brassard in 1984 (building on work of Wiesner). Other protocols followed.

6. Primer on quantum crypto
Coin-flipping,
Bit commitment,
Quantum money,
Secret sharing, …
Quantum key distribution
Delegated quantum computation
Quantum random
number generation

7. Primer on quantum crypto
A basic goal: Prove that a prescribed protocol is indistinguishable from a simple ”ideal” process.
(Despite adversarial behavior & imperfect hardware.) … …

8. Primer on quantum crypto
A basic goal: Prove that a prescribed protocol is indistinguishable from a simple ”ideal” process.
(Despite adversarial behavior & imperfect hardware.)
Perfect RNG … …

9. Pictures from categorical QM
state effect process trace uniform state
composition tensor product
duplication
equality test
state effect process trace
state effect process trace
state effect process trace
copy equality test uniform
distribution
copy equality test uniform
distribution
composition tensor product
composition tensor product
composition tensor product

10. New features for cryptography
Diagrams Sets of linear operators
For example,
denotes the set of quantum processes (completely positive trace non-increasing maps) from register R to register Q.

11. New features for cryptography
Approximation chains
If A and B are diagrams, then
A B
denotes that for every element a of A, there is an element of B that is within distance from a.
(protocol) (process 1) (process2) … (ideal process)

12. An initial exercise
It is known that you can generate an unlimited amount of randomness by alternating between two Bell experiments. We reproved part of this result.
Example step:

13. An initial exercise
We computer-verified the main approximation sequence using Globular.

14. “Parallel Self-Testing of the GHZ State with a Proof by Diagrams”
(Breiner, Kalev, Miller)

15. The Greenberger-Horne-Zeilinger game
The GHZ game (shown) can only be won using the GHZ state (McKague 10):
The GHZ state is self-testing.
Inputs
(xyz)
Score if
abc = 1
abc = -1
000 +1 -1
011
101 a b
c 𝜖 {-1, 1} x y z

16. Parallel Self-Testing
In our work, we prove self-testing of N copies of the GHZ state.
This is the first parallel result that goes beyond 2 players, and it has potential cryptographic applications [Hillery 99].

17. Let A, B, and C denote the quantum systems held by Alice, Bob, and Charlie.
Let L denote their initial state.
High-level view

18. High-level view
The goal is to construct three isometries, acting on A, B, C separately, which bring L close to a GHZN state.
Apply
isometries d
Prepare
state L
“States are within distance
d.”
Prepare
GHZN
Prepare
junk state

19. An Example Step

20. Future Directions

21. Next Frontiers Computer software to build & verify picture-proofs.
(Globular, Quantomatic, TikZiT, …?)
Introduce more advanced categorical tricks.
Unify quantum crypto with other fields through category theory:
- Other topics within quantum.
- Classical cryptography.

22. Picture-proofs for quantum cryptography based on categorical quantum mechanics
Carl A. Miller
Fellow, Joint Center for Quantum Information and Computer Science (QuICS)
Mathematican, National Institute of Standards and Technology (NIST)
Co-authors: Spencer Breiner (NIST), Amir Kalev (QuICS),
and Neil J. Ross (Dalhousie University)
February 4, 2019