Wigner tomography of multispin quantum states
David Leiner Robert Zeier Steffen J. Glaser
quant-ph







[]david.leiner@tum.de

[]zeier@tum.de

[]steffen.glaser@tum.de



Technische Universität München, Department Chemie, Lichtenbergstrasse 4, 
85747 Garching, Germany







We study the tomography of multispin quantum states
in the context of finite-dimensional Wigner representations.
An arbitrary operator can be completely characterized and visualized
using multiple shapes assembled from linear combinations of spherical harmonics
[A. Garon, R. Zeier, and S. J. Glaser, Phys. Rev. A 91, 042122 (2015)].
We develop a general methodology to experimentally recover 
these shapes by measuring expectation values of rotated 
axial spherical tensor operators and provide
an interpretation in terms of fictitious multipole potentials.
Our approach is experimentally demonstrated 
for quantum systems consisting of up to three spins 
using nuclear magnetic resonance spectroscopy.  










Wigner tomography of multispin quantum states
    Steffen J. Glaser
    January 15, 2018
=============================================








§ INTRODUCTION


Optical homodyne tomography can be applied to experimentally measure
the quantum state of light <cit.>.
One thereby recovers 
an infinite-dimensional Wigner function <cit.>
as a classically motivated phase-space representation, 
providing a useful tool for the characterization and visualization of 
quantum-optical systems <cit.>.
This results in an advantageous dualism between measurement scheme
and phase-space representation, which we would like to transfer to the case of 
finite-dimensional, coupled spin systems.

One important representation of finite-dimensional quantum systems 
relies on discrete Wigner functions 
<cit.>.
But we will restrict ourselves to continuous representations
in order to naturally reflect the inherent rotational symmetries
of spins. Individual spins are faithfully described by their
magnetization vector (or Bloch vector), which, however, neglects
relevant parts of the full density matrix in the case of multiple, coupled spins.
These missing parts include zero- and multiple-quantum or antiphase
coherence  as well as spin alignment <cit.>,
which are partially characterized by visual approaches based 
on single-transition operators <cit.>.

We will follow the general strategy of Stratonovich <cit.>
which specifies criteria for the definition of 
continuous Wigner functions for finite-dimensional quantum systems. 
The case of single spins is widely studied in the literature
<cit.>,
and visualizations for multiple spins have been considered 
in <cit.> with
various degrees of generality. However, until very recently, it was not
clear <cit.> if a general Wigner representation also exists
for arbitrary, coupled spin systems, and even the case
of three coupled spins 1/2 was open.

Fortunately, a general Wigner representation for 
characterizing and visualizing arbitrary coupled spin systems has been developed
in <cit.>. It is based on mapping arbitrary operators to a set of spherical functions
which are denoted as droplets, while preserving crucial features of the quantum system.
The characteristic shapes of these droplets can be interpreted as the result
of an abstract mapping, but we also ask in this paper how they are related to experimentally measurable
quantities. The general Wigner representation introduced
in <cit.> is denoted as
the representation (discrete representation of
operators for spin systems), 
and its basics are recalled in Sec. <ref> where important properties
are also summarized.

In this paper, we theoretically develop 
a tomography scheme for spherical functions of arbitrary multispin quantum states.
We study experimental schemes to reconstruct 
the generalized Wigner representation of a given density operator 
(representing mixed or pure quantum states). 
Extensions to quantum process tomography <cit.>
as given by the experimental reconstruction of entire 
propagators (e.g., representing quantum gates) are, however, beyond the scope of this paper.
Our scheme is particularly tailored to
the Wigner representation of <cit.>, for which 
an interpretation in terms of
fictitious multipole potentials is provided.
We will focus on systems consisting 
of spins 1/2, even though our approach is applicable to arbitrary spin numbers. 
We also provide explicit experimental protocols for our Wigner tomography scheme
and demonstrate its feasibility using 
nuclear magnetic resonance (NMR) experiments.
Motivated by our experiments, most  of the discussed examples consider only the traceless part
of the density matrix. 

This paper is organized as follows.  A brief summary of the 
DROPS representation is presented in Sec. <ref>.
Our general methodology for sampling spherical functions
of multispin operators is introduced in Sec. <ref>, which also
states the main technical results for the Wigner tomography.
Section <ref> provides a physical interpretation
of spherical functions 
in terms of fictitious multipole potentials. The performed NMR experiments
are summarized in Sec. <ref>, and Sec. <ref>
discusses the use of temporal averaging.
The precise experimental scheme and its implementation on a spectrometer
are detailed in Secs. <ref> and <ref>.
We conclude by 
summarizing and discussing theoretical 
and experimental 
aspects, while also contrasting our paper with other 
tomography approaches.
Further details are deferred to the Appendices.







§ VISUALIZATION OF OPERATORS USING SPHERICAL FUNCTIONS

We summarize the approach of <cit.>
to obtain a Wigner representation of arbitrary operators A 
in coupled spin systems 
using multiple spherical functions, which is
based on a general one-to-one mapping from
spherical tensor operators to spherical harmonics.
An operator

    A = ∑_ℓ∈ L A^(ℓ)

is decomposed according to a suitable set L of labels ℓ (i.e. quantum numbers)
inducing a bijective mapping
between the components A^(ℓ) 
and spherical functions f^(ℓ)=f^(ℓ)(θ,ϕ).
These spherical functions
can be plotted together as seen in the example of Fig. <ref>(a) where
the corresponding mapping is highlighted.
This provides a pictorial representation of the operator A, which
conserves important properties  and symmetries 
depending on the chosen label set L.

footnote1
[footnote]
The components  T_jm of 
varying rank j∈{0,1,…,2s} and 
order m∈{-j,…,j} form a complete orthonormal 
matrix basis.  In particular, one obtains the matrix basis
T_00 = 
(
[ 1 0; 0 1 ])/√(2),
T_1,-1 = 
(
[ 0 0; 1 0 ]),
T_10 = 
(
[  1  0;  0 -1 ])/√(2),
T_11 = 
(
[  0 -1;  0  0 ])
for a single spin with spin number s=1/2.
tensor



footnote1
[footnote]Spherical harmonics
Y_jm(θ,ϕ)=r(θ,ϕ) exp[iη(θ,ϕ)]
(and spherical functions) are plotted 
throughout this work by mapping their spherical coordinates θ and ϕ to the radial 
part r(θ,ϕ) and phase η(θ,ϕ).
spherical



The components A^(ℓ) 
and the spherical functions f^(ℓ)=f^(ℓ)(θ,ϕ)
can be further split up into their multipole contributions  A_j^(ℓ) and 
f_j^(ℓ)=f^(ℓ)_j(θ,ϕ) depending on the ranks j∈ J(ℓ)
occuring for each label ℓ as shown in Fig. <ref>(b), i.e.,

    A^(ℓ)  = ∑_j ∈ J(ℓ) A_j^(ℓ)  and  
    f^(ℓ)  = ∑_j ∈ J(ℓ) f_j^(ℓ).

