Nathaniel Johnston
@njohnston.ca
Associate Professor of Mathematics at Mount Allison University
Interested in quantum information theory, Conway's Game of Life, recreational mathematics, and mathematics pedagogy.
🔗 https://njohnston.ca/
▶️ https://www.youtube.com/@NathanielMath
Interested in quantum information theory, Conway's Game of Life, recreational mathematics, and mathematics pedagogy.
🔗 https://njohnston.ca/
▶️ https://www.youtube.com/@NathanielMath
New preprint today with Benjamin Lovitz, @vrusso.bsky.social, and Jamie Sikora: arxiv.org/abs/2609.17411
We have been trying to extend our older paper (arXiv:2311.17047) since late 2023, but we were stuck on the proof of what is now Theorem 2 in the new paper for *years*.
We have been trying to extend our older paper (arXiv:2311.17047) since late 2023, but we were stuck on the proof of what is now Theorem 2 in the new paper for *years*.
Sharp bounds for perfect quantum state classification beyond antidistinguishability
A multiset of pure quantum states is said to be k-learnable if there is a measurement strategy that always narrows an unknown sample drawn from the list down to one of at most k candidates. The parame...
arxiv.org
September 16, 2026 at 11:28 AM
New preprint today with Benjamin Lovitz, @vrusso.bsky.social, and Jamie Sikora: arxiv.org/abs/2609.17411
We have been trying to extend our older paper (arXiv:2311.17047) since late 2023, but we were stuck on the proof of what is now Theorem 2 in the new paper for *years*.
We have been trying to extend our older paper (arXiv:2311.17047) since late 2023, but we were stuck on the proof of what is now Theorem 2 in the new paper for *years*.
New preprint up today with Benjamin Lovitz: arxiv.org/abs/2609.11849
We construct positive-partial-transpose states with Schmidt number near n - sqrt(2n), versus the previously best-known constructions with Schmidt number of roughly n/2.
We construct positive-partial-transpose states with Schmidt number near n - sqrt(2n), versus the previously best-known constructions with Schmidt number of roughly n/2.
PPT states of almost maximal Schmidt number
We construct PPT states on $\mathbb{C}^m \otimes\mathbb{C}^n$ that have Schmidt number asymptotically approaching the smaller local dimension. More specifically, we construct a PPT state with Schmidt ...
arxiv.org
September 11, 2026 at 12:26 PM
New preprint up today with Benjamin Lovitz: arxiv.org/abs/2609.11849
We construct positive-partial-transpose states with Schmidt number near n - sqrt(2n), versus the previously best-known constructions with Schmidt number of roughly n/2.
We construct positive-partial-transpose states with Schmidt number near n - sqrt(2n), versus the previously best-known constructions with Schmidt number of roughly n/2.
It seems the rush to use ChatGPT Sol to solve all our open problems has hit quantum information theory. A problem from the field that has been open for 25 years was just solved 3 times, by 3 separate groups, in 2 days:
arxiv.org/abs/2607.23416
arxiv.org/abs/2607.24309
arxiv.org/abs/2607.24479
arxiv.org/abs/2607.23416
arxiv.org/abs/2607.24309
arxiv.org/abs/2607.24479
A partial-trace matrix inequality and Werner-state distillability
Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bi...
arxiv.org
August 2, 2026 at 9:40 AM
It seems the rush to use ChatGPT Sol to solve all our open problems has hit quantum information theory. A problem from the field that has been open for 25 years was just solved 3 times, by 3 separate groups, in 2 days:
arxiv.org/abs/2607.23416
arxiv.org/abs/2607.24309
arxiv.org/abs/2607.24479
arxiv.org/abs/2607.23416
arxiv.org/abs/2607.24309
arxiv.org/abs/2607.24479
New preprint with Sarah Plosker and my student, Luis Varona: "Enumeration of Laplacian integral and {-1,0,1}-diagonalizable graphs"
(description in replies)
arxiv.org/abs/2607.06336
(description in replies)
arxiv.org/abs/2607.06336
Enumeration of Laplacian integral and {-1,0,1}-diagonalizable graphs
A graph with Laplacian matrix $L$ is called Laplacian integral if the eigenvalues of $L$ are all integers, and it is called $\{-1,0,1\}$-diagonalizable if $L$ has a full set of eigenvectors with entri...
