Complementary observables

One the more puzzling aspects of the standard measurement axiom of quantum mechanics is that the way the state “collapses” depends on the quantity you choose to measure. (Any quantity that you can measure is called an observable). In the “which path” Double slit experiment, one measures the position of the particle to be in one path or the other. We saw that the interference disappears when information about the position of the particle is unambiguously transferred to another system, a process we are calling “micromeasurement”. The disappearance of interference arose because of entanglement that was generated between the position of the particle and the position of the mirror. What happens if we measure something other than position?

Measuring phase

We will retain the Mach-Zehnder setup from the double slit section, but now we will be interested in the phase difference between the components of the wave packet in the two paths. Recall that “phase” refers to the position of the peaks and troughs of a wave. The relative alignment of the peaks and troughs of two waves is the phase difference. Phase is measured as an angle. Shifting the phase by \(180^\circ\) (or \(\pi\) radians) swaps the peaks and the troughs, and shifting the phase by \(360^\circ\) (or \(2\pi\) radians) leaves the wave unchanged. A good way to measure the phase difference of two waves is to look at their interference when they overlap. The particular location of constructive and destructive interference reveals the phase difference. Specifically, if the phase difference changes by \(180^\circ\) (\(\pi\) radians), the constructive and destructive interference will be swapped.

Interference of two wave packet components with different phase is shown in Fig. 5.1. We start by constructing single-particle wave functions from two components: \(\Psi_a\), representing a wave packet centered at \(x=10\), and \(\Psi_b\), representing a wave packet centered at \(x=-10\). The top panel shows a wave function \(\Psi_0 = \Psi_a + \Psi_b\) that is a superposition summing the two components [1]. The middle panel instead shows the difference \(\Psi_\pi = \Psi_a - \Psi_b\). (The subscript denotes the phase difference between the two components, in radians.) The two initial wave functions can be distinguished by comparing them at the gray dashed lines. On the left side, the red and blue wave functions are opposite, while on the right side they are the same. The bottom panel shows the amplitude squared for both wave functions (initially the two are the same, so only one is visible.) The video in Fig. 5.1 ends when the two components are fully interfering. Now we can see the difference in the bottom panel. The blue plot (corresponding to \(|\Psi_0|^2\)) shows interference fringes in exactly the opposite positions as the red plot (corresponding to \(|\Psi_\pi|^2\)). Where one has a peak, the other has zero value.

Figure 5.1

Video example of the interference between components of a wave packet, for two opposite phase differences. Top and middle panels show real and imaginary components of \(\Psi_0 = \Psi_a + \Psi_b\) and \(\Psi_\pi = \Psi_a - \Psi_b\), respectively. Bottom panel shows amplitude squared for both wave functions.

The two observables – which path the particle is in, and the phase between the two components of the particle – are known as complementary observables. This means that measuring one of them precludes measurement of the other. Using the standard measurement axiom, this is explained by saying that the measurement of one quantity causes the state to collapse into one where any information about the other has been erased. In the Mach-Zehnder interferometer, one would say that measurement of the path collapses the state such that the wave packet is only traveling along one path. Since the other component no longer exists, we cannot find out what the phase difference was. On the other hand, we saw in the previous section that even without invoking this axiom of collapse, the interference disappears when information about the path is transferred. If the interference disappears, then of course we cannot use it to determine the phase.

The opposite should also occur — if we measure the phase difference between the two paths, any information about which path the particle was in should be erased. As before, we will see how this comes about in a specific example just from the behavior of quantum waves, and with no need for any “collapse” via the measurement axiom.

Though Fig. 5.1 allowed us to visually inspect the phase difference revealed by the interference, we need to come up with some way of performing a micromeasurement of the phase difference. That is, we need to transfer information about the phase difference to a second quantum particle. We will make use of the fact that the change in the interference pattern translates the phase difference information into position of the wave (the positions of the peaks and zeros of the interference). We could try to borrow the idea from the previous position micromeasurement: have the second particle reflect (or not) off of the first. Loosely speaking, we could shoot a very small particle at a spot where the interference has a peak at one phase difference and a zero at the other phase difference. This, however, is quite difficult to simulate. For one thing, we would need the wave packet of the second particle to be much smaller than the wavelength of the first, and it is hard to simulate things on such different length scales. For another thing, we would have to do it in two dimensions, which would require a four-dimensional plot to visualize the two-particle wave function.