Finally, the rank-j multipole contributions [see Fig. <ref>(c)]

    A^(ℓ)_j  = ∑_m=-j^j c_jm^(ℓ) T_jm^(ℓ)  and  
    f^(ℓ)_j  = ∑_m=-j^j c_jm^(ℓ) Y_jm,

can be decomposed into
components of irreducible spherical
tensor operators T_jm^(ℓ) <cit.>
 and the corresponding spherical harmonics  Y_jm <cit.>
of order m with -j≤ m≤ j. Note the identical
expansion coefficients c_jm^(ℓ) in Eq. (<ref>).
The dualism in Eqs. (<ref>) and (<ref>) exploits the well-known correspondence between 
irreducible tensor operators and spherical harmonics <cit.>.
In summary, an operator A is mapped to a set of spherical functions
f^(ℓ), each of which is referred to as a droplet identified by ℓ.
The whole representation (and its visualization)
was introduced in <cit.> and is denoted as
the representation (discrete representation of
operators for spin systems), and it lends 
itself to interactively exploring the dynamics of multispin systems, e.g., by 
use of the free application <cit.>.

footnote1
[footnote]
Cartesian operators for single spins are 
I_x:=σ_x/2, I_y:=σ_y/2, and I_z:=σ_z/2, where
the Pauli matrices are
σ_x=(
[ 0 1; 1 0 ]),
σ_y=(
[  0 -i;  i  0 ]), and
σ_z=(
[  1  0;  0 -1 ]). For n spins, one has the operators
I_k η := ⊗_s=1^n I_a_s 
where a_s is equal to η for s=k 
and is zero otherwise; note I_0:=
([ 1 0; 0 1 ]).
cart



footnote1
[footnote]Hermitian operators
lead to positive and negative values which 
are shown in red (dark gray) and green (light gray).
colors





The example presented in Fig. <ref> 
uses one particular version of this representation 
which relies on the tensor operator basis
as defined in <cit.>, which is characterized by the 
linearity of the basis operators, the involved subsystem, 
and auxiliary criteria, such as permutation symmetry. 
For coupled spins 1/2, operators are first decomposed in this basis
according to the set of involved spins, e.g.,
one introduces the labels {k} and {kl}
for linear and bilinear operators 
acting on a subset of one or two spins numbered by k and l, and so forth.
Secondly, the LISA basis for operators acting on three or more spins
needs to also distinguish symmetry properties under permutations, i.e., 
combined labels such as “{klm},τ_p" are used, where the permutation 
symmetry type τ_p is given by a Young tableau <cit.>. Finally, 
further ad hoc labels are necessary for operators involving six or more spins.
Arbitrary operators of a coupled spin system can be uniquely represented using
this tensor operator basis. Additional details for the visualization technique 
are given in <cit.> which also discusses
alternative labeling approaches for representations.


The presented Wigner representation can be applied to general mixed 
quantum states as represented by the density operator, and it 
is not limited to pure quantum states as given by 
a state function. In fact, it can be used to 
represent arbitrary operators of spin systems: examples include Hermitian operators 
as Hamiltonians or
density operators representing observables as well as
non-Hermitian operators such as propagators or general quantum gates
<cit.>.

The
Wigner representation using the basis is particularly attractive for the 
visualization and analysis of quantum states in magnetic resonance spectroscopy 
<cit.> and quantum information processing <cit.> and its properties 
have been discussed in <cit.>.

(a) The location of the droplets 
informs about which and how many spin operators
are involved in a given quantum state and what symmetries 
under particle exchange are present [c.f. Fig. <ref>(a)].

(b) The shape and colors of the droplets reflect 
spectroscopically important properties. For example,  states with defined 
coherence order p <cit.> can be recognized by their axially symmetric 
shape, and the magnitude as well as sign of the coherence order p 
are represented by the number and direction of “rainbows” per  revolution around the 
z axis [see Figs. <ref> and <ref>]. This also allows us to 
recognize characteristic states, such as inphase and anti-phase 
coherences. 

(c) Furthermore, our representation directly 
depicts information about reduced density matrices and thereby conveys information related to entanglement 
measures, which would have to be first computed from the density-matrix description
via partial traces. In particular, the size of the droplets corresponding to linear 
terms [positioned in the vertices of the triangle in Fig. <ref>(a)] 
provides information on the amount of bipartite entanglement measured by 
the concurrence (see Sec. IV E in <cit.>). 
This is an example of the fact that relevant information is often determined
already by a subset of all droplets in the basis. The basis 
thereby offers a more structured picture than the density matrix, 
even though the number of droplets grows rapidly with increasing
number of spins. However, as pointed out in <cit.>, this number
grows less rapidly than the number of density matrix elements.

(d) The droplets rotate  under 
non-selective pulses in a natural way. In combination with the 
characteristic droplet shapes, this property makes it in many cases possible  
to design experiments that transfer  a given initial state into a desired 
target state without detailed calculations. Beyond merely interpreting 
the occurring characteristic shapes as a result of an abstract mapping, it 
is interesting to ask whether they are connected to experimentally 
measurable quantities.