arxiv.org
July 8, 2026 at 1:31 PM
New preprint with Sarah Plosker and my student, Luis Varona: "Enumeration of Laplacian integral and {-1,0,1}-diagonalizable graphs"
(description in replies)
arxiv.org/abs/2607.06336
(description in replies)
arxiv.org/abs/2607.06336
Reposted by Nathaniel Johnston
New preprint with @njohnston.ca: "Distinguishability of locally diagonal orthogonally invariant quantum states" Optimal measurements preserve LDOI structure, reducing optimization from n^4 to O(n^2) variables. For two-qubit cases: LOCC = SEP = PPT.
arxiv.org/abs/2604.12808
arxiv.org/abs/2604.12808
April 15, 2026 at 11:16 AM
New preprint with @njohnston.ca: "Distinguishability of locally diagonal orthogonally invariant quantum states" Optimal measurements preserve LDOI structure, reducing optimization from n^4 to O(n^2) variables. For two-qubit cases: LOCC = SEP = PPT.
arxiv.org/abs/2604.12808
arxiv.org/abs/2604.12808
Huge shout-out to authors who put humour, even very mild humour, in their papers. You keep me awake.
August 25, 2025 at 8:57 PM
Huge shout-out to authors who put humour, even very mild humour, in their papers. You keep me awake.
New paper published today! "A hierarchy of eigencomputations for polynomial optimization on the sphere", with Benjamin Lovitz: link.springer.com/article/10.1...
A hierarchy of eigencomputations for polynomial optimization on the sphere - Mathematical Programming
We introduce a convergent hierarchy of lower bounds on the minimum value of a real form over the unit sphere. The main practical advantage of our hierarchy over the real sum-of-squares (RSOS) hierarch...
link.springer.com
July 21, 2025 at 5:29 PM
New paper published today! "A hierarchy of eigencomputations for polynomial optimization on the sphere", with Benjamin Lovitz: link.springer.com/article/10.1...
New paper published today! "The factor width rank of a matrix", with Shirin Moein and Sarah Plosker: www.sciencedirect.com/science/arti...
The factor width rank of a matrix
A matrix is said to have factor width at most k if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single k×k…
www.sciencedirect.com
April 3, 2025 at 12:48 PM
New paper published today! "The factor width rank of a matrix", with Shirin Moein and Sarah Plosker: www.sciencedirect.com/science/arti...
Happy belated pi day! Had a midterm in my Vector Calculus class yesterday, so I asked my students to compute some vector line integrals along pi: www.desmos.com/calculator/u...
#ITeachMath #MathsToday
#ITeachMath #MathsToday
Desmos | Graphing Calculator
www.desmos.com
March 15, 2025 at 3:25 PM
Happy belated pi day! Had a midterm in my Vector Calculus class yesterday, so I asked my students to compute some vector line integrals along pi: www.desmos.com/calculator/u...
#ITeachMath #MathsToday
#ITeachMath #MathsToday
Now, two months later, Musk says that "Grok 3 is becoming superhuman" because Grok 3 obtained just as good as solution (i.e., an absolutely terrible non-solution) to this Putnam problem. Unreal.
February 28, 2025 at 8:24 PM
Now, two months later, Musk says that "Grok 3 is becoming superhuman" because Grok 3 obtained just as good as solution (i.e., an absolutely terrible non-solution) to this Putnam problem. Unreal.
