Once upon a time, I worked with a postdoc who shaped my views of mathematical physics, research, and life. Each week, I’d email him a PDF of the calculations and insights I’d accrued. He’d respond along the lines of, “Thanks so much for your notes. They look great! I think they’re mostly correct; there are just a few details that might need fixing.” My postdoc would point out the “details” over espresso, at a café table by a window. “Are you familiar with…?” he’d begin, and pull out of his back pocket some bit of math I’d never heard of. My calculations appeared to crumble like biscotti.
Some of the math involved CPTP maps. “CPTP” stands for a phrase little more enlightening than the acronym: “completely positive trace-preserving”. CPTP maps represent processes undergone by quantum systems. Imagine preparing some system—an electron, a photon, a superconductor, etc.—in a state I’ll call ““. Imagine turning on a magnetic field, or coupling one electron to another, or letting the superconductor sit untouched. A CPTP map, labeled as , represents every such evolution.
“Trace-preserving” means the following: Imagine that, instead of switching on the magnetic field, you measured some property of . If your measurement device (your photodetector, spectrometer, etc.) worked perfectly, you’d read out one of several possible numbers. Let denote the probability that you read out the possible number. Because your device outputs some number, the probabilities sum to one: . We say that “has trace one.” But you don’t measure ; you switch on the magnetic field. undergoes the process , becoming a quantum state . Imagine that, after the process ended, you measured a property of . If your measurement device worked perfectly, you’d read out one of several possible numbers. Let denote the probability that you read out the possible number. The probabilities sum to one: . “has trace one”, so the map is “trace preserving”.
Now that we understand trace preservation, we can understand positivity. The probabilities are positive (actually, nonnegative) because they lie between zero and one. Since the characterize a crucial aspect of , we call “positive” (though we should call “nonnegative”). turns the positive into the positive . Since maps positive objects to positive objects, we call “positive”. also satisfies a stronger condition, so we call such maps “completely positive.”**
So I called my postdoc. “It’s almost right,” he’d repeat, nudging aside his espresso and pulling out a pencil. We’d patch the holes in my calculations. We might rewrite my conclusions, strengthen my assumptions, or prove another lemma. Always, we salvaged cargo. Always, I learned.
I no longer email weekly updates to a postdoc. But I apply what I learned at that café table, about entanglement and monotones and complete positivity. “It’s almost right,” I tell myself when a hole yawns in my calculations and a week’s work appears to fly out the window. “I have to fix a few details.”
Am I certain? No. But I remain positive.
*Experts: “Trace-preserving” means .
**Experts: Suppose that ρ is defined on a Hilbert space and that is defined on . “ is positive” means
To understand what “completely positive” means, imagine that our quantum system interacts with an environment. For example, suppose the system consists of photons in a box. If the box leaks, the photons interact with the electromagnetic field outside the box. Suppose the system-and-environment composite begins in a state defined on a Hilbert space . acts on the system’s part of state. Let denote the identity operation that maps every possible environment state to itself. Suppose that changes the system’s state while preserves the environment’s state. The system-and-environment composite ends up in the state . This state is positive, so we call “completely positive”:
First comment! Also, a personal note. I’ve seen the frustration in students when they first take QM and they realize there are so many things to explore about its principles. One sacrifices generality for a coherent introduction. It is just reasonable to go for incremental understanding and it might be too soon at that point to talk about mixed states and quantum transformations. But it is fun to tease them a bit with this kind of topics. It not only sparks their curiosity but makes them realize QM is a field that continuously challenges the student and the professor. The only choice is then, as you say, to remain positive. Thanks for the article!
Thanks for your thoughts, Sergio! At least your students have an advantage, having an instructor who notices what challenges them and who tries to motivate them. If your students need more “teasing . . . with this kind of topics,” feel free to send them to Quantum Frontiers!
thanks Nicole food for thought here!
we have to remember that there is no boundary between the classical and quantum read Bell’s papers especially ‘Introduction to the hidden variable question.’
Read Feynman on spinning plate and wobble it is fundamental and so deep. What is spin in particles? It’s not classical spin is it so what exactly is it? Any thoughts let me know help me please!
Carlton Caves’ on-line notes “Completely positive maps, positive maps, and the Lindblad form” (2002) provide a student-friendly survey of the differential limit of CPTP maps (to my knowledge this material is not found elsewhere in the literature).
And thank you, Nicole, for your outstanding work and sustained commitment to quantum community-building.
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