§ SAMPLING SPHERICAL FUNCTIONS OF MULTI-SPIN OPERATORS

We explain now how the shape of spherical functions 
can be characterized by
suitable chosen spherical samples.
This will be particularly relevant for spherical functions representing
spin operators as discussed in Sec. <ref>
for which these spherical samples
can be experimentally measured.
One obtains
a reconstruction method for the quantum state in terms of spherical functions.
In the general case, the associated rank-j components g_j(θ,ϕ) 
of an arbitrary spherical function g(θ,ϕ) are determined by
its scalar product with 
rotated versions R_αβ Y_j0(θ,ϕ)
of axial spherical harmonics Y_j0(θ,ϕ), which have 
rank j and order zero. Given two spherical functions h(θ,ϕ) and g(θ,ϕ), 
we recall the definition of their scalar product 
⟨ h(θ,ϕ)| g(θ,ϕ)⟩_L^2 := ∫_θ=0^π∫_ϕ=0^2 π 
h^∗(θ, ϕ)   g(θ, ϕ) sinθ  dθ  dϕ.
The rotation operator R_αβ
acts on a spherical function h(θ,ϕ)
by first rotating it around the y axis by a polar angle β
and then rotating the result around the z axis by an azimuthal angle α, i.e., 
R_αβ h(θ, ϕ):=
h[R^-1_αβ(θ, ϕ)]=h(θ-β, ϕ-α). 
After these preparations, the mathematical result underpinning our reconstruction method
states that the value of the rank-j component g_j(β, α) 
is proportional to the scalar product of R_αβ Y_j0(θ, ϕ) with g(θ, ϕ):



Consider a spherical function g(θ,ϕ)=∑_j g_j(θ,ϕ).
The rank-j components
g_j(β,α) for angles β and α
can be obtained
from the scalar products

    g_j(β, α)   =   s_j ⟨ R_αβ Y_j0(θ,ϕ) | g(θ,ϕ)⟩_L^2

with s_j:=√((2j+1)/(4π)).








Assuming that an operator A is represented by a set of spherical functions 
f^(ℓ)(θ,ϕ), we can apply Result <ref> by setting 
g(θ,ϕ):= f^(ℓ)(θ,ϕ) for each label ℓ separately. 
We extend Result <ref> such that the spherical rank-j components 
f^(ℓ)_j(β, α) can also be recovered by comparing the operator 
A directly with rotated axial tensor operators ℛ_αβ T_j0^(ℓ). 
Consequently, the values of the rank-j spherical components f^(ℓ)_j(β, α)
can be experimentally measured for any combination
of polar angles  β 
and azimuthal angles α.
Here, ℛ_αβ C := ℜ_αβ C ℜ^-1_αβ
describes the rotation of an n-spin operator C
where the simultaneous rotation ℜ_αβ=
e^- i α F_z e^- i β F_y of all spins is defined using the  
total spin operators  F_z= ∑_k=1^n I_kz and 
F_y= ∑_k=1^n I_ky <cit.>.   We recall the scalar product 
⟨ C| B⟩= ( C^† B) for operators 
C and B as well as the definition of the expectation value 
⟨ B ⟩_ρ=(ρ B)
of an operator B if the state of the spin system is given by
the density matrix ρ. Our result for
recovering rank-j droplet components of an operator can now be stated as follows:



Consider a multi-spin operator A which is represented by a set of
spherical functions f^(ℓ)(θ,ϕ)=∑_j ∈ J(ℓ) f_j^(ℓ)(θ,ϕ).
For each label ℓ, the rank-j component f_j^(ℓ)(β,α)
can be experimentally measured 
for arbitrary angles β and α via the scalar products

    f_j^(ℓ)(β, α)   =   s_j ⟨ℛ_αβ T_j0^(ℓ) | A⟩.

If the density matrix ρ of a spin system can be prepared to be identical to the operator A,
the rank-j droplet components are given by the expectation values

    f^(ℓ)_j(β, α)=  
    s_j ⟨ℛ_αβ T^(ℓ)_j0⟩_ρ.



The proofs of Results <ref> and <ref> are deferred to 
Appendices <ref> and <ref>.
Equation (<ref>) implies that the rank-j droplet components f^(ℓ)_j(β, α) 
for a density matrix ρ can be calculated from the expectation values of rotated
axial tensor operators ℛ_αβ T^(ℓ)_j0. 
Result <ref> shows that
one can retrace the shapes of the spherical functions f^(ℓ)(β, α)
representing any operator that can be mapped onto the density matrix
if one experimentally measures f^(ℓ)(β, α) for sufficiently many angles β and α.



§ DROPLETS AS MULTIPOLE POTENTIALS

The methodology of Wigner tomography as presented in Sec. <ref>
can be motivated by relating spherical functions to physical multipole potentials.
Section <ref> details connections to dipole potentials,
which is then generalized to fictitious multipole potentials in Sec. <ref>.
This allows us to interpret the proposed Wigner tomography
as measuring a fictitious potential using axial multipole sensors
(see  Sec. <ref>).




 §.§ Spherical functions and dipole potentials





The most direct physical interpretation of spherical functions is found for Hermitian 
single-spin terms <cit.>

    ρ_k =m_x I_kx+m_y I_ky+m_z I_kz

of the density matrix
with (possibly time-dependent) real coefficients m_x, m_y,
and m_z.
The corresponding spherical function f^{k}(θ, ϕ)
is now related to a magnetic 
dipole potential.
The operator ρ_k associated with spin k is interpreted as a 
magnetization vector  (or Bloch vector) M⃗= (m_x , m_y , m_z )^T 
the components of which are 
proportional to the expectation values of the spin operators I_kx, 
I_ky, and I_kz.
An actual (time-dependent) magnetic dipole  d⃗∝M⃗
creates a detectable signal  in an NMR experiment
by inducing a voltage in a detection coil.
It is associated  with a scalar dipole potential V_1(r⃗)
at r⃗=r⃗ (sinθcosϕ,  0.1emsinθsinϕ, 0.1em cosθ), where 
θ and ϕ are polar and azimuthal angles, respectively.
At a constant distance r⃗ 
from the dipole, the potential V_1(r⃗) is proportional to the scalar 
product d⃗·r⃗ <cit.>. In the case of an axial
dipole d⃗_ax∝ (0,0,1)^T oriented along the z axis, 
the dipole potential is proportional to the axial spherical harmonic
Y_10(θ, ϕ)=√(3/(4 π))cosθ as detailed
in Fig. <ref>(a)-(c).
For a general dipole
d⃗=R_αβ d⃗_ax∝
(sinβcosα, sinβsinα, cosβ)^T,
the dipole potential V_1(r⃗) ∝R_αβ 
Y_10(θ, ϕ) is rotated accordingly as shown in 
Fig. <ref>(d)-(f). Recall that  R_αβ denotes a rotation 
around the y axis 
by a polar angle β
followed by one around the z axis by an azimuthal angle α.