We can more simply implement a phase micromeasurement by making use of the confined waves discussed in the previous section (Quantized waves). In that section, we saw an example where there were just two allowed frequencies for confined waves in a well. Now we will let the particle in the well interact with a second particle that is in a state described by a travelling wave packet, not confined by the well. One possible interaction of the wave packet with the confined wave can be viewed as tilting the shape of the well. This has the effect of shifting the frequencies of the confined waves. We illustrate this effect in Fig. 5.2. The top panel shows the well with a tilt, whereas the middle panel shows the same untilted well shown in Fig. 4.4. Also as in Fig. 4.4, the initial wave function is the superposition of the sum of both frequencies. The bottom panel shows the amplitude squared for both cases. Again, as we saw in Fig. 4.4, the amplitude squared oscillates back and forth within the well. But now, because the tilt of the well has altered the frequencies, they slosh back and forth at different rates. At the point that the video ends, the two different cases are almost completely opposite.

Figure 5.2

Video example of the effect of an interaction on confined waves in a well.

The effect illustrated in Fig. 5.2 gives us a way to measure the phase difference between two components of a wave packet. We just need an interaction such that when the wave packet overlaps the well, the confined wave frequencies are shifted.

But let’s take a moment to address a few issues with our idea to use confined waves to detect the phase difference. One issue is that we still need the well to be narrow compared to the wavelength of the traveling wave packet so that it can be positioned right at the peak or trough of the interference. As mentioned before, it is hard to simulate things at two very different length scales. To get around this, we will not directly simulate the waves confined in a narrow well, but just use what we have already learned about the behavior of the confined waves to simulate the evolution of the state more abstractly as a two-level quantum system. Another issue is that the assumption of a well violates our self-imposed rule against invoking “barriers” of unspecified origin. In principle, we could get around this by replacing the “well” with the attractive potential of a third particle. If we make that third particle very heavy, then we can approximate it as being fixed in position (as we verified by the heavy mirrors and beam splitter in Figs. 3.2, 3.3, and 3.4.) But actually, what we are describing now is just an atom. The the third heavy particle is the positively charged nucleus of the atom, which is much heavier than a negatively charged electron bound to it. And in the case of atoms, the frequency shift from tilting the “well” is known as the Stark effect. Of course, atoms are three-dimensional objects so there are some differences in the waves describing an electron in an atom and the 1D waves shown here, but the essential ideas are the same.

Henceforth, for convenience, we will refer to the well with confined waves as an “atom.” We’ll start off with the particle confined in the atom in the superposition where the two waves are added. We’ll refer to this as the “\(|+\rangle\)” state of the atom. (If you haven’t seen it, don’t worry about the \(|...\rangle\) notation — it is just the conventional way to denote that we are talking about a quantum state.) Then what we are looking for is: when the atom interacts with the wave packet, does the atom stay in the \(|+\rangle\) state, or evolve into the \(|-\rangle\) state, where the two waves are subtracted?

For the experts

We will assume, for the sake of simplicity that the two energy eigenstates of the atom, \(|0\rangle\) and \(|1\rangle\) are degenerate (alternatively we could transform to a rotating frame). Taking a contact interaction between the wave packet and the point-like atom, we write an interaction Hamiltonian \(H_{int} = g \delta(x - x_0) \otimes \sigma_z\), where \(g\) sets the strength of the interaction, \(x_0\) is the position of the atom, and \(\sigma_z\) is a Pauli matrix. The atom is initially in the state \(|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}\). If the wave packet interference is such that its wave function is zero at \(x_0\), then there is no interaction and the atom stays in the state \(|+\rangle\). If the wave function of the wave packet is not zero at \(x_0\), then the atom’s state will evolve in time as the degeneracy is lifted, and the wave packet will scatter off of a delta-function potential. If the two components of the wave packet with momentum \(\pm \hbar k_0\) interfere constructively at \(x_0\), then it turns out that setting \(g = m/\hbar k_0\) causes a near-optimal rotation from \(|+\rangle\) to \(|-\rangle\) (becoming more optimal as the width of the packet increases, and the spread of momenta decreases.) Note that the potential \(V = \pm (m/\hbar k_0)\delta(x)\) has 50% transmission for a particle with momentum \(\hbar k_0\).

As mentioned above, in order to use the atom as a detector of the wave packet phase difference, we will assume we have an interaction that shifts the atom frequencies, depending on the overlap of the atom with the wave packet. If there is no overlap, then the atom will just sit there in the state \(|+\rangle\). This is the case when the wave packet is far away from the atom. But it is also the case when the atom is positioned precisely at a trough of destructive interference. On the other hand, if the position of the atom coincides with a peak of constructive interference, then the frequencies shift and the atom’s state starts to evolve towards the \(|-\rangle\) state.

