August | 2019 | Quantum Frontiers

If only my coauthors and I had quarreled.

I was working with Tony Bartolotta, a PhD student in theoretical physics at Caltech, and Jason Pollack, a postdoc in cosmology at the University of British Columbia. They acted as the souls of consideration. We missed out on dozens of opportunities to bicker—about the paper’s focus, who undertook which tasks, which journal to submit to, and more. Bickering would have spiced up the story behind our paper, because the paper concerns disagreement.

Quantum observables can disagree. Observables are measurable properties, such as position and momentum. Suppose that you’ve measured a quantum particle’s position and obtained an outcome $x$ . If you measure the position immediately afterward, you’ll obtain $x$ again. Suppose that, instead of measuring the position again, you measure the momentum. All the possible outcomes have equal probabilities of obtaining. You can’t predict the outcome.

The particle’s position can have a well-defined value, or the momentum can have a well-defined value, but the observables can’t have well-defined values simultaneously. Furthermore, if you measure the position, you randomize the outcome of a momentum measurement. Position and momentum disagree.

How should we quantify the disagreement of two quantum observables, $\hat{A}$ and $\hat{B}$ ? The question splits physicists into two camps. Pure quantum information (QI) theorists use uncertainty relations, whereas condensed-matter and high-energy physicists prefer out-of-time-ordered correlators. Let’s meet the camps in turn.

Heisenberg intuited an uncertainty relation that Robertson formalized during the 1920s,

$\Delta \hat{A} \, \Delta \hat{B} \geq \frac{1}{i \hbar} \langle [\hat{A}, \hat{B}] \rangle$ .

Imagine preparing a quantum state $| \psi \rangle$ and measuring $\hat{A}$ , then repeating this protocol in many trials. Each trial has some probability $p_a$ of yielding the outcome $a$ . Different trials will yield different $a$ ’s. We quantify the spread in $a$ values with the standard deviation $\Delta \hat{A} = \sqrt{ \langle \psi | \hat{A}^2 | \psi \rangle - \langle \psi | \hat{A} | \psi \rangle^2 }$ . We define $\Delta \hat{B}$ analogously. $\hbar$ denotes Planck’s constant, a number that characterizes our universe as the electron’s mass does.

$[\hat{A}, \hat{B}]$ denotes the observables’ commutator. The numbers that we use in daily life commute: $7 \times 5 = 5 \times 7$ . Quantum numbers, or operators, represent $\hat{A}$ and $\hat{B}$ . Operators don’t necessarily commute. The commutator $[\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}$ represents how little $\hat{A}$ and $\hat{B}$ resemble 7 and 5.

Robertson’s uncertainty relation means, “If you can predict an $\hat{A}$ measurement’s outcome precisely, you can’t predict a $\hat{B}$ measurement’s outcome precisely, and vice versa. The uncertainties must multiply to at least some number. The number depends on how much $\hat{A}$ fails to commute with $\hat{B}$ .” The higher an uncertainty bound (the greater the inequality’s right-hand side), the more the operators disagree.

Heisenberg and Robertson explored operator disagreement during the 1920s. They wouldn’t have seen eye to eye with today’s QI theorists. For instance, QI theorists consider how we can apply quantum phenomena, such as operator disagreement, to information processing. Information processing includes cryptography. Quantum cryptography benefits from operator disagreement: An eavesdropper must observe, or measure, a message. The eavesdropper’s measurement of one observable can “disturb” a disagreeing observable. The message’s sender and intended recipient can detect the disturbance and so detect the eavesdropper.

How efficiently can one perform an information-processing task? The answer usually depends on an entropy $H$ , a property of quantum states and of probability distributions. Uncertainty relations cry out for recasting in terms of entropies. So QI theorists have devised entropic uncertainty relations, such as

$H (\hat{A}) + H( \hat{B} ) \geq - \log c. \qquad (^*)$

The entropy $H( \hat{A} )$ quantifies the difficulty of predicting the outcome $a$ of an $\hat{A}$ measurement. $H( \hat{B} )$ is defined analogously. $c$ is called the overlap. It quantifies your ability to predict what happens if you prepare your system with a well-defined $\hat{A}$ value, then measure $\hat{B}$ . For further analysis, check out this paper. Entropic uncertainty relations have blossomed within QI theory over the past few years.

Pure QI theorists, we’ve seen, quantify operator disagreement with entropic uncertainty relations. Physicists at the intersection of condensed matter and high-energy physics prefer out-of-time-ordered correlators (OTOCs). I’ve blogged about OTOCs so many times, Quantum Frontiers regulars will be able to guess the next two paragraphs.

Consider a quantum many-body system, such as a chain of qubits. Imagine poking one end of the system, such as by flipping the first qubit upside-down. Let the operator $\hat{W}$ represent the poke. Suppose that the system evolves chaotically for a time $t$ afterward, the qubits interacting. Information about the poke spreads through many-body entanglement, or scrambles.

Imagine measuring an observable $\hat{V}$ of a few qubits far from the $\hat{W}$ qubits. A little information about $\hat{W}$ migrates into the $\hat{V}$ qubits. But measuring $\hat{V}$ reveals almost nothing about $\hat{W}$ , because most of the information about $\hat{W}$ has spread across the system. $\hat{V}$ disagrees with $\hat{W}$ , in a sense. Actually, $\hat{V}$ disagrees with $\hat{W}(t)$ . The $(t)$ represents the time evolution.

The OTOC’s smallness reflects how much $\hat{W}(t)$ disagrees with $\hat{V}$ at any instant $t$ . At early times $t \gtrsim 0$ , the operators agree, and the OTOC $\approx 1$ . At late times, the operators disagree loads, and the OTOC $\approx 0$ .

Different camps of physicists, we’ve seen, quantify operator disagreement with different measures: Today’s pure QI theorists use entropic uncertainty relations. Condensed-matter and high-energy physicists use OTOCs. Trust physicists to disagree about what “quantum operator disagreement” means.

I want peace on Earth. I conjectured, in 2016 or so, that one could reconcile the two notions of quantum operator disagreement. One must be able to prove an entropic uncertainty relation for scrambling, wouldn’t you think?

You might try substituting $\hat{W}(t)$ for the $\hat{A}$ in Ineq. ${(^*)}$ , and $\hat{V}$ for the $\hat{B}$ . You’d expect the uncertainty bound to tighten—the inequality’s right-hand side to grow—when the system scrambles. Scrambling—the condensed-matter and high-energy-physics notion of disagreement—would coincide with a high uncertainty bound—the pure-QI-theory notion of disagreement. The two notions of operator disagreement would agree. But the bound I’ve described doesn’t reflect scrambling. Nor do similar bounds that I tried constructing. I banged my head against the problem for about a year.

The sky brightened when Jason and Tony developed an interest in the conjecture. Their energy and conversation enabled us to prove an entropic uncertainty relation for scrambling, published this month.¹ We tested the relation in computer simulations of a qubit chain. Our bound tightens when the system scrambles, as expected: The uncertainty relation reflects the same operator disagreement as the OTOC. We reconciled two notions of quantum operator disagreement.

As Quantum Frontiers regulars will anticipate, our uncertainty relation involves weak measurements and quasiprobability distributions: I’ve been studying their roles in scrambling over the past three years, with colleagues for whose collaborations I have the utmost gratitude. I’m grateful to have collaborated with Tony and Jason. Harmony helps when you’re tackling (quantum operator) disagreement—even if squabbling would spice up your paper’s backstory.

¹Thanks to Communications Physics for publishing the paper. For pedagogical formatting, read the arXiv version.

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Monthly Archives: August 2019

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