A scalar dipole potential V_1(r⃗)=V_1 (θ, ϕ) can be indicated by its 
values on the surface of a  sphere by encoding its sign by the color
and its magnitude by the brightness [see Fig. <ref>(b) and (e)].
Alternatively, its magnitude can be represented 
by the distance from the origin as in Fig. <ref>(c) and (f),
where dipole potentials are shown as 
a positive, red (dark gray) sphere and a negative, green (light gray) one
which touch each other at the origin.
This characteristic shape arises as
V_1 (θ, ϕ) is proportional   
to the projection of the dipole d⃗ onto r⃗ as depicted in  Fig. <ref>.
Also, the vector from the center of the negative sphere to the positive one
is collinear with d⃗.






In summary, a single-spin axial spherical tensor operator T_10^{k} = √(2) I_kz
is mapped to the 
axial spherical harmonics Y_10(θ, ϕ), and ρ_k from  Eq. (<ref>)
is mapped to

    f^{k}(θ, ϕ) = M⃗ R_αβ Y_10(θ, ϕ) / √(2),

where β=  atan(m_z/√(m^2_x+m^2_y) ), α=  atan (m_y/m_x), and
M⃗=√(m^2_x+m^2_y+m^2_z).
Although the direct correspondence between spherical functions
and actual physical dipole potentials appears to be limited to the case
of single-spin terms, it suggest the following interpretation for other
spherical functions presented here.



 §.§ Fictitious multipole potentials

Any spherical function f^(ℓ)(θ, ϕ) can be regarded as the 
potential V^(ℓ)(r⃗) of a fictitious charge distribution σ^(ℓ) (r⃗)
localized in a small volume close to the origin, i.e., σ^(ℓ)(r⃗) is 
non-zero only  for | r⃗| ≪ 1. At a radius of | r⃗| = 1,
the potential  can be expressed 
as a sum V^(ℓ)(θ, ϕ)=∑_j V^(ℓ)_j(θ, ϕ)
of different 2^j-pole potentials V^(ℓ)_j(θ, ϕ).
Although a large number of multipole potentials  might be required in general,
only a moderate number of components with different rank j appear 
for up to three spins in the representation of Sec. <ref> <cit.>. 
Fictitious multipole potentials sufficient to completely describe
the potential V^(ℓ)(θ, ϕ)
are detailed in Table <ref>: 
one has monopoles (2^0=1), dipoles (2^1=2), quadrupoles (2^2=4), and octupoles (2^3=8).
For a two-spin droplet with label “{kl}”, only ranks j of zero, one, and two occur
which correspond to fictitious monopole, dipole, and quadrupole potentials,
whereas the fully symmetric three-spin 
droplet with label “{ 123},τ_1” has only rank-1 and rank-3 components
associated with dipole and octupole terms.






 §.§ Axial multipole sensors


Based on the provided interpretation of droplet functions as fictitious multipole potentials,
the results of Sec. <ref> on how to 
experimentally measure spherical functions of spin operators
can be mapped to the analogous problem of measuring an unknown electrostatic potential.
This analogy is complete for Hermitian spin operators with
real-valued spherical functions <cit.>.
Suppose we would like to determine an unknown (real-valued) electrostatic 
potential V(θ, ϕ) at a radius r⃗=1
that is created by an object located in the interior of a unit sphere.
An electric point charge q at position r⃗ in an electric potential V(r⃗)
has a  potential energy U_pot(r⃗)=q V(r⃗).
Given an electrostatic potential V(θ, ϕ),
the electrostatic potential energy of a (nonconducting) unit sphere
with the surface charge distribution σ(θ,ϕ) is given by

    U_pot(β, α) 
    = ∫_θ=0^π∫_ϕ=0^2 π σ(θ,ϕ)  V(θ, ϕ)
     sinθ dθ dϕ,

which is equivalent to
U_pot(β, α) =  ⟨σ(θ,ϕ) | V(θ, ϕ)⟩_L^2
for real valued σ(θ,ϕ)
as in Eq. (<ref>) of  Sec. <ref>.




The 2^j-pole components V_j(θ,ϕ) of an unknown 
multipole potential V(θ, ϕ) can be sampled by a set of axial 2^j-pole sensors, 
each consisting of a charge distribution 
σ_j0(θ,ϕ) proportional to the axial spherical harmonics Y_j0(θ, ϕ).
Each individual sample V_j(β,α) can be determined using
Eq. (<ref>)  by measuring the potential energy 

    V_j(β, α)∝  U_pot(β,α) =  ⟨R_αβ σ_j0(θ, ϕ) |   V(θ, ϕ)⟩

of the axial 2^j-pole sensor rotated by the polar angle β around 
the y axis followed by a rotation by the azimuthal angle α around the z axis. 
The full electrostatic 2^j-pole potential V_j(θ, ϕ) can be recovered
by systematically incrementing β and α.
In summary, the analogy between real-valued spherical functions
and multipole potentials helps to better understand our results of Sec. <ref>
on the measurement of spherical functions. It can also
be extended in a straight-forward manner to non-Hermitian spin operators 
by considering complex, fictitious multipole potentials.









§ SUMMARY OF NMR EXPERIMENTS 


Building on the previous sections, we demonstrate the Wigner
tomography of various prepared density-matrix components in spin-1/2 
systems using nuclear magnetic resonance. Experimental details
are deferred to Secs. <ref> and <ref> where
the precise experimental scheme and its implementation on a spectrometer
are discussed. The experiments were performed on
one-, two-, and three-spin systems. 
The shapes of the spherical functions are recovered
for the  prepared Cartesian products operators listed in Table <ref>,
where also their respective experimental reconstruction errors are given.
Experimental and theoretical results for the reconstruction 
are visually compared for four examples in Fig. <ref>.
For the rightmost example of 4I_1xI_2yI_3z in Fig. <ref>, multiple droplets 
corresponding to different permutation symmetries are necessary to completely describe 
the quantum operator,  as outlined in Sec. <ref> (see Fig. <ref>
and <cit.>).