The trick to making the atom sensor work will be to set the strength of the interaction (how much the frequencies shift for a given wave function overlap), such that complete constructive interference for a given wave packet results in a perfect switch from \(|+\rangle\) to \(|-\rangle\). This amounts to a calibration of our (micro)measurement device. It turns out that this calibration depends only on the speed of the wave packet. A faster wave packet requires a stronger interaction, since it overlaps the atom for less time.

Before we demonstrate the atom phase sensor, we need to remind ourselves about how we describe the quantum state of two particles. In general, we cannot describe them separately. When we described two wave packets, we specified amplitudes for the particles to be at pairs of positions, and we had to visualize the state of two particles in 1D using a 2D plot (see Quantum waves). This allows for the all-important entanglement. We have simplified things with our atom sensor, since we aren’t actually looking at the wave function of the particle in the atom. We have instead just boiled all those complicated waves down to two distinct possibilities of interest: \(|+\rangle\) and \(|-\rangle\). As such, the combined state of the wave packet and the atom sensor must be described as amplitudes for the position of the wave packet, paired with either \(|+\rangle\) or \(|-\rangle\). Now we don’t need a full 2D plot — we can just use two 1D plots (or you could think of this as a 2D plot where one axis only has two pixels since we have distilled that system to just two states). So one 1D plot will represent a wave function for the wave packet that is paired with (or correlated with) the \(|+\rangle\) state of the atom, and the other 1D plot will represent the wave function for the wave packet that is paired with the \(|-\rangle\) state of the atom.

At last, a demonstration of the atom phase sensor. First, we’ll look at the more boring case, where the sensor does nothing. In Fig. 5.3, the wave packet begins in the state \(\Psi_\pi\), which is a wave packet on the left minus a wave packet on the right. The dashed lines on the right and left sides are guides to help us see that two wave packet components have opposite sign. The atom sensor is located at \(x=0\), as indicated by the center dashed line. As per the plan, the atom is initially in the state \(|+\rangle\). The top panel of Fig. 5.3 shows the wave packet paired with \(|+\rangle\), and the bottom panel shows the wave packet paired with \(|-\rangle\). Because the initial state of the atom is just \(|+\rangle\), only the top panel has a nonzero wave function to start. Here, we are showing both the real and imaginary parts of the wave function (thin solid and dotted lines), along with the amplitude squared (thick line) all on the same plot to save some space.

Figure 5.3

Video example of the atom phase sensor interacting with a wave packet \(\Psi_\pi\) with two out-of-phase components. The top panel shows the real part (thin solid), imaginary part (dotted), and amplitude squared (thick) of the wave function paired with \(|+\rangle\), and the bottom panel shows the same quantities paired with \(|-\rangle\). Atom sensor is located at the central dashed line. Left and right dashed lines are guides to help distinguish the relative sign of the packets on both sides.

When the two components of the out-of-phase wave packet combine in Fig. 5.3, we see the usual interference pattern, with a zero value right at the central dashed line. This is what we wanted, so that there is no interaction with the atom sensor, which remains in the state \(|+\rangle\). Indeed, we see that the wave packet in the top panel evolves as we have seen before (e.g. Fig. 1.6 or Fig. 5.1), and (almost) nothing happens in the bottom panel corresponding to \(|-\rangle\). (A small wiggle appears in the bottom panel due to imperfect calibration of the sensor. A wider packet compared to the wavelength improves the performance.) But ignoring that small wiggle in the bottom panel, the final state is not entangled — we can consider the wave packet with its final wave function separately from the atom in state \(|+\rangle\). Note that the final state of the wave packet still has opposite phase in the two packets (check the sign of the wave function at the left and right dashed lines.)

Now let’s look at the other case: when we start the wave packet with the same phase on both sides. Figure 5.4 starts with the wave packet in the state \(\Psi_0\), which is a wave packet on the left plus a wave packet on the right. We can verify that the two wave packets have the same phase by comparing the sign at the left and right dashed lines. As always, we start the atom sensor in the state \(|+\rangle\), so the wave function is initially entirely in the upper panel.

