# Identical twins and quantum entanglement

“If I had a nickel for every unsolicited and very personal health question I’ve gotten at parties, I’d have paid off my medical school loans by now,” my doctor friend complained. As a physicist, I can somewhat relate. I occasionally find myself nodding along politely to people’s eccentric theories about the universe. A gentleman once explained to me how twin telepathy (the phenomenon where, for example, one twin feels the other’s pain despite being in separate countries) comes from twins’ brains being entangled in the womb. Entanglement is a nonclassical correlation that can exist between spatially separated systems. If two objects are entangled, it’s possible to know everything about both of them together but nothing about either one. Entangling two particles (let alone full brains) over tens of kilometres (let alone full countries) is incredibly challenging. “Using twins to study entanglement, that’ll be the day,” I thought. Well, my last paper did something like that.

In theory, a twin study consists of two people that are as identical as possible in every way except for one. What that allows you to do is isolate the effect of that one thing on something else. Aleksander Lasek (postdoc at QuICS), David Huse (professor of physics at Princeton), Nicole Yunger Halpern (NIST physicist and Quantum Frontiers blogger), and I were interested in isolating the effects of quantities’ noncommutation (explained below) on entanglement. To do so, we first built a pair of twins and then compared them

Consider a well-insulated thermos filled with soup. The heat and the number of “soup particles” inside the thermos are conserved. So the energy and the number of “soup particles” are conserved quantities. In classical physics, conserved quantities commute. This means that we can simultaneously measure the amount of each conserved quantity in our system, like the energy and number of soup particles. However, in quantum mechanics, this needn’t be true. Measuring one property of a quantum system can change another measurement’s outcome.

Conserved quantities’ noncommutation in thermodynamics has led to some interesting results. For example, it’s been shown that conserved quantities’ noncommutation can decrease the rate of entropy production. For the purposes of this post, entropy production is something that limits engine efficiency—how well engines can convert fuel to useful work. For example, if your car engine had zero entropy production (which is impossible), it would convert 100% of the energy in your car’s fuel into work that moved your car along the road. Current car engines can convert about 30% of this energy, so it’s no wonder that people are excited about the prospective application of decreasing entropy production. Other results (like this one and that one) have connected noncommutation to potentially hindering thermalization—the phenomenon where systems interact until they have similar properties, like when a cup of coffee cools. Thermalization limits memory storage and battery lifetimes. Thus, learning how to resist thermalization could also potentially lead to better technologies, such as longer-lasting batteries.

One can measure the amount of entanglement within a system, and as quantum particles thermalize, they entangle. Given the above results about thermalization, we might expect that noncommutation would decrease entanglement. Testing this expectation is where the twins come in.

Say we built a pair of twins that were identical in every way except for one. Nancy, the noncommuting twin, has some features that don’t commute, say, her hair colour and height. This means that if we measure her height, we’ll have no idea what her hair colour is. For Connor, the commuting twin, his hair colour and height commute, so we can determine them both simultaneously. Which twin has more entanglement? It turns out it’s Nancy.

Disclaimer: This paragraph is written for an expert audience. Our actual models consist of 1D chains of pairs of qubits. Each model has three conserved quantities (“charges”), which are sums over local charges on the sites. In the noncommuting model, the three local charges are tensor products of Pauli matrices with the identity (XI, YI, ZI). In the commuting model, the three local charges are tensor products of the Pauli matrices with themselves (XX, YY, ZZ). The paper explains in what sense these models are similar. We compared these models numerically and analytically in different settings suggested by conventional and quantum thermodynamics. In every comparison, the noncommuting model had more entanglement on average.

Our result thus suggests that noncommutation increases entanglement. So does charges’ noncommutation promote or hinder thermalization? Frankly, I’m not sure. But I’d bet the answer won’t be in the next eccentric theory I hear at a party.

# Mo’ heights mo’ challenges – Climbing mount grad school

My wife’s love of mountain hiking and my interest in quantum thermodynamics collided in Telluride, Colorado.

We spent ten days in Telluride, where I spoke at the Information Engines at the Frontiers of Nanoscale Thermodynamics workshop. Telluride is a gorgeous city surrounded by mountains and scenic trails. My wife (Hasti) and I were looking for a leisurely activity one morning. We chose hiking Bear Creek Trail, a 5.1-mile hike with a 1092-foot elevation. This would have been a reasonable choice… back home.

Telluride’s elevation is 8,750 feet (ten times that of our hometown’s). This meant there was nothing leisurely about the hike. Ill-prepared, I dragged myself up the mountain in worn runners and tight jeans. My gasps for breath reminded me how new heights (a literal one in this case) could bring unexpected challenges – A lesson I’ve encountered many times as a graduate student.

I completely squandered my undergrad. One story sums it up best. I was studying for my third-year statistical mechanics final when I realized I could pass the course without writing the final. So, I didn’t write the final. After four years of similar negligence, I somehow graduated, certain I’d left academics forever. Two years later, I rediscovered my love for physics and grieved about wasting my undergraduate. I decided to try again and apply for grad school. I learned Canada had 17 Universities I could apply to with a 70 average; each one rejected me.