§ TEMPORAL AVERAGING


Re-using our experimental data as summarized 
in Sec. <ref>, we can also highlight how temporal averaging <cit.>
is used to emulate the preparation of quantum operators.
The direct experimental preparation of Hermitian operators would be also
possible, but we have chosen temporal averaging for its
simplicity and convenience. The experimental values are shown in Fig. <ref>,
while the corresponding theoretical predictions are given in
Fig. <ref>.
The Cartesian operators
2I_1xI_2x, 2I_1yI_2y, 2I_1xI_2y, and 
2I_1yI_2x had been sequentially measured
and are now combined in 
Fig. <ref>(a) to form the 
double quantum operators 

    DQ_x := I_1xI_2x - I_1yI_2y  and  DQ_y := I_1xI_2y + I_1yI_2x.

Their characteristic shapes reflect the fact that they have 
coherence order p=2 and are invariant under
non-selective rotations around the z axis by an integer multiple of 180 deg
<cit.>.
This is in contrast to
single-quantum operators such as the linear operators I_x
or I_y [see <ref>(c)] 
or the bilinear operator  2 I_1z I_2x 
[see Fig. <ref>]  which 
are only invariant under non-selective rotations around the z 
axis by an integer multiple of 360 deg <cit.>. 






In general, p-quantum operators 
are invariant under non-selective rotations around the z axis by an integer multiple of 
360/p deg, and their spherical functions illustrate this symmetry.
Figure <ref>(b) exemplifies the invariance under 120 deg rotations around 
the z axis in the case of p=3 for the triple-quantum
operator 

    TQ_x    := I_1xI_2xI_3x-I_1xI_2yI_3y-I_1yI_2xI_3y-I_1yI_2yI_3x,
    similar to the case ofTQ_y    :=I_1yI_2xI_3x+I_1xI_2yI_3x+I_1xI_2xI_3y-I_1yI_2yI_3y.

Up to experimental imperfections, 
only the spherical function f^τ_1 
contributes to the operator TQ_x.

Finally, we also consider temporal averaging 
for non-Hermitian operators which obviously cannot 
be directly prepared in experiments (see also Sec. <ref>).
Figure <ref>(c) presents the non-Hermitian operators
I_1^+ = I_1x+iI_2y, 
I_1^+I_2^+ = DQ_x + i DQ_y, and
I_1^+I_2^+I_3^+  = TQ_x + i TQ_y.
These operators have the respective
coherence orders p of 1, 2, and 3 which results
in donut-shaped spherical functions the colors of which
cycle through one, two, or three rainbows <cit.>.






§ WIGNER TOMOGRAPHY USING NMR 


We complement our results in Sec. <ref> and describe 
the experimental scheme for an NMR-based implementation of our
Wigner tomography. Recall that Eq. (<ref>) of Result <ref> 
provides an approach for measuring an arbitrary operator. This can be translated 
into the diagram of Fig. <ref>(a): The operator A is decomposed into
its components A_j^(ℓ) which are mapped by the Wigner transformation W
to spherical samples f_j^(ℓ)(β,α). The spherical samples
can be recovered using Eq. (<ref>). Very similarly, Fig. <ref>(b)
depicts the equivalent measurement procedure for density matrices which
relies on Eq. (<ref>) of Result <ref>.




Before proceeding to the NMR-based scheme, it is important 
to emphasize that only traceless operators can be measured 
in NMR experiments <cit.>, which rules out the 
identity component of a density matrix. However, the traceless part of 
a density matrix covers most important features, and it is
sufficient  to calculate the time evolution and 
all relevant expectation values. Hence, we will 
consider in the following only the  traceless part which is 
for simplicity also denoted by ρ.






Further complications arise from the fact that signatures of 
Cartesian product operators <cit.>
that contain only a single transverse Cartesian operator I_ka with a∈{ x,  y} can be measured 
directly; examples are I_ka, 2I_kaI_lz, and 4I_kaI_lz I_mz.
This complication can be resolved in two steps:  
Firstly, any traceless operator can be decomposed into (Hermitian) Cartesian product operators
C^(ℓ,n)_j.
This decomposition of relevant axial tensors 

    T^(ℓ)_j0=  
     ∑_n r^(ℓ,n)_j C^(ℓ,n)_j

with respect to real coefficients r^(ℓ,n)_j
is given in Table <ref>.
For example,
the axial tensor operator T_10^{ k} acting on the k th spin 
decomposes directly into
the Cartesian product operator 
C_1^({k},1)=I_kz
with  the coefficient r_1^({k},1)=√(2).



Secondly, the Cartesian product operators have to be transformed
into NMR-measurable ones

    M^(ℓ,n)_j= 𝒰^(ℓ,n)_j C^(ℓ,n)_j := U^(ℓ,n)_j C^(ℓ,n)_jU^(ℓ,n)†_j.

The unitary operators U^(ℓ,n)_j
can be experimentally realized
using radio-frequency pulses and evolution periods under couplings
as discussed in Sec. <ref> and 
their explicit form is detailed in 
App. <ref>. Combining both steps leads to an
indirect approach for measuring spherical functions of density operators, 
as schematically outlined in Fig. <ref>(c). 
The density matrix is equivalently rotated inversely
in contrast to  Fig. <ref>(a)-(b) where the axial tensor operator T_j0^(ℓ)
is rotated. The complete measurement scheme is formalized 
along the lines of Result <ref>:



Consider a density operator ρ which is represented by a set of
spherical functions f^(ℓ)(θ,ϕ)=∑_j ∈ J(ℓ) f_j^(ℓ)(θ,ϕ).
For each label ℓ, the rank-j  component f_j^(ℓ)(β,α)
can be  measured for arbitrary angles β and α
by determining the expectation values 

    f^(ℓ)_j(β, α)
        = s_j   ∑_n  0.1em r^(ℓ,n)_j 0.1em ⟨ M^(ℓ,n)_j⟩_ρ̃̃̃^(ℓ,j,n)of suitable operators M^(ℓ,n)_j as in Eq. (<ref>), whereρ̃̃̃^(ℓ,j,n)    = 𝒰^(ℓ,n)_j ρ̃= U^(ℓ,n)_j ρ̃  U^(ℓ,n)†_j, 
    ρ̃    =  R^-1_αβ ρ
    =ℜ^-1_αβ ρ ℜ_αβ,

and 
ℜ_αβ=exp(- i α∑_k=1^n I_kz) exp(- i β∑_k=1^n I_ky).