Figure 5.4

Video example of the atom phase sensor interacting with a wave packet \(\Psi_0\) with two in-phase components. The top panel shows the real part (thin solid), imaginary part (dotted), and amplitude squared (thick) of the wave function paired with \(|+\rangle\), and the bottom panel shows the same quantities paired with \(|-\rangle\). Atom sensor is located at the central dashed line. Left and right dashed lines are guides to help distinguish the relative sign of the packets on both sides.

In the in-phase case shown in Fig. 5.4, the behavior is quite different when the two wave packet components overlap. Now the wave function is not zero at the atom’s position, and the interaction has an effect on the state. Specifically, we see the wave function disappear from the top panel and reappear in the bottom panel. This means that the state of the atom is rotating from \(|+\rangle\) to \(|-\rangle\). Once the wave packet components are no longer overlapping, the wave function is essentially all in the bottom panel (again ignoring the slight wiggle now left in the top panel). As in the out-of-phase case, the final state is also not entangled. We could consider the final wave packet and the atom in the state \(|-\rangle\) separately. We also note that the two components of the final wave packet are still in phase with each other (look again at the sign of the wave function at the left and right dashed lines.)

The atom phase sensor works! It did just what we wanted: when the initial wave packet components had opposite phase the atom stayed in the state \(|+\rangle\), and when they had the same phase the atom switched to the state \(|-\rangle\), as if it were a pointer on a measurement device swinging over from “out-of-phase” to “in-phase”. Now let’s take a closer look at how the phase micromeasurement compares to the position micromeasurement, and to our notion of measurements in general.

Repeated measurements

First let’s think about the case where the particle is initially in a state where the quantity we are measuring is well-defined. For a position measurement, the well-defined state would be one with a single wave packet component. For the phase measurement, the well-defined state is the case we have been looking at where we have two components with some phase difference between them.

We haven’t really discussed position micromeasurement starting from a state with well-defined position, because that case is almost trivial. We skipped right to the more complicated case where the initial state was a superposition of two positions. But to take an example of this, we can consider a wave packet for “particle 1” starting at \(x=0\) and either moving to the left or the right. If we position a “mirror” particle to the left, then the two particles will collide if particle 1 is moving to the left, and will not collide if particle 1 is moving to the right. Whether the mirror particle recoils or not is then a micromeasurement of whether particle 1 is on the left side or the right side. Just as in the Double slit example, the mirror particle must be light enough that we can unambiguously distinguish the recoil case from the no recoil case.

One feature that we expect for a measurement is that you get the same result if you repeat the measurement. If we start in a state with some quantity well-defined, then measure that quantity, then measure it again, the two measurement outcomes should be consistent with each other (unsurprisingly). We can see that this is the case both for the position and phase micromeasurements. In the case of position, if we place a second mirror particle somewhere on the left side so that particle 1 hits it after bouncing off of the first mirror, then if the particle is indeed going to the left, both mirrors will recoil. Or if the particle goes to the right, neither mirror recoils. In the case of phase, we saw that after the interaction with the atom sensor, the wave packet components still had the same phase difference. That means we could reflect each of the components off of mirrors (heavy mirrors so that they don’t recoil and accidentally measure position) and have them overlap a second time, interacting with a second atom phase sensor. We should get the same result — if the first atom sensor gave the result \(|+\rangle\), so will the second (and vice versa).

But what if we start in a state with a quantity not well-defined, then perform a repeated measurement of that quantity? By “not well-defined,” we mean in a superposition of two distinctly separated values. So for position, this is the case we looked at in the Double slit example. The particle is in a superposition of two distinct positions. As discussed there, the micromeasurement, unlike a normal measurement, does not just give a single result. Instead, it gives an entangled state where both results have occurred. Specifically, we can find a state where the particle is in path A and the mirror has recoiled, and also the particle is in path B and the mirror has not recoiled. What if we then place a second mirror particle, just as we did in the previous paragraph, so that a particle in path A would hit it? Then the component of the state with the particle in path A and mirror 1 recoiled would also cause mirror 2 to recoil. And the component of the state with particle in path B and mirror 1 unrecoiled would also leave mirror 2 unrecoiled. So the micromeasurement again does not yield a single result, but we see that the results are correlated within each component of the entangled state.