I was ecstatic to eventually be accepted into a master’s of math. But, the high didn’t last. Learning math and physics from popular science books and PBS videos is very different from studying for University exams. If I wanted to keep this opportunity, I had to learn how to study.

16 months later, I graduated with a 97 average and an offer to do a master’s of physics at the University of Waterloo. I would be working at the Institute for Quantum Computing (IQC), supervised by Raymond Laflamme (one of the co-founders of the field of quantum computing). My first day at IQC felt surreal. I had become an efficient student and felt ready for the IQC. But, like the bear creek trail, another height would bring another challenge. Ultimately, grad school isn’t about getting good grades; it’s about researching. Raymond (Ray) gave me my first research project, and I was dumbfounded about where to start and too insecure about asking for help.

With Ray and Jonathan Halliwell’s (professor at Imperial College London and guitarist-extraordinaire) guidance, I published my first paper and accepted a Ph.D. offer from Ray. After publishing my second paper, I thought it would be smooth sailing through my Ph.D..  Alas, I was again mistaken. It’s also not enough to solve problems others give you; you need to come up with some problems on your own. So, I tried. I spent the first 8-months of my Ph.D. pursuing a problem I came up with, and It was a complete dud. It turns out the problems also need to be worth solving. For those keeping track, this is challenge number four.

I have now finished the third year of my Ph.D. (two, if you don’t count the year I “took off” to write a textbook with Ray and superconducting-qubit experimentalist, Prof. Chris Wilson). During that time, Nicole Yunger Halpern (NIST physicist, my new co-advisor, and Quantum Frontiers blogger) introduced me to the field of quantum thermodynamics. We’ve published a paper together (related blog post and Vanier interview) and have a second on the way. Despite that, I’m still grappling with that last challenge. I have no problem finding research questions that would be fun to solve. However, I’m still not sure which ones are worth solving. But, like the other challenges, I’m hopeful I’ll figure it out.

While this lesson was inspiring, the city of Telluride inspired me the most. Telluride is at a local minimum elevation, surrounded by mountains. Meaning there is virtually nowhere to go but up. I’m hoping the same is true for me.

# Building a Koi pond with Lie algebras

When I was growing up, one of my favourite places was the shabby all-you-can-eat buffet near our house. We’d walk in, my mom would approach the hostess to explain that, despite my being abnormally large for my age, I qualified for kids-eat-free, and I would peel away to stare at the Koi pond. The display of different fish rolling over one another was bewitching. Ten-year-old me would have been giddy to build my own Koi pond, and now I finally have. However, I built one using Lie algebras.

The different fish swimming in the Koi pond are, in many ways, like charges being exchanged between subsystems. A “charge” is any globally conserved quantity. Examples of charges include energy, particles, electric charge, or angular momentum. Consider a system consisting of a cup of coffee in your office. The coffee will dynamically exchange charges with your office in the form of heat energy. Still, the total energy of the coffee and office is conserved (assuming your office walls are really well insulated). In this example, we had one type of charge (heat energy) and two subsystems (coffee and office). Consider now a closed system consisting of many subsystems and many different types of charges. The closed system is like the finite Koi pond with different charges like the different fish species. The charges can move around locally, but the total number of charges is globally fixed, like how the fish swim around but can’t escape the pond. Also, the presence of one type of charge can alter another’s movement, just as a big fish might block a little one’s path.

Unfortunately, the Koi pond analogy reaches its limit when we move to quantum charges. Classically, charges commute. This means that we can simultaneously determine the amount of each charge in our system at each given moment. In quantum mechanics, this isn’t necessarily true. In other words, classically, I can count the number of glossy fish and matt fish. But, in quantum mechanics, I can’t.

So why does this matter? Subsystems exchanging charges are prevalent in thermodynamics. Quantum thermodynamics extends thermodynamics to include small systems and quantum effects. Noncommutation underlies many important quantum phenomena. Hence, studying the exchange of noncommuting charges is pivotal in understanding quantum thermodynamics. Consequently, noncommuting charges have emerged as a rapidly growing subfield of quantum thermodynamics. Many interesting results have been discovered from no longer assuming that charges commute (such as these). Until recently, most of these discoveries have been theoretical. Bridging these discoveries to experimental reality requires Hamiltonians (functions that tell you how your system evolves in time) that move charges locally but conserve them globally. Last year it was unknown whether these Hamiltonians exist, what they look like generally, how to build them, and for what charges you could find them.

Nicole Yunger Halpern (NIST physicist, my co-advisor, and Quantum Frontiers blogger) and I developed a prescription for building Koi ponds for noncommuting charges. Our prescription allows you to systematically build Hamiltonians that overtly move noncommuting charges between subsystems while conserving the charges globally. These Hamiltonians are built using Lie algebras, abstract mathematical tools that can describe many physical quantities (including everything in the standard model of particle physics and space-time metric). Our results were recently published in npj QI. We hope that our prescription will bolster the efforts to bridge the results of noncommuting charges to experimental reality.

In the end, a little group theory was all I needed for my Koi pond. Maybe I’ll build a treehouse next with calculus or a remote control car with combinatorics.