A detailed derivation of Eq. (<ref>) is provided in Appendix <ref>. 
In summary, the rank-j components f^(ℓ)_j of spherical functions f^(ℓ) 
representing the density matrix ρ can be sampled in NMR experiments by 
transforming the density operator ρ to the states ρ̃̃̃^(n) and 
then measuring a set of expectation values of suitable
operators ⟨ M^(ℓ,n)_j⟩_ρ̃̃̃^(ℓ,j,n). The explicit form 
of the Cartesian operators C^(ℓ,n)_j, their 
NMR-measurable forms  M^(ℓ,n)_j, and 
the transformations U^(ℓ,n)_j for up to three
spins is given in App. <ref>.



The approach of Result <ref> can be extended to non-Hermitian operators,
even though these cannot be prepared directly in an experiment. We apply
temporal averaging <cit.> as already discussed in Sec. <ref>.
Any operator A can be expressed as a complex linear combination
A=  ∑_i c^(i)ρ^(i) of Hermitian operators ρ^(i).
As the representation is linear, we can sample the traceless 
part of any operator A using the spherical functions

    f^(ℓ)_j(β, α)
       =   ∑_i c^(i) f^(i,ℓ)_j(β, α) 
       = ∑_i c^(i)   s_j   ⟨ R_αβ T^(ℓ)_j0⟩_ρ^(i)
       =∑_i c^(i)   s_j  ∑_n   r^(ℓ,n)_j ⟨ M^(ℓ,n)_j ⟩_ρ̃̃̃^(ℓ,j,n,i).

In an experiment,  temporal averaging 
of Hermitian operators ρ^(i)
can be implemented by sequentially measuring spherical functions for each operator
ρ^(i) and linearly combining the results
f^(i,ℓ)_j(β, α)=s_j   ⟨ R_αβ T^(ℓ)_j0⟩_ρ^(i) 
= s_j  ∑_n   r^(ℓ,n)_j 0.1em ⟨ M^(ℓ,n)_j ⟩_ρ̃̃̃^(ℓ,j,n,i)
as illustrated in Fig. <ref>(d).










§ EXPERIMENTAL IMPLEMENTATION


After outlining the experimental scheme for Wigner tomography
in Sec. <ref>, we present now the details for the experimental implementation
which results in the spherical functions in Figs. <ref>-<ref>.
We start by describing the molecules and experimental setting 
in Sec. <ref>. We continue in Sec. <ref> with the experimental protocol,
and we finally discuss experimental errors in Sec. <ref>.





 §.§ Molecules and experimental setting




In order to simplify the experiments, the linear and bilinear Cartesian product 
operators have been prepared and measured using 
respective single-spin and two-spin samples: The single-spin sample
was prepared by dissolving
5% H_2O in D_2O, which resulted in a sample containing about
10% HDO, i.e.  water molecules in which one of the ^1H spins is replaced by 
deuterium (^2D) [see Fig. <ref>(a)]. In case of two spins,
we have used a 10% sample of chloroform dissolved in fully deuterated DMSO-d6, 
where the ^1H spin  and the ^13C spin of each chloroform molecule 
form a system consisting of two coupled heteronuclear spins 1/2 [see Fig. <ref>(b)].
A three-spin sample consisting 
of 2-^13C-2-fluoromalonic-acid-diethyl-ester dissolved in CD_3CN
[see Fig. <ref>(c)]
was utilized  for the preparation and  reconstruction of trilinear operators. 
All liquid samples were measured in 5 mm Shigemi NMR tubes at room temperature (298 K)
in  a 14.1 T magnet
using a Bruker Avance III 600 spectrometer.








 §.§ Experimental protocol


Our experimental protocol is composed of 
five main building blocks (see Fig. <ref>).
In the first block 𝒫, the desired density operator ρ is
prepared starting from the initial thermal equilibrium density operator
which in the high-temperature limit is 
proportional to <cit.>
ρ_th = 
∑_k=1^Nγ_kI_kz,
where γ_k denotes the gyromagnetic ratio of the kth nuclear spin. 
This requires
unitary transformations which are created by pulses and evolution periods under the 
effect of couplings and frequency offsets as well as non-unitary transformations which are
implemented by pulsed magnetic-field gradients. The explicit pulse sequences 
are discussed in App. <ref>.
Table <ref> in Sec. <ref> summarizes
all the different Cartesian product operators which have been experimentally prepared.





The second block consists of the rotation ℛ^-1_α_rβ_r
which rotates the prepared  density operator ρ 
into ρ̃
in order to probe the corresponding spherical functions 
f_j^(ℓ)(β_r, α_r) for different  polar angles β_r and azimuthal angles α_r
using axial tensor operators, i.e.,  axial multipole sensors)
(see Result <ref>). The rotation ℛ^-1_α_r β_r 
is implemented by rf pulses [β_r]_α_r-π/2
with flip angle β_r and phase (α_r-π/2)
which are simultaneously applied to all spins.

The unitary transformations 𝒰_j^(ℓ,n)
[see Eq. (<ref>) of Result <ref>]
are applied in the third block in order to transform the density matrix ρ̃ into
directly detectable Cartesian product operators
for the various linear, bilinear, 
and trilinear operators (see Table <ref> in Sec. <ref>).
The specific experimental implementation of the unitary transformations 𝒰^(ℓ,n)_j 
consists of rf pulses with flip angle π/2
as detailed in App. <ref>.




In the fourth block, the NMR signal is measured in an acquisition period Acq
which has a duration of 5.7 ms (one spin), 
11.4 ms (two spins), and 14.9 ms (three spins). In the last block,
a relaxation delay RD with a duration of 
7 s (one spin), 10 s (two spins), and 15 s (three spins)
recovers the initial equilibrium state ρ_th,


In the tomography experiment, all blocks are repeated multiple times 
(see Fig. <ref>).
The outer loop A runs over all possible droplets ℓ∈ L. Loop B  runs over all 
ranks j contributing to the droplet ℓ. Loop C 
cycles through
all Cartesian product operators  C^(ℓ,n)_j [see Eq. (<ref>)] appearing
in the decomposition of the axial tensor operator T^(ℓ)_j0 (see Table <ref> in 
Sec. <ref>). Finally, the discretized angles 
β_r∈{0,  15, 30, … , 180} and 
α_r∈{0,  15, 30, … , 360} (both in degrees)
are incremented in the innermost loops D and E.
Although not explicitly indicated in Fig. <ref>, one further loop is necessary for
a temporal averaging scheme [see  Eq. (<ref>)].

