Continuing an apparent recent theme on this blog, I’d like to discuss a particular application of phase retrieval, also known as phaseless reconstruction or reconstruction without phase. Specifically, I wish to discuss a relationship with a particular problem in quantum mechanics, but this requires a brief introduction to the field of study. Luckily, I’ve read quite a bit on the topic over the past week, so I am ready to discuss in some detail, although I must warn the reader that I am not an expert by any means—I defer to Scott Aaronson on all things quantum.
Quantum mechanics concerns physical systems which operate on a particularly small scale; specifically, when the action (energy times time) involved is on the order of Plank’s constant erg-sec. While classical mechanics fails to predict reality in this small-scale domain, quantum mechanics both predicts reality and agrees with classical mechanics as the scale grows. But quantum mechanics gains prediction accuracy at the price of intuition. Luckily for me, I lack any semblance of physical intuition, even at the macroscopic level (here’s an example where my intuition fails), and so this isn’t much of a sacrifice.
A quantum mechanical system is completely characterized by its state, defined to be a unit vector in some separable complex Hilbert space , and two states are considered equivalent if they are equal up to a global phase factor. This Hilbert space can be finite- or infinite-dimensional, depending on the system, but it’s often . We shall consider the state to be unknown to the experimenter. In order to gather information about the state, the experimenter performs a measurement using an observable, which is defined to be a self-adjoint operator on . The outcome of the measurement is a random variable whose distribution depends on the state and the observable, say . In the finite-dimensional setting, this probability distribution is relatively straightforward: the outcome of the measurement is the eigenvalue of with probability , where denotes the orthogonal projection onto the eigenspace of corresponding to . Note that this is, indeed, a probability distribution, as is a unit vector and the eigenspaces of the self-adjoint operator are mutually orthogonal.
When is infinite-dimensional, the spectral theorem guarantees that is unitarily equivalent to some multiplication operator on , where is some measure space. Say the multiplication operator is multiplication by . With this, we can build a projection-valued measure: for any measurable set , unitarily map back the operation of multiplication by the indicator function to form the orthogonal projection on . Then if we measure the state with the observable , the probability that the measurement outcome is in is given by . We will continue to express our projections in this way, since the analysis also works in the finite-dimensional setting.
Immediately after the measurement, the state of the system is changed (or, makes a “quantum leap”) to , where is the outcome of the measurement; this new state is well-defined provided the outcome satisfies , which occurs with probability one. Notice that if the experimenter measures the same system with the same observable twice in a row, then both outcomes will be the same with probability one, since
this is known as the projection postulate. Actually, the probability could be slightly less than one since the state of the system will conceivably have time to evolve a little between measurements; this evolution is governed by Schrodinger’s equation:
where denotes the Hamiltonian operator, given by a weighted sum of the Laplacian and a system-specific potential energy field. When , this indicates that needs to be twice continuously differentiable; it is unclear how this could be maintained after a quantum leap to, say, a Dirac delta function—but then again, I haven’t studied the topic for very long (yet).
Now that we have some understanding of quantum mechanics, I return to the goal of this entry: to discuss phase retrieval. Specifically, I turn to the following paper:
In this paper, Vogt considers two particular observables called position and momentum. In , position is the multiplication operator defined by for every and . This makes intuitive sense because the position measurement outcome will have a probability density function defined by for all . However, this is an example where the end state of a quantum leap is apparently a Dirac delta function—here, the location of the point mass is identified by the measurement outcome. Also concerning is the fact that apparently maps some functions, such as those which decay like , out of ; but observables only need to be densely defined, so perhaps that’s not a problem. The other observable considered is called momentum , which is less intuitive (at least to me). Here, the operation is differentiation, which is a multiplication operator in the Fourier domain. Specifically, momentum can be decomposed as .
Vogt’s paper is specifically concerned with the distributions of with respect to position and momentum. We already established that the distribution with respect to position is . Considering the decomposition of , the distribution with respect to momentum is , up to scale factors. As indicated by the title of his paper, Vogt demonstrated that the position and momentum distributions together fail to determine the state . Specifically, he constructed a state such that its point-wise complex conjugate shares the same position and momentum distributions. Note that this is true for any for which is even, as ; we just need to guarantee that , which we can impose by making sure is not real everywhere.
What this indicates is that there is not enough “phaseless information” in the position and momentum distributions to completely determine the state. Vogt’s analysis directly caries over to the finite-dimensional setting, where is a diagonal matrix with and is the circulant finite difference matrix. In this setting, Vogt’s result states that phaseless measurements of with the identify and Fourier bases fail to determine up to a global phase factor. But he also suggests that the state might be recovered with slightly more information. Specifically, he notes that by letting and evolve according to Schrodinger’s equation, their position and momentum distributions appear to become different, meaning the time operator may distinguish states which were previously indistinguishable.
Letting time pass is an example of a physical symmetry operation which, in accordance with Wigner’s theorem, corresponds to a unitary (or sometimes an anti-unitary) operator on the quantum state Hilbert space . I conjecture that there exists a physical symmetric operation which corresponds to a unitary operator on such that is completely determined by the point-wise absolute values of , , and . In the finite-dimensional setting, this would produce injective phaseless measurements of -dimensional vectors, which appears to be near-optimal.