Bloch State Tomography: Methods and Insights

• Bloch state tomography is a reconstruction protocol that maps measured observables into Bloch coordinates to infer quantum states or momentum-dependent eigenfunctions.
• It employs methods like maximum-likelihood estimation and Fisher information metrics to ensure the reconstructed density matrix or state adheres to physical constraints.
• The technique spans qubit, condensed-matter, and temporal domains, enabling the extraction of Berry curvature, Chern numbers, and multi-time correlators for deep quantum insights.

Bloch state tomography denotes reconstruction protocols in which an unknown quantum object is represented by Bloch coordinates and inferred from measurement data. For a qubit, the target state is commonly written as

$$\rho=\frac12\bigl(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z\bigr)$$

or, equivalently, as

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$

with physically allowed region given by the Bloch ball $\|\vec r\|\le1$; for a two-band Bloch problem, the cell-periodic eigenstate can be written relative to a reference basis as

$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$

In these settings, reconstructed Bloch data are used either to obtain a physical density matrix or to determine Berry curvature, the quantum metric tensor, Chern numbers, Wilson–Zak-loop eigenvalues, and $\mathbb Z_2$ invariants (Fujiwara et al., 2016, Li et al., 2015, Yi et al., 2023).

1. State spaces, parametrizations, and measured quantities

In single-qubit tomography, the Bloch or Stokes coordinates are expectation values of Pauli observables. If Pauli-$k$ is measured on a qubit state $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$, the two outcomes occur with probabilities

$$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$

For conventional qubit state tomography, one measures each Pauli $\sigma_i$ exactly $N$ times; if $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$0 and $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$1 are observed, the empirical Stokes vector is

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$2

and this empirical vector may lie outside the Bloch ball (Fujiwara et al., 2016).

The same Bloch-vector formalism underlies constrained hardware realizations. In a singlet–triplet double quantum dot, prepared states and effective measurements are parameterized as

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$3

so that the Born probabilities are linear in Bloch components: $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$4 This formulation allows stochastic and systematic errors to be incorporated directly into the reconstructed state and measurement model (Takahashi et al., 2013).

In Bloch-band tomography, the reconstructed object is not a single density operator but the cell-periodic Bloch function over the Brillouin zone. For the lowest band in a two-band manifold one writes

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$5

where the two real angles $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$6 parametrize the state up to a global gauge. The measurement task is then to recover amplitudes and phases of $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$7 as functions of quasimomentum, rather than only three expectation values at one point in Hilbert space (Li et al., 2015).

A common source of confusion is that “Bloch state tomography” is used in both senses. In qubit work it reconstructs a state inside the Bloch ball; in condensed-matter and cold-atom work it reconstructs momentum-dependent Bloch eigenfunctions. The shared structure is the use of Bloch coordinates as experimentally accessible parameters.

2. Information-geometric reconstruction of qubit states

For conventional Pauli tomography, the joint model of outcomes is the product distribution

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$8

On the open cube $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$9, the Fisher information metric of this model is

$\|\vec r\|\le1$0

so the Stokes-parameter space is treated as a Riemannian manifold endowed with a dual-flat statistical-manifold structure (Fujiwara et al., 2016).

The maximum-likelihood estimate is obtained by minimizing the Kullback–Leibler divergence from the empirical distribution: $\|\vec r\|\le1$1 When the empirical vector already satisfies $\|\vec r\|\le1$2, the maximum-likelihood estimate coincides with the empirical estimate. When the empirical vector lies outside the Bloch ball, the maximum-likelihood estimate is the unique point on the boundary sphere that is orthogonal, with respect to the Fisher metric, to the geodesic joining the empirical point to the sphere. In this formulation, data processing based on the maximum-likelihood method is an orthogonal projection onto the Bloch sphere with respect to the Fisher metric rather than a Euclidean radial projection (Fujiwara et al., 2016).

For randomized tomography with measurement counts $\|\vec r\|\le1$3 and weights $\|\vec r\|\le1$4, the metric becomes

$\|\vec r\|\le1$5

If the raw estimate is outside the physical region, the reconstruction reduces to solving

$\|\vec r\|\le1$6

for a Lagrange multiplier $\|\vec r\|\le1$7. The proposed algorithm transforms this into a single-variable root-finding problem

$\|\vec r\|\le1$8

which can be solved by Newton–Raphson. The reported convergence is quadratic from a reasonable initial guess, typically in $\|\vec r\|\le1$9–$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$0 iterations, and the overall cost is $$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$1 independent of sample size (Fujiwara et al., 2016).

The statistical interpretation is explicit. The Fisher metric stretches directions with large variance and compresses those with small variance, so the correction respects the statistical reliability of each coordinate. The method guarantees a physical density matrix, is statistically consistent and efficient, achieves the Cramér–Rao bound asymptotically, is unbiased in the large-$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$2 limit, and minimizes the KL divergence to the empirical distribution (Fujiwara et al., 2016).