The whole protocol allows us to 
determine expectation 
values ⟨ M_j^(ℓ,n)⟩_ρ̃̃̃^(ℓ,j,n),
which are normalized and range between 1 and -1.
As illustrated in Fig. <ref>, the spherical functions
f^(ℓ)(θ,ϕ)
can be reconstructed
by plotting the spherical samples 
f^(ℓ)(β_r,α_r) for all angles
 β_r ∈{0,…,π} 
and α_r ∈{0,…,2π} at a distance | f^(ℓ)(β_r,α_r) | from the origin. 
The phase 
φ^(ℓ)   (β_r, α_r)=  atan ( Im{   f^(ℓ)(β_r, α_r)   }   /  Re{   f^(ℓ)(β_r, α_r)   }     ) 
is color coded. For example, the spherical function of the Hermitian operator I_1x is given by a
real function, and the positive and 
negative values of f^(ℓ)(θ,ϕ) are indicated by the colors red  (dark gray) and green (light gray), 
respectively.






 §.§ Experimental errors

A reasonable match between the experimentally reconstructed and 
theoretical predicted spherical functions is found in Figs. <ref>-<ref>. 
Deviations are attributed to experimental imperfections, such as the 
finite experimental signal-to-noise ratio, finite accuracy of pulse calibration, 
B_0 and B_1 inhomogeneity <cit.>, 
pulse shape distortions due to the amplifiers and the finite bandwidth of the resonator <cit.>,  
relaxation losses during the preparation and detection blocks,
partial saturation of the signal due to a finite relaxation period between scans,
radiation damping effects <cit.>, and
truncation effects in the automated integration and comparison of the spectra.
We quantify these deviations by 
the root-mean-square difference between experiment and theory
averaged over all measured angles β_r and α_r.
The resulting errors for the prepared and measured Cartesian product operators 
are summarized in Table <ref> of Sec. <ref>.
In order to minimize the effects of field 
inhomogeneities, Shigemi tubes were used in the experiments in order to reduce the sample volume. 
We have tested replacing 
the simple rectangular pulses [β_r]_α_r-π/2 
in the implementation of  the rotations
ℛ^-1_α_r β_r
with  composite pulses <cit.>, which,
however, did not result in an improved performance.




§ CONCLUSION


We have theoretically developed and experimentally demonstrated a Wigner tomography scheme the mapping of which 
multi-spin operators and spherical functions is based on <cit.>.
Our approach reconstructs the relevant spherical functions
by measuring expectation values of rotated axial tensor operators, i.e. 
axial multipole sensors. It is universally applicable and 
not restricted to NMR methodologies or particles with spin 1/2.
A reasonable match between theoretical predictions 
and NMR experiments was found. 

Our theoretical analysis provides a simple physical interpretation of the individual 
spherical functions in terms of fictitious multipole potentials
which can be sampled locally. The objective was to 
experimentally recover the three-dimensional shapes of the spherical
functions for each of the prepared operators. In particular, 
we have not used any a priori information on what shapes
to expect. A large number of 
sampling points is necessary to recover the shapes in sufficient detail.
As the simple rectangular grid of sampling points
in the space of polar 
and azimuthal angles used here is highly anisotropic
(i.e. more densely concentrated at the poles), 
a straight-forward improvement is to choose 
more isotropic sampling strategies such as Lebedev grids <cit.>.
Moreover, one could use interpolation methods or even adaptive sampling schemes
which increase the sampling density in areas where the spherical function varies 
more strongly. A more quantitative analysis of the complexity of
our proposed tomography approach in terms
of the number of individual measurements could rely first on a detailed 
empirical account on how the precision measured by the fidelity depends on 
the sampling strategy and its density. From the theoretical side, it is clear that
the number of measurements for any complete tomography of a quantum state
will scale exponentially in the number of qubits (or spins). However, relevant
information can be reconstructed in our approach even from a subset 
of the droplets as discussed under (c) in Sec. <ref>.
A  more accurate understanding of concrete sampling schemes 
and their optimization building on 
our reconstruction method will have to be addressed in future work.

Beyond the optical tomographic methods mentioned at the start of the 
introduction, tomographic approaches play obviously an important part in 
almost any experiment  in quantum information or quantum physics 
in general. We will now shortly discuss some selected results from the literature.
The proposed reconstruction procedure for spherical functions
can be directly extended to other types of spherical representations
such as multipole operators <cit.> or the 
so-called PROPS representation which 
is based on products of single-spin representations <cit.>.
Our scheme can also be compared 
to
<cit.> which introduces Wigner functions built from 
products of single-spin representations, and a corresponding
raster scan method in <cit.> utilizes the 
probability to find the rotated system in each of the basis states 
of a Stern-Gerlach-type experiment.
In <cit.>,
a similar tomographic method 
based on
filtered back projections (in analogy to planar inverse Radon 
transforms used e.g. in medical imaging)
is used 
in Stern-Gerlach-type experiments. 
However, complementary to these tomographic reconstruction schemes
for density matrices relying on Stern-Gerlach-type experiments,
our scheme is based on projections of operators onto rotated 
axial tensor operators or experimentally accessible expectation 
values of transformed axial tensor operators.
The measurement of spherical harmonics components of
electromagnetic near-field radiation using specifically designed loop 
antennas as sensors <cit.>
is closely related to our interpretation of droplets as multipole potentials
(see Sec. <ref>).

Further tomographic approaches 
have been established in <cit.>
with applications to molecular systems. 
In those works, the quantum state given as 
the wave functions of an excited state
is obtained by decomposing the wave function in a series of basis functions 
and the expansion coefficients are acquired
by calculating a set of Fourier integrals 
from the detected signal. Also, the wave packet is 
reconstructed in <cit.>
by computing the overlap of the state with well-defined reference states for different time intervals.