Let’s look at the analogous situation for phase measurements. So we will start in a state that does not have a well-defined phase, and try performing two repeated phase measurements. Taking a cue from the position case, we will start in a state that is a superposition of two waves with distinctly different well-defined phases. Our states with well-defined phase are \(\Psi_0 = \Psi_a + \Psi_b\) and \(\Psi_\pi = \Psi_a - \Psi_b\) above. A superposition where we add the two together simply yields \(\Psi_0 + \Psi_\pi = 2 \Psi_a\). (We first introduced this sort of basis switcheroo back at the beginning of the Classical waves section.) Since we don’t care about the overall amplitude, we’ll just ignore the factor of two. Our state with a superposition of distinctly different phases is just the packet with a well-defined position! Just like how our state with two distinctly different positions is our state with well defined phase. This is the idea of complementary observables.

For the experts

We have defined two complementary observables so far. The two observables have binary outcomes: two positions (\(\Psi_a\), \(\Psi_b\)), two phases (\(\Psi_a \pm \Psi_b\)). Those familiar with qubits will note that there should be a third complementary observable, with outcomes \(\Psi_a \pm i\Psi_b\). Indeed we could extend our discussion to probe this observable as well. Neither micromeasurement described so far distinguishes between the two states \(\Psi_a \pm i\Psi_b\). It should be clear why the position micromeasurement does not measure this other phase difference. As for the phase micromeasurement, recall that it distinguishes phases via the peaks and zeros of the interference pattern. To measure the difference between \(\Psi_a \pm \Psi_b\), we have placed the atom at a position where the interference is either a peak or a zero. If we try to measure \(\Psi_a \pm i\Psi_b\), we would find that both states yield 50% interference at the atom position (one at the rising edge of a fringe, one at the falling edge). Thus the two states are not distinguished by the sensor, which only depends on the overlap with the wave function at the atom position. If we want to make a sensor to distinguish \(\Psi_a \pm i\Psi_b\), we should just shift the position of the atom sensor by \(\lambda/8 = \pi/4k\). Now it will be sensitive to the difference between \(\Psi_a \pm i\Psi_b\) and not the other two observables.

With this picture, we see that the three complementary observables map directly onto the usual picture of a two-level system or qubit. We can picture the state of a particle as a vector on the Bloch sphere with the states \(\Psi_a\) and \(\Psi_b\), \(\Psi_a \pm \Psi_b\), and \(\Psi_a \pm i\Psi_b\) located along three mutually perpendicular axes. The micromeasurements are only deterministically sensitive to one pair of orthogonal states. We are free to view each of these pairs of orthogonal states as eigenvectors of some \(2 \times 2\) Hermitian operator. And if we wish to assign some real values to the outcomes of the measurements, we can construct those operators so that the measurement outcomes are the eigenvalues. This yields the observable operator called for in the standard quantum measurement axiom.

So let’s find out what happens when we attempt to measure the phase difference on a state that does not have a well-defined phase. Figure 5.5 shows the initial single wave packet \(\Psi_a\) interacting with the atom phase sensor. In a way, it is obvious that this initial state does not have a well-defined phase difference, since there is no other wave packet component with which to compare phases. But we have seen mathematically that this single wave packet can be viewed as a superposition of a pair of packets that are in-phase and a pair of packets that are out-of-phase.

Figure 5.5

Video example of the atom phase sensor interacting with a single wave packet component \(\Psi_a\), which can be viewed as a superposition of in-phase and out-of-phase wave packet pairs. The top panel shows the real part (thin solid), imaginary part (dotted), and amplitude squared (thick) of the wave function paired with \(|+\rangle\), and the bottom panel shows the same quantities paired with \(|-\rangle\). Atom sensor is located at the central dashed line. Left and right dashed lines are guides to help distinguish the relative sign of the packets on both sides.

So what happened when the single wave packet interacted with the atom sensor in Fig. 5.5? Well, exactly what should have happened. We started in an equal superposition of \(\Psi_0\) and \(\Psi_\pi\), and the result was an equal superposition of the results following from those states, as seen in Fig. 5.3 and Fig. 5.4. Now the final state is an entangled state. Just as with the position case, the micromeasurement has not provided a single outcome, but an entangled state representing both outcomes occurring. The final state shown in Fig. 5.5 represents out-of-phase wave packets paired with \(|+\rangle\), and in-phase wave packets paired with \(|-\rangle\). If we look carefully at the positions of peaks and troughs relative to the dashed lines in Fig. 5.5, we can see that in the top panel, there is a peak in the real part on the left, and a trough on the right. In the bottom panel, there are peaks in both the imaginary parts (dotted lines) on both sides. This indicates that, in addition to the top panel being out of phase, and the bottom panel in phase, there is a \(90^\circ\) (\(\pi/2\) radian) phase shift in the bottom panel relative to the top.