3. Experimental architectures for qubit Bloch tomography

Qubit implementations differ primarily in how Bloch components are mapped onto experimentally accessible observables. The reconstruction formulas remain Bloch-linear, but the measurement primitives range from projective Pauli readout to reflection spectroscopy, weak pointer shifts, and continuous Rabi scans.

Platform	Measurement primitive	Reconstruction relation
Pauli tomography	$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$3 projective counts	$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$4
Singlet–triplet qubit	Native $$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$5 readout plus control pulses	$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$6
Kerr parametric oscillator	Reflection at a resonant line	$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$7
NV-center RQST	X–Rabi and Y–Rabi scans	$$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$8, $$u_k^1=\cos(\theta_k/2)\,|1\rangle+\sin(\theta_k/2)e^{i\phi_k}|2\rangle.$$9
Weak-measurement protocol	Three simultaneous weak couplings	$\mathbb Z_2$0

For the singlet–triplet qubit in a double quantum dot, initialization and readout are native only in the computational $\mathbb Z_2$1 basis, identified approximately with $\mathbb Z_2$2. Additional effective $\mathbb Z_2$3- and $\mathbb Z_2$4-measurements are generated either by preparing approximate eigenstates of those axes or by rotating an unknown state onto the $\mathbb Z_2$5 axis before native readout, using the control Hamiltonian

$\mathbb Z_2$6

A minimal tomographically complete set requires at least four independent states and three independent two-outcome measurements. The cited procedure also gives a self-consistent treatment of SPAM errors, free precession under a Lindblad model, and joint fitting of the unknown state and POVM parameters (Takahashi et al., 2013).

For Kerr parametric oscillators encoded in coherent states $\mathbb Z_2$7 and $\mathbb Z_2$8, a weak single-photon probe and reflection measurement provide a one-to-one mapping from the reflection coefficient to the population difference $\mathbb Z_2$9 when the probe is tuned to a dominant transition. Pre-rotations $k$0 and $k$1 map $k$2 and $k$3 onto the same $k$4 readout channel, leading to

$k$5

The reported simulations give average tomography fidelity $k$6 over the Bloch sphere, and $k$7 with optimized ramp shapes and maximum-likelihood post-processing (Suzuki et al., 2022).

Rabi-based quantum state tomography replaces discrete calibration pulses by continuous X–Rabi and Y–Rabi scans. After fitting photoluminescence traces to decaying sinusoids, the Bloch components are obtained either from amplitudes or directly from fitted phases: $k$8 For the NV-center electron spin, the reported average fidelity is $k$9 over more than $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$0 measurements on different Bloch-sphere states, with a maximum fidelity of $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$1; the same method is extended to the dark $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$2 nuclear spin via conditional gates and electron-spin readout (Shukla et al., 2024).

Weak-measurement tomography provides a different limit. With three independent pointer couplings

$\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$3

and no post-selection discard, the first-order pointer shifts satisfy

$\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$4

All three Bloch coordinates are thus extracted from one joint weak interaction, and the variance obeys

$\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$5

This scheme is explicitly described as using all data rather than retaining only a post-selected subensemble (Wu, 2012).

These examples show that restriction to a single native basis does not preclude full tomography. The essential requirement is a calibrated mapping from measured signals to Bloch coordinates.

4. Bloch-band tomography and reciprocal-space geometry

In Bloch-band systems, tomography reconstructs quasimomentum-dependent cell-periodic eigenfunctions. Under a constant external force $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$6, the quasimomentum evolves as

$\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$7

and, in the limit of an infinitely large force, the band evolution becomes purely geometric. The corresponding Wilson-line operator is

$\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$8

with $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$9. In a lattice whose relevant bands span the same two-dimensional Hilbert space at all $$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$0, this simplifies to

$$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$1

The Wilson line therefore encodes both intra-band Berry phases and interband couplings (Li et al., 2015).

The tomography protocol in the honeycomb-lattice experiment reconstructs the lowest-band state at each $$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$2 through two measurements. First, a strong-force transport from a reference point $$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$3 to $$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$4 yields the band-1 population

$$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$5

which determines $$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$6. Second, a Ramsey-type interferometer in quasimomentum space yields the phase

$$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$7

and hence the full spinor

$$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$8

From the reconstructed wave functions, one computes Berry curvature, Chern numbers, Wilson–loop spectra, and, in multiband generalizations, $$p_k(+1)=\tfrac12(1+r_k),\qquad p_k(-1)=\tfrac12(1-r_k).$$9 indices (Li et al., 2015).

A distinct implementation in a two-dimensional optical Raman lattice performs a complete Bloch-state tomography by direct spin-resolved measurements. The eigenstate in the first Brillouin zone is restricted to

$\sigma_i$0

so the two plane-wave states act as north and south poles of an effective Bloch sphere. Stern–Gerlach time-of-flight yields $\sigma_i$1, while a $\sigma_i$2 momentum-transferring Raman pulse with controlled phase $\sigma_i$3 gives

$\sigma_i$4

Choosing $\sigma_i$5 with $\sigma_i$6 or $\sigma_i$7 yields $\sigma_i$8 or $\sigma_i$9, respectively (Yi et al., 2023).