We want to also contrast our Wigner tomography approach
to the so-called spherical tensor analysis (STA) method
developed by Suter and Pearson <cit.> and 
van Beek et al. <cit.>. Both Wigner tomography and STA experiments have a 
similar structure. First, a preparation block (called an excitation sequence in STA) 
is used to prepare a density operator. In a second step, 
rotations around several axes are applied in a rotation block.
Finally, the density matrix is transformed 
(by a reconversion sequence in STA)
into a detectable basis and the signal is 
detected during an acquisition period.
Despite these similarities and the fact that both methods are based on 
characteristic properties of spherical tensor operators under rotations, 
the desired data and therefore also the details of the experiments 
differ considerably. 
The goal of STA is not to measure a density operator represented by 
spherical functions, but to decompose the detectable signal at a later time  
into individual signal components depending on 
occurring ranks j and orders m
in the current density matrix.
In the standard form of STA <cit.>, the detected signal 
of a density matrix is not characterized or decomposed
in terms of additional quantum numbers or labels ℓ.
While a single reconversion sequence is used in STA, 
different pulse sequences are applied 
in the Wigner tomography in order to
transform operators to directly detectable ones 
(see  Fig. <ref>(c), (d) and Table <ref>).
As a final difference, a rotation block in STA uses rotations for 
three Euler angles α, β, and γ,
whereas in the Wigner tomography only two Euler
angles α and β are necessary.

Lastly, our Wigner tomography can be seen as a stepping
stone along the path to identifying 
and characterizing operators in terms
of expansion coefficients for a suitable chosen basis.
One can consider various different bases such as 
simple matrix coefficients, a spherical tensor basis <cit.>
(as the one used here), 
or a Cartesian product basis 
<cit.>.
In this context of quantum state tomography, the shapes of 
spherical functions recovered in the Wigner tomography 
clearly contain highly redundant information,
but they also provide information about random or systematic errors of the tomography process itself.
One can also envision Wigner tomography as a component 
of a more general approach where
one would like
to optimize the number and location of samples for
achieving a desired fidelity 
and robustness against experimental errors <cit.>, or
recover a physical density matrix and estimate experimental errors
(see, e.g., <cit.> and references therein).






This work was supported in part by the Excellence Network of Bavaria (ENB) through ExQM. R.Z. 
and  S.J.G.  acknowledge  support  from  the  Deutsche Forschungsgemeinschaft (DFG) through
Grant No. Gl 203/7-2. We thank Raimund Marx for providing the samples. 
The experiments were performed at the Bavarian NMR Center at the Technical University of Munich.










§ PROOF OF RESULT <REF>

We detail now the arguments leading to our reconstruction formula for a  
spherical function g(θ,ϕ) as stated in Result <ref>.
This result relies on projections of 
rotated spherical harmonics. 



We use the notation introduced in Sec. <ref>. The angles θ and ϕ
indicate generic argument values of a spherical function g(θ,ϕ), but the angles β and α refer to
specific argument values. First, the right hand side 
s_j ⟨ R_αβ Y_j0(θ,ϕ) | g(θ,ϕ)⟩_L^2
of Eq. (<ref>) in Result <ref> is 
rewritten as 
s_j ⟨ R_αβ Y_j0(θ,ϕ) | g_j(θ,ϕ)⟩_L^2,
where the familiar orthogonality relation
⟨ Y_jm(θ,ϕ) |  Y_j'm'(θ,ϕ)⟩_L^2 = δ_jj'δ_mm'
of spherical harmonics (see  p. 68 of <cit.>) implies the relation
⟨ g_j(θ,ϕ) |  g_j'(θ,ϕ)⟩_L^2 = δ_jj' for the
rank-j parts g_j(θ,ϕ) in the decomposition g(θ,ϕ)=∑_j g_j(θ,ϕ).
Second, 
one obtains that
s_j ⟨ R_αβ Y_j0(θ,ϕ) | g_j(θ,ϕ)⟩_L^2
=s_j ⟨  Y_j0(θ,ϕ) |
R_αβ^-1
g_j(θ,ϕ)⟩_L^2 holds, which can be deduced from
the invariance
⟨ R_αβ Y_jm(θ,ϕ) | 
R_αβ Y_j'm'(θ,ϕ)⟩_L^2
= ⟨ Y_jm(θ,ϕ) | 
Y_j'm'(θ,ϕ)⟩_L^2
under rotations. The last relation
is easily verified using
the formula R_αβ Y_jm(θ,ϕ)=
Y_jm(θ-β,ϕ-α) and
a change of variables in the integral defining the scalar product
(see Sec. <ref>). Finally, 
R_αβ^-1
g_j(θ,ϕ) is expanded into a linear combination 
∑_m'=-j^j c_jm'(α,β) 
Y_jm'(θ,ϕ) 
of spherical harmonics
[Even though our arguments do not rely on explicitly knowing
the expansion coefficients c_jm'(α,β), it might be 
instructive to specify the expansion coefficients in terms
of the widely used Wigner-D matrices D_m'm^j(α,β,γ) 
<cit.>.
Let us also 
assume that the rank-j
parts
g_j(θ,ϕ) 
are expanded into
g_j(θ,ϕ)=∑_m=-j^jc̃_jm Y_jm(θ,ϕ)
using certain coefficients c̃_jm.
The theory of Wigner-D matrices <cit.>
implies that
R_αβ^-1  Y_jm(θ,ϕ)  =∑_m'=-j^j
D_m'm^j(0,-β,-α)  Y_jm'(θ,ϕ).
Hence, we obtain the formula
R_αβ^-1  g_j(θ,ϕ) 
=∑_m=-j^jc̃_jm×∑_m'=-j^j
D_m'm^j(0,-β,-α)  Y_jm'(θ,ϕ)
which shows
that c_jm'(α,β) = ∑_m=-j^jc̃_jm
D_m'm^j(0,-β,-α).].
BL81,SR73
It follows that 
the right hand side of Eq. (<ref>)
is given by
s_j ⟨  Y_j0(θ,ϕ) |
∑_m'=-j^j c_jm'(α,β) 
Y_jm'(θ,ϕ)
⟩_L^2=s_j  c_j0(α,β).







































































The left hand side of Eq. (<ref>)
is transformed into g_j(β,α)=R_αβ^-1
g_j(0,0) = ∑_m'=-j^j c_jm'(α,β) Y_jm'(0,0)
= s_j  c_j0(α,β), where  the formula
Y_jm'(0,ϕ) = s_j δ_m'