Gently yoking yin to yang

The architecture at the University of California, Berkeley mystified me. California Hall evokes a Spanish mission. The main library consists of white stone pillared by ionic columns. A sea-green building scintillates in the sunlight like a scarab. The buildings straddle the map of styles.

So do Berkeley’s quantum scientists, information-theory users, and statistical mechanics.

The chemists rove from abstract quantum information (QI) theory to experiments. Physicists experiment with superconducting qubits, trapped ions, and numerical simulations. Computer scientists invent algorithms for quantum computers to perform.

Few activities light me up more than bouncing from quantum group to info-theory group to stat-mech group, hunting commonalities. I was honored to bounce from group to group at Berkeley this September.

What a trampoline Berkeley has.

The groups fan out across campus and science, but I found compatibility. Including a collaboration that illuminated quantum incompatibility.

Quantum incompatibility originated in studies by Werner Heisenberg. He and colleagues cofounded quantum mechanics during the early 20th century. Measuring one property of a quantum system, Heisenberg intuited, can affect another property.

The most famous example involves position and momentum. Say that I hand you an electron. The electron occupies some quantum state represented by | Psi rangle. Suppose that you measure the electron’s position. The measurement outputs one of many possible values x (unless | Psi rangle has an unusual form, the form a Dirac delta function).

But we can’t say that the electron occupies any particular point x = x_0 in space. Measurement devices have limited precision. You can measure the position only to within some error varepsilon: x = x_0 pm varepsilon.

Suppose that, immediately afterward, you measure the electron’s momentum. This measurement, too, outputs one of many possible values. What probability q(p) dp does the measurement have of outputting some value p? We can calculate q(p) dp, knowing the mathematical form of | Psi rangle and knowing the values of x_0 and varepsilon.

q(p) is a probability density, which you can think of as a set of probabilities. The density can vary with p. Suppose that q(p) varies little: The probabilities spread evenly across the possible p values. You have no idea which value your momentum measurement will output. Suppose, instead, that q(p) peaks sharply at some value p = p_0. You can likely predict the momentum measurement’s outcome.

The certainty about the momentum measurement trades off with the precision varepsilon of the position measurement. The smaller the varepsilon (the more precisely you measured the position), the greater the momentum’s unpredictability. We call position and momentum complementary, or incompatible.

Heisenberg - xkcd

https://www.xkcd.com/1473/

You can’t measure incompatible properties, with high precision, simultaneously. Imagine trying to do so. Upon measuring the momentum, you ascribe a tiny range of momentum values p to the electron. If you measured the momentum again, an instant later, you could likely predict that measurement’s outcome: The second measurement’s q(p) would peak sharply (encode high predictability). But, in the first instant, you measure also the position. Hence, by the discussion above, q(p) would spread out widely. But we just concluded that q(p) would peak sharply. This contradiction illustrates that you can’t measure position and momentum, precisely, at the same time.

But you can simultaneously measure incompatible properties weakly. A weak measurement has an enormous varepsilon. A weak position measurement barely spreads out q(p). If you want more details, ask a Quantum Frontiers regular; I’ve been harping on weak measurements for months.

Blame Berkeley for my harping this month. Irfan Siddiqi’s and Birgitta Whaley’s groups collaborated on weak measurements of incompatible observables. They tracked how the measured quantum state | Psi (t) rangle evolved in time (represented by t).

Irfan’s group manipulates superconducting qubits.1 The qubits sit in the physics building, a white-stone specimen stamped with an egg-and-dart motif. Across the street sit chemists, including members of Birgitta’s group. The experimental physicists and theoretical chemists teamed up to study a quantum lack of teaming up.

Phys. & chem. bldgs

The experiment involved one superconducting qubit. The qubit has properties analogous to position and momentum: A ball, called the Bloch ball, represents the set of states that the qubit can occupy. Imagine an arrow pointing from the sphere’s center to any point in the ball. This Bloch vector represents the qubit’s state. Consider an arrow that points upward from the center to the surface. This arrow represents the qubit state | 0 rangle. | 0 rangle is the quantum analog of the possible value 0 of a bit, or unit of information. The analogous downward-pointing arrow represents the qubit state | 1 rangle, analogous to 1.

Infinitely many axes intersect the sphere. Different axes represent different observables that Irfan’s group can measure. Nonparallel axes represent incompatible observables. For example, the x-axis represents an observable sigma_x analogous to position. The y-axis represents an observable sigma_y analogous to momentum.

Tug-of-war

Siddiqi lab, decorated with the trademark for the paper’s tug-of-war between incompatible observables. Photo credit: Leigh Martin, one of the paper’s leading authors.

Irfan’s group stuck their superconducting qubit in a cavity, or box. The cavity contained light that interacted with the qubit. The interactions transferred information from the qubit to the light: The light measured the qubit’s state. The experimentalists controlled the interactions, controlling the axes “along which” the light was measured. The experimentalists weakly measured along two axes simultaneously.

Suppose that the axes coincided—say, at the x-axis hatx. The qubit would collapse to the state | Psi rangle = frac1 sqrt2  ( | 0 rangle + | 1 rangle ), represented by the arrow that points along hatx to the sphere’s surface, or to the state | Psi rangle = frac1 sqrt2  ( | 0 rangle - | 1 rangle ), represented by the opposite arrow.

0 deg

(Projection of) the Bloch Ball after the measurement. The system can access the colored points. The lighter a point, the greater the late-time state’s weight on the point.

Let hatx' denote an axis near hatx—say, 18° away. Suppose that the group weakly measured along hatx and hatx'. The state would partially collapse. The system would access points in the region straddled by hatx and hatx', as well as points straddled by - hatx and - hatx'.

18 deg

Finally, suppose that the group weakly measured along hatx and haty. These axes stand in for position and momentum. The state would, loosely speaking, swirl around the Bloch ball.

90 deg

The Berkeley experiment illuminates foundations of quantum theory. Incompatible observables, physics students learn, can’t be measured simultaneously. This experiment blasts our expectations, using weak measurements. But the experiment doesn’t just destroy. It rebuilds the blast zone, by showing how | Psi (t) rangle evolves.

“Position” and “momentum” can hang together. So can experimentalists and theorists, physicists and chemists. So, somehow, can the California mission and the ionic columns. Maybe I’ll understand the scarab building when we understand quantum theory.2

With thanks to Birgitta’s group, Irfan’s group, and the rest of Berkeley’s quantum/stat-mech/info-theory community for its hospitality. The Bloch-sphere figures come from http://www.nature.com/articles/nature19762.

1The qubit is the quantum analog of a bit. The bit is the basic unit of information. A bit can be in one of two possible states, which we can label as 0 and 1. Qubits can manifest in many physical systems, including superconducting circuits. Such circuits are tiny quantum circuits through which current can flow, without resistance, forever.

2Soda Hall dazzled but startled me.