It is all well and good to say that of course Fig. 5.5 came out like it did based on the idea of superposition, but the question remains: why did the wave function split up like that? In the cases of well-defined phase in Fig. 5.3 and Fig. 5.4, it appeared that the wave packets pretty much just traveled right past the atom sensor, either switching the atom’s state or not. But in this case, it seems that the single wave packet is evenly split into two. Something is going on that is similar to Newton’s law of action and reaction: when the wave packet causes a change to the atom’s state, the atom’s state also causes a change to the wave packet. Specifically, when the wave packet causes a shift in the atom’s frequencies, the atom acts as a narrow barrier for the wave packet. As we have seen before (e.g. Fig. 1.5 or Fig. 3.2), a narrow barrier can split a wave packet into two — one component that reflects, and another that tunnels through.

You might think I have very carefully arranged the situation so that the atom sensor gets flipped from \(|+\rangle\) to \(|-\rangle\) for different phases, and also splits a single packet exactly 50/50. But in fact, calibrating the system to one of those behaviors automatically guarantees the other. Otherwise we would get inconsistent results when considering different superpositions. We might think it would be nice to have a phase sensor that just passively measures the phase without affecting the wave packet. But this is not possible — if we make the interaction weaker, then the atom state will not change all the way from \(|+\rangle\) to \(|-\rangle\).

Now for the second measurement of phase. Let’s take the end result of Fig. 5.5, reflect the packets on both sides back towards the center, and swap the atom sensor out with a fresh one initialized to the state \(|+\rangle\). We can think of this initial state as an entangled superposition of two things: A. wave packet in the state \(\Psi_\pi\), atom 1 in its final state \(|+\rangle\), and atom 2 in its initial state \(|+\rangle\); and B. wave packet in the state \(\Psi_0\), atom 1 in its final state \(|-\rangle\), and atom 2 in its initial state \(|+\rangle\). Now because we have swapped out atom 1 its state will no longer change, and we just need to think about how the wave packet interacts with atom 2. Let’s consider the components A and B separately, then add them back together. In the component A, we know that the out-of-phase state \(\Psi_\pi\) leaves the atom unchanged, and so in that component, atom 2 will remain in the state \(|+\rangle\). On the other hand in component B, the in-phase state \(\Psi_0\) will switch the atom 2 state to \(|-\rangle\). Putting them back together we obtain the final entangled state with two parts: A. \(\Psi_\pi\), \(|+\rangle\), and \(|+\rangle\); and B. \(\Psi_0\), \(|-\rangle\), and \(|-\rangle\), where we are specifying the states of the wave packet, atom 1, and atom 2, respectively. Again, as with repeated position measurements, we find that the repeated phase micromeasurement leaves us in an entangled state with both outcomes occurring, but with the repeat outcomes correlated with each other.

Complementary measurements

Now we turn to the interesting case of repeated measurement of two different complementary observables. As mentioned at the beginning of this section, we expect that a measurement of one observable will somehow erase any information about a complementary observable. For example, if we start in a state with well-defined position, carry out a phase measurement, then carry out a position measurement, we should not be able to learn what the original position was. Or alternatively, we could start with a state with well-defined phase difference, make a position measurement, then a phase measurement. Again, we should not be able to find out the original phase.

We’ll start with the first case: starting with well-defined position, measuring phase, then measuring position. According to the standard measurement axiom, one would say that the phase measurement “collapses” the initial state at random to one with a well-defined phase (and thus a not well-defined position.) The subsequent position measurement then chooses at random from the two or more positions in the new state. Now we will see in our picture of micromeasurements, we replace “collapse” with the formation of entangled states.

In our example of micromeasurements, the initial state with well-defined position is a single wave packet. We just saw in Fig. 5.5 what happens when we perform a micromeasurement of phase on this state: the single wave packet splits evenly into two spatially separated components. The two components are out of phase when paired with the \(|+\rangle\) state of the atom phase sensor, and in phase when paired with the \(|-\rangle\) state. But next, we want to measure the position, and for that the phase doesn’t matter. Ignoring the phase, the effect of the atom sensor is very similar to the beam splitter that we saw back in Fig. 3.2 in the Double slit section. In fact, the rest of the position measurement follows exactly as shown in Figs. 3.3 and 3.5. The only difference is that we should remember that the state has two entangled components corresponding to the two states of the atom sensor. The second measurement is shown in Fig. 5.6. The left and right panels shows the position micromeasurement correlated with \(|+\rangle\) and \(|-\rangle\), respectively. Both panels look the same, because the two components affect the position measurement in the same way — one part of the wave packet causes the mirror to recoil, and the other doesn’t. In the end, there are four entangled components, two of which have the mirror recoiled, and two that have the mirror not recoiled. Remember that we started with a single wave packet. If we had performed the micromeasurement of position initially, the mirror would have either definitely recoiled or not, transferring information about the particle’s position. Instead, we get half of the wave function components causing recoil and half causing no recoil, so the recoil of the mirror does not provide information about the original position.

