Uniqueness of phase retrieval from three measurements

Abstract

In this paper, we consider the question of finding an as small as possible family of operators $(T_j)_{j\in J}$ on $L^2({\mathbb {R}})$ that does phase retrieval: every $\varphi$ is uniquely determined (up to a constant phase factor) by the phaseless data $(|T_j\varphi |)_{j\in J}$. This problem arises in various fields of applied sciences where usually the operators obey further restrictions. Of particular interest here are so-called coded diffraction patterns where the operators are of the form $T_j\varphi ={\mathcal F}[m_j\varphi ]$, ${\mathcal F}$ the Fourier transform and $m_j\in L^\infty ({\mathbb {R}})$ are “masks”. Here we explicitly construct three real-valued masks $m_1,m_2,m_3\in L^\infty ({\mathbb {R}})$ so that the associated coded diffraction patterns do phase retrieval. This implies that the three self-adjoint operators $T_j\varphi ={\mathcal F}[m_j{\mathcal F}^{-1}\varphi ]$ also do phase retrieval. The proof uses complex analysis. We then show that some natural analogs of these operators in the finite dimensional setting do not always lead to the same uniqueness result due to an under-sampling effect.

Appendix

The purpose of this section is to provide some background on Wright’s conjecture, as formulated in Conjecture 1.1.

We begin with introducing the main objects and notions appearing in quantum mechanics that we need here. The space of all possible states of a quantum mechanical system is represented by $L^2(\mathbb {R})$. A state $\psi \in L^2(\mathbb {R})$ is also called a wave function. Two wave functions $\psi$ and $\varphi$ are considered equivalent if they agree up to multiplication by a unimodular constant.

Quantities of a system that can be measured are called observables and represented by densely-defined self-adjoint operators on $L^2(\mathbb {R})$. The expected value of the state $\psi \in D(A)$ in the observable A is defined as

$$\begin{aligned} E_\psi (A)={\left\langle {\psi ,A\psi }\right\rangle }. \end{aligned}$$

The two following examples are essential in this paper. Let $u,v\in L^\infty ({\mathbb {R}})$ be real valued. To u and v associate the following two observables¹

$$\begin{aligned} M_u\varphi =u\varphi \qquad \text{ and }\quad {\mathcal M}_v \varphi ={\mathcal F}^*\bigl [v{\mathcal F}[\varphi ]\bigr ]. \end{aligned}$$

Then

$$\begin{aligned} E_\psi (M_u)=\int _{\mathbb {R}}u(x)|\psi (x)|^2\,\text{ d }x \end{aligned}$$

and

$$\begin{aligned} E_\psi ({\mathcal M}_v)=\int _{\mathbb {R}}v(\xi )|\widehat{\psi }(\xi )|^2\,\text{ d }\xi . \end{aligned}$$

Here, we keep the convention of notation in mathematics where the position variable is denoted by x and the momentum variable is denoted by $\xi$ instead of p.

Let ${\mathcal B}$ be the set of Borel subsets of ${\mathbb {R}}$. It is then obvious that $|\psi (x)|$ is uniquely determined by

$$\begin{aligned} \mathcal {E}_Q=\big \{E_\psi (M_{\textbf{1}_B})\big \}_{B\in {\mathcal B}} \end{aligned}$$

which is called the distribution of the state $\psi$ with respect to position since

$$\begin{aligned} \mathcal {E}_Q=\big \{\Vert \textbf{1}_B(Q)\psi \Vert \big \}_{B\in {\mathcal B}} \end{aligned}$$

where $\textbf{1}_B(Q)$ are the spectral projections associated to the position operator.

On the other hand $|\widehat{\psi }(\xi )|$ is uniquely determined by distribution of the state $\psi$ with respect to momentum:

$$\begin{aligned} \mathcal {E}_P:=\{E_\psi ({\mathcal M}_v)\,:\ v=\textbf{1}_B,\ B \text{ a } \text{ Borel } \text{ set }\} =\{\Vert \textbf{1}_B(P)\psi \Vert ,\ B \text{ a } \text{ Borel } \text{ set }\}:=\mathcal {S}_P \end{aligned}$$

where $\textbf{1}_B(P)$ are the spectral projections associated to the momentum operator.

In a footnote to the Handbuch der Physik article on the general principle of wave mechanics [30], W. Pauli asked whether a wave function $\psi$ is uniquely determined (up to a constant phase factor) by one of the equivalent quantities

• the Pauli data $(|\psi |,|\widehat{\psi }|)$;

• $\big \{\Vert \textbf{1}_B(Q)\psi \Vert \big \}_{B\in {\mathcal B}}$, $\big \{\Vert \textbf{1}_B(P)\psi \Vert \big \}_{B\in {\mathcal B}}$;

• $\big \{E_\psi (M_{\textbf{1}_B})\big \}_{B\in {\mathcal B}}$, $\big \{E_\psi ({\mathcal M}_{\textbf{1}_B})\big \}_{B\in {\mathcal B}}$.

The question can also be found e.g. in the book by H. Reichenbach [31] and in Busch & Lahti [5].

As mentioned in the introduction, it is known that in general the Pauli data does not uniquely determine the state $\psi$ (up to a constant phase factor).

It is then natural to ask whether there exists a set of observables $(A_j)_{j\in J}$ (preferably including position and momentum or at least having a physical meaning) such that the associated sets built from spectral projections

$$\begin{aligned} \mathcal {E}_j:=\big \{\Vert \textbf{1}_B(A_j)\psi \Vert \big \}_{B\in {\mathcal B}} =\big \{E_\psi \bigl (\textbf{1}_B(A_j)\bigr )\big \}_{B\in {\mathcal B}} \end{aligned}$$

uniquely determine every state $\psi$.

Using the spectral theorem, to a self-adjoint operator $A_j$ we can associate a unitary operator $U_j$ and a multiplication operator $M_j$ on a space $L^2(\mu _j)$ such that $A_j=U_j^*M_jU_j$. Then the data $\mathcal {E}_j$, $j\in J$ uniquely determine $|U_j\psi |$, $j\in J$. This then directly leads to Wright’s Conjecture 1.1 and to its relaxation 1.2: find a set of measures $\mu _j$ and unitary operators $U_j:L^2({\mathbb {R}}^d)\rightarrow L^2(\mu _j)$ such that $|U_j\psi |=|U_j\varphi |$, $j\in J$, implies that $\psi$ and $\varphi$ are equivalent up to a constant phase factor.

A relaxed version is to find a set $\{T_j\}_{j\in J}$ of bounded self-adjoint (or even only bounded) operators on $L^2({\mathbb {R}})$ such that $|T_j\psi |=|T_j\varphi |$, $j\in J$, implies that $\psi$ and $\varphi$ are equivalent up to a constant phase factor. The data $|T_j\psi |$ can also be interpreted as an expectation of the state $\psi$ with respect to a family of observables. To be more precise, to a bounded operator T, we may associate the self-adjoint operator $A_u=T^*M_uT$ with $u\in L^\infty ({\mathbb {R}})$ real valued. Then

$$\begin{aligned} {\left\langle {\psi ,A_u\psi }\right\rangle }=\int _{\mathbb {R}}u(x)|T\psi (x)|^2\,\text{ d }x \end{aligned}$$

so that $|T\psi |$ is uniquely determined by

$$\begin{aligned} {\mathcal E}_T:=\big \{E_\psi (T^*M_{\textbf{1}_B}T)\big \}_{B\in {\mathcal B}}. \end{aligned}$$

However, it does not seem possible to reformulate this family of measurements in terms of spectral projections associated to a single self-adjoint operator.