Friday, January 12, 2018

Tensor squares entangle.

Reading (arXiv:1208.0928) about quantum error correction, I was very confused for a while when the paper described a tensor product of measurement operators as "simultaneous" measurement. Actually, a tensor product A⊗B corresponds to measuring the arithmetic product of an A measurement value and a B measurement value.

The context involves a pair of particles which we will assume to be electrons for simplicity. Let Xn measure the spin of the nth electron with respect to the x-axis. Applying Xn will put the nth electron's spin in the +x or -x direction; the corresponding macroscopic observation will be +1 or -1 (ignoring physical units). Let Yn be the analog of Xn for the y-axis. If we measure with Xn then Yn then Xn again, then the first and second Xn measurements merely have a 1/2 chance of being the same. In general, if spin of an electron is along one axis and we measure its spin with respect to another axis, we effectively destroy information. (Technically, the information is not destroyed. But recovering it is like unscrambling an egg.)

The above information loss is algebraically manifested as Xn and Yn not commuting. However, these operators do anti-commute: XnYn=-YnXn. Therefore, assuming our two electrons' spin states are independent of each other (which is approximately true if the electrons are not too close together), a little algebra shows that the tensor products X=X1X2 and Y=Y1Y2 do commute. Physically, this means that if we measure with X then Y then X again, the two X measurements will agree with probability 1, not 1/2. If we physically interpret a tensor product as simply performing two measurements at the same time, then this makes no sense.

But X actually measures the product of a potential X1 measurement value and a potential X2 measurement value. This will be +1 if the X1 and X2 both output +1 or both output -1, and will be -1 otherwise. In other words, X is measuring merely whether the two particles have the same or opposite spin with respect to the x-axis. Unlike an Xn, the act of measuring X does not align either particle's spin to the x-axis (unless it was already there). Instead, applying X merely changes the joint state of the particles such that the results of potential future X1 and X2 measurements are now either perfectly correlated or perfectly anti-correlated, depending on whether X measured +1 or -1.

The bottom line is that, by measuring with X and then Y, that is, by measuring with respect to each of two perpendicular axes merely whether our two particles have the same or opposite spins, we put the two particles into one of four maximally entangled joint states with the very nice property that repeated measurements of X and Y will preserve the joint state of the electrons. If the X output changes or the Y output ever changes when performing these repeated measurements, that indicates outside "noise." This is a simple instance of quantum error detection for two qubits. With more electrons, the paper explains how to achieve quantum error correction.

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