Figure 5.6

Video example of a position micromeasurement on the two components resulting from a micromeasurement of phase on the state \(\Psi_a\).

So, did the phase measurement somehow erase the information about the initial position of the particle? Not exactly – it is just harder to access now. The final micromeasurement of position “failed” in that the mirror did not deterministically recoil, or not. But the information about the initial position of the wave packet was still contained in the state. Specifically, it is contained in the particular phases of the different parts of the wave function. The state after the first phase measurement is the final state of Fig. 5.5. If we had chosen the other well-defined position for the initial state (coming in from the right), then the two packets correlated with \(|+\rangle\) would be swapped. In this way, the state retains information about the initial position. Can we perform some micromeasurement to transfer that information to another system? Not easily. If we are measuring phase differences, the only way to distinguish the two initial states is by some process that compares the phases of the packets correlated with \(|+\rangle\) to those correlated with \(|-\rangle\). That is, after the phase measurement the position information is no longer encoded in any one system (neither particle 1, nor the atom) but instead is encoded in the relative phase between multiple entangled components of the state. So no micromeasurement of either particle 1 or the atom can transfer information about the original position. We would need to involve both the atom and the particle in the process of extracting the information. In a sense, we would need to undo the entanglement created by the phase measurement. This will be somewhat more clear when we consider the next case.

Now let’s look at the opposite case: we start with an initial state with well-defined phase, then measure position, then measure phase. We’ll start with the state \(\Psi_0\), with two spatially separated in-phase components. The initial measurement of position is exactly what was depicted in Fig. 3.5 for the “which path” measurement in the double slit experiement. We show the same thing again here in Fig. 5.7. As before, the state winds up as two entangled components: the wave packet reflected and the mirror recoiled, and the wave packet not reflected and the mirror undisturbed. If we had instead started with the out-of-phase state \(\Psi_\pi\), Fig. 5.7 would look the same, because we don’t show the phases in the plots of amplitude squared. But the difference would be that the two blobs would be out of phase.

Figure 5.7

Video example of a position micromeasurement on the state \(\Psi_0\).

Starting now from the final state of Fig. 5.7, we want to carry out a phase measurement on the particle. As usual, we place the atom phase sensor at the position where the two components of particle 1 overlap. The whole idea of the atom phase sensor was based on an interaction with the interference fringes when two wave packet components overlap. But as we know well by now, a “which path” micromeasurement causes the interference fringes to disappear. So it is no surprise then that we can no longer measure the original phase difference of the two components.

The situation is depicted in Fig. 5.8, with the white dotted lines indicating the position of the atom phase sensor. Now we are depicting the state of two particles (particle 1, and the mirror) as well as an atom phase sensor. Therefore, the state must be described by triples: the position of particle 1, the position of the mirror, and the state of the atom, either \(|+\rangle\) or \(|-\rangle\). As such, we can visualize it using two 2D plots, one paired with \(|+\rangle\) and one paired with \(|-\rangle\). Since the atom sensor starts in the state \(|+\rangle\), the wave function is initially shown only in the left panel.

Figure 5.8

Video example of a phase micromeasurement on the two-particle state resulting from a micromeasurement of position on the state \(\Psi_0\).

Recall that we are showing the positions of both particles (particle 1, and the mirror particle) in one dimension. When the two yellow blobs in the figure move past each other in the horizontal direction, the particle 1 components are overlapping in space. But they do not interfere because they are correlated with the two spatially separate components of the mirror particle.