I'm teaching Vector Calculus this semester (for the first time, somehow!) and making lecture videos to accompany the course. The first video is now up, with about 3 per week planned (35 to 40 total): www.youtube.com/watch?v=VbDE...
The videos make huge use of @desmos.com
#ITeachMath #EduSky
The videos make huge use of @desmos.com
#ITeachMath #EduSky
Vector Calculus - Lecture 1: Paths and Curves
YouTube video by Nathaniel Johnston
www.youtube.com
January 8, 2025 at 12:43 PM
I'm teaching Vector Calculus this semester (for the first time, somehow!) and making lecture videos to accompany the course. The first video is now up, with about 3 per week planned (35 to 40 total): www.youtube.com/watch?v=VbDE...
The videos make huge use of @desmos.com
#ITeachMath #EduSky
The videos make huge use of @desmos.com
#ITeachMath #EduSky
Happy New Year! Just like every year, there were tons of fantastic discoveries and theorems proved in Conway's Game of Life in 2024. This is a thread for my three favourites (and the context behind them to try to convince you that they're interesting). 🧵
#MathSky
#MathSky
January 1, 2025 at 2:19 AM
Happy New Year! Just like every year, there were tons of fantastic discoveries and theorems proved in Conway's Game of Life in 2024. This is a thread for my three favourites (and the context behind them to try to convince you that they're interesting). 🧵
#MathSky
#MathSky
There's been a bunch of claims (mostly on X) that ChatGPT did great on this year's Putnam math competition. Let's do a thread to talk about it! 🧵
#MathSky
#MathSky
December 20, 2024 at 2:55 AM
There's been a bunch of claims (mostly on X) that ChatGPT did great on this year's Putnam math competition. Let's do a thread to talk about it! 🧵
#MathSky
#MathSky
Reposted by Nathaniel Johnston
It seems like a tradition is emerging here, and it is out duty to maintain it. So here is my part announcing the publication of our review in semidefinite programming for characterizing quantum correlations @dulwichquantum.bsky.social
journals.aps.org/rmp/abstract...
journals.aps.org/rmp/abstract...
December 14, 2024 at 4:07 PM
It seems like a tradition is emerging here, and it is out duty to maintain it. So here is my part announcing the publication of our review in semidefinite programming for characterizing quantum correlations @dulwichquantum.bsky.social
journals.aps.org/rmp/abstract...
journals.aps.org/rmp/abstract...
The Putnam math competition happened this past weekend! Made a video of how to work through question A1, which (not surprisingly) was the most "rote" of the bunch. I thought questions A6 and B1 were really neat too. #ITeachMath
www.youtube.com/watch?v=ccu-...
www.youtube.com/watch?v=ccu-...
2024 Putnam Math Competition - Question A1 - Solutions to 2a^n + 3b^n = 4c^n
YouTube video by Nathaniel Johnston
www.youtube.com
December 10, 2024 at 12:27 PM
The Putnam math competition happened this past weekend! Made a video of how to work through question A1, which (not surprisingly) was the most "rote" of the bunch. I thought questions A6 and B1 were really neat too. #ITeachMath
www.youtube.com/watch?v=ccu-...
www.youtube.com/watch?v=ccu-...
It looks like Overleaf's new AI writing tools don't like the notes that I make to myself while writing papers.
December 5, 2024 at 4:21 PM
It looks like Overleaf's new AI writing tools don't like the notes that I make to myself while writing papers.
Had a recent research project where we had to evaluate this hideous sum. Convolution to the rescue! Made a video to talk about how it works (or at least how some similar but simpler sums work):
www.youtube.com/watch?v=aIj6...
www.youtube.com/watch?v=aIj6...
December 5, 2024 at 11:40 AM
Had a recent research project where we had to evaluate this hideous sum. Convolution to the rescue! Made a video to talk about how it works (or at least how some similar but simpler sums work):
www.youtube.com/watch?v=aIj6...
www.youtube.com/watch?v=aIj6...