Once $N$0 are known, the reconstructed lowest-band spinor is obtained from

$N$1

The quantum geometric tensor is then

$N$2

with real part $N$3 and imaginary part related to the Berry curvature by

$N$4

For the reported detuning $N$5, numerical integration on a grid of approximately $N$6 $N$7-points gives

$N$8

The same data were used to test the pointwise inequality

$N$9

and the global bound

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$00

This is a direct example in which Bloch-state tomography accesses both topology and quantum geometry rather than only band populations (Yi et al., 2023).

The reciprocal-space version of Bloch-state tomography therefore differs conceptually from qubit tomography. The reconstructed object is a gauge-fixed wave function over the Brillouin zone, and geometric quantities are derived from momentum-space derivatives of that wave function.

5. Minimality, confidence regions, and scaling

Minimal qubit tomography is achieved with four outcomes. The tetrahedron measurement, a qubit SIC-POVM, has elements

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$01

where the four Bloch vectors $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$02 point to the vertices of a regular tetrahedron and satisfy

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$03

For a double-slit qubit of light, this minimal tomography can be implemented on a single detection plane. Detecting at four carefully chosen positions $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$04 realizes the required projectors, and the reconstruction is

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$05

The reported smallest solution uses $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$06 and $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$07 for the Gaussian-slit geometry (Sogamoso et al., 2017).

Reliable estimation requires more than a point estimator. For single-qubit tomography, prior-free confidence regions can be constructed directly in Bloch space. With empirical estimate $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$08 from Pauli-axis counts, total number of copies $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$09, and confidence level $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$10, one obtains the confidence region

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$11

where

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$12

This region contains the true Bloch vector with probability at least $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$13, independently of any prior assumption on the distribution of possible states (Christandl et al., 2011).

At the opposite end of the design space are scalable many-qubit protocols. Fourier-style tomography from single-pulse X/Y Bloch rotations applies

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$14

to an $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$15-qubit register, then measures in the laboratory $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$16 basis. Because each off-diagonal density-matrix element acquires a unique multi-qubit beat frequency, the state is recovered by a discrete-Fourier inversion over the measured times. At the operating point

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$17

the total tomography variance obeys

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$18

while the purity estimator satisfies

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$19

The scaling is still exponential in $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$20—an unavoidable fact of full QST—but the reported base is reduced from $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$21 to $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$22 for full tomography, and to $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$23 for purity estimation (Yanay et al., 2021).

These results delimit three distinct desiderata: symmetry and minimality, as in tetrahedral SIC measurements; rigorous uncertainty quantification, as in confidence-region tomography; and reduced exponential prefactors for many-qubit reconstruction.

6. Temporal Bloch tomography and generalized state formalisms

Bloch tomography has also been extended from states at one time to “states over time.” For a system of local dimension $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$24 monitored at times $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$25, an operator basis

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$26

allows any operator on the temporal Hilbert space $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$27 to be expanded as

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$28

with coefficients

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$29

In temporal Bloch tomography, the reconstructed object is a temporal state $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$30, generally non-positive, specified by such multi-time correlators (Jia et al., 8 Jan 2026).

The operational data are temporal Kirkwood–Dirac quasiprobabilities. For a process

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$31

one defines right, left, and doubled temporal KD distributions. Their real parts are temporal Margenau–Hill quasiprobabilities. For two-outcome $$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$32 measurements with projectors

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$33

the corresponding correlators assemble into a Bloch-state expansion

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$34

and the spatiotemporal Born rule becomes

$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$35

This gives a direct reconstruction route from experimentally accessible quasiprobabilities to temporal Bloch states (Jia et al., 8 Jan 2026).

The same work describes interferometric schemes for measuring the characteristic function of the temporal KD distribution using an ancilla qubit, controlled unitaries, and ancilla Pauli-$$\rho(\xi)=\frac12\bigl(I+\xi_1\sigma_1+\xi_2\sigma_2+\xi_3\sigma_3\bigr),$$36 readout. For general CPTP dynamics, each channel is first Stinespring-dilated and the same interferometric logic is applied. The formal relationships among existing temporal-state constructions are then made explicit: the doubled KD Bloch state is exactly the doubled-density-operator of Jia–Kaszlikowski; the two-time MH Bloch state coincides with the pseudo-density-operator of Fitzsimons et al.; the KD Bloch state is generally non-Hermitian; the MH state is Hermitian but not positive; and only the doubled MH state has the correct positivity/marginal structure to be called a full spatiotemporal density operator (Jia et al., 8 Jan 2026).

This extension changes the scope of Bloch tomography from static state estimation to process-aware reconstruction of multi-time correlations. In that sense, the Bloch expansion functions not only as a visualization device but as an operator basis for a broad class of reconstruction problems.

Across these formulations, Bloch state tomography is defined less by a single measurement technology than by a common reconstruction strategy: measured observables are converted into Bloch coordinates, those coordinates are constrained by physical or geometric structure, and the resulting reconstruction is used either to infer a density matrix, a momentum-space eigenfunction, or a temporal state.