We can understand what is going on in Fig. 5.8 by separately thinking about the two components of the mirror particle, one recoiling and the other on its original trajectory. In this figure, the mirror particle is not interacting with anything (we have switched off the interaction with particle 1 for simplicity). The motion of the mirror particle gives rise to the vertical motion of the blobs — one going up and the other going down. Since the mirror particle is not interacting with anything, this motion will remain constant. The horizontal motion of the blobs represents the two components of particle 1 approaching each other. When they cross at the atom phase sensor, the two blobs do not overlap due to the entanglement with the mirror particle. Instead, each blob acts like a single wave packet interacting with the atom sensor. The upwards moving blob is like a wave packet approaching from the right, and the downwards moving blob is like a wave packet approaching from the left. Then, they do just what we have seen when we measure the phase of a single wave packet: they split into two equal components, an in-phase pair correlated with the atom state \(|-\rangle\) and an out-of-phase pair correlated with \(|+\rangle\). This happens for both the upwards-going and downwards-going mirror components, resulting in the end with four pairs of in-phase or out-of-phase wave packet components.

It might look like the final state in Fig. 5.8 is not entangled — it has the symmetric pattern of blobs that we associated with separable states (e.g. Fig. 2.2), but we are neglecting to take into account the different phases of the blobs. Because it is not visible in the plot of amplitude squared, we label the relative phases of the different packets at the end of the video in Fig. 5.8. We see that the waves making up these different blobs are not actually identical. In the \(|+\rangle\) panel, there are two pairs of out-of-phase packets, and in the \(|-\rangle\) panel, there are two pairs of in-phase packets. (The fact that the \(|-\rangle\) packets all have a \(\pi/2\) phase shift happens because of the atom sensor, and is not important.) The important point is that the initial state had a particular phase difference (0), and had we just measured that state with an atom sensor we would have found that the atom state deterministically ended up as \(|-\rangle\). But in the final state of Fig. 5.8, there are equal numbers of components correlated with the \(|+\rangle\) and \(|-\rangle\) states. The phase micromeasurement has failed to transfer information about the original phase difference to the atom.

But again, we can ask if the position micromeasurement somehow erased the original phase information. In the final state of Fig 5.7, the two entangled blobs still have the original phase difference between them. Again, the phase is still encoded in the overall state, just not in any one particle. This makes it more difficult to retrieve, in that no single micromeasurement on a given particle can obtain the information. This is a difference between the standard measurement axiom and our micromeasurements. The standard axiom says that when a “measurement” occurs, the state “collapses” and the complementary information vanishes instantly. In the micromeasurement example, we see that the information is not gone, but that the micromeasurement that would have worked initially to obtain the information no longer works. Moreover, we see that because the information is now shared between multiple entangled particles, we are going to need to involve multiple particles to get the information out.

Specifically, in order to retrieve the phase information following the position micromeasurement, we need to get the two yellow blobs to cross. That is the whole idea behind the phase micromeasurement. At the end of Fig 5.7 or the beginning of Fig 5.8 we see that the blobs are heading away from each other in the vertical direction as the two components of the mirror diverge, and thus do not overlap. One way to get them to cross would be to reflect both components of the mirror particle (with a heavy mirror to avoid additional entanglement) to reverse their trajectories. If we time it just right, then the components of the mirror particle will overlap at the same time as the components of particle 1, which means that the yellow blobs overlap, and will interfere. In this case, we at least have a chance of getting the phase micromeasurement to work. So we see that we have to involve both particles in the process, and that it relies on getting the timing just right.

So what gives rise to the difference between a micromeasurement and a standard axiomatic measurement? Here’s one idea. We are claiming that when the information is shared between two entangled particles, it is difficult though possible to transfer that information to a third particle. What if the information is shared between three particles? It would stand to reason that the information is still there, but that it would be even more tricky to extract. We would have to “disentangle” three particles. And as we increase the number of particles over which the information is shared, it will become vastly more difficult to arrange a scheme to get the information out. Perhaps this explains the difference: it is not that the information is really gone in a standard measurement, but that so many particles become involved in a real laboratory measurement that there is no hope of ever “disentangling” them to get the information about a complementary observable out.

A final thought for this section: Note that in analyzing various situations above, we often found it useful to consider some entangled components separately. Looking at our two-particle 2D plots, we can often consider each yellow blob separately as they move or reflect. The only place where this would lead us astray is when two blobs overlap, and thus interfere. Considering them separately, we would miss the interference. Of course, we get the interference back if we add the blobs back together after considering them separately. But if we are confident that two blobs will never overlap, we don’t need to add them back together and can just consider them to be completely separate, as if they exist in different worlds.

Footnotes

[1]

Aficionados of quantum mechanics might notice that we haven’t bothered normalizing these wave functions. Normalization is only needed to make Born’s rule come out right, and since one of the main points here is to avoid Born’s rule - we don’t need to worry about normalization!