Overleaf's down, so I'm cancelling research for the day. Everyone go home and play Slay the Spire.
It is my honest opinion that Slay the Spire is one of the greatest games of all time. Greater than infinite replay value — it gets better the more you play.
And please don't say Balatro, that game is pretty good but not as good. Just my opinion.
And please don't say Balatro, that game is pretty good but not as good. Just my opinion.
December 3, 2024 at 2:41 PM
Overleaf's down, so I'm cancelling research for the day. Everyone go home and play Slay the Spire.
Me, a moron: I don't need a local TeX installation, Overleaf works so well and is so easy!
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
December 3, 2024 at 1:05 PM
Me, a moron: I don't need a local TeX installation, Overleaf works so well and is so easy!
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
There are lots of standard examples of binary operations that are associative but not commutative (e.g., matrix multiplication). But I don’t know of a better example of the opposite (i.e., commutative but not associative) than Infinite Craft: neal.fun/infinite-cra...
Infinite Craft
An endless crafting game
neal.fun
December 2, 2024 at 10:08 PM
There are lots of standard examples of binary operations that are associative but not commutative (e.g., matrix multiplication). But I don’t know of a better example of the opposite (i.e., commutative but not associative) than Infinite Craft: neal.fun/infinite-cra...
Having links to relevant lecture videos included in the margin of the lecture notes is one of the cooler things I’ve seen. Those are gorgeous.
If you're curious about quantum computing and want to learn the basics, check out the "Quantum Quest" created by me and Michael Walter!
www.quantum-quest.org/material
www.quantum-quest.org/material
The Quantum Quest - Material
www.quantum-quest.org
November 30, 2024 at 11:35 AM
Having links to relevant lecture videos included in the margin of the lecture notes is one of the cooler things I’ve seen. Those are gorgeous.
The On-Line Encyclopedia of Integer Sequences (OEIS) is hiring someone to manage the crazy number of sequences that it has to review. If you're a US resident with a math PhD, give it a look! Applications due Jan. 25, 2025.
neilsloane.com/doc/OEIS.ME....
neilsloane.com/doc/OEIS.ME....
November 25, 2024 at 6:33 PM
The On-Line Encyclopedia of Integer Sequences (OEIS) is hiring someone to manage the crazy number of sequences that it has to review. If you're a US resident with a math PhD, give it a look! Applications due Jan. 25, 2025.
neilsloane.com/doc/OEIS.ME....
neilsloane.com/doc/OEIS.ME....
I wrote a review of Jane Hawkins' new book "The Mathematics of Cellular Automata", which is now online.
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
November 25, 2024 at 12:18 PM
I wrote a review of Jane Hawkins' new book "The Mathematics of Cellular Automata", which is now online.
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
Fun fact: my advanced linear algebra textbook has a page about exactly this! Screenshot attached.
November 22, 2024 at 10:49 AM
Fun fact: my advanced linear algebra textbook has a page about exactly this! Screenshot attached.
Back making math videos! #ITeachMath #EduSky
I'm teaching calculus this semester, so I'm filling in some of the gaps in my calculus playlist. Animations made with #Manim.
www.youtube.com/watch?v=Afc4...
I'm teaching calculus this semester, so I'm filling in some of the gaps in my calculus playlist. Animations made with #Manim.
www.youtube.com/watch?v=Afc4...
Constrained optimization: how to find the maximal area of a Norman window
YouTube video by Nathaniel Johnston
www.youtube.com
November 21, 2024 at 2:51 PM
Back making math videos! #ITeachMath #EduSky
I'm teaching calculus this semester, so I'm filling in some of the gaps in my calculus playlist. Animations made with #Manim.
www.youtube.com/watch?v=Afc4...
I'm teaching calculus this semester, so I'm filling in some of the gaps in my calculus playlist. Animations made with #Manim.
www.youtube.com/watch?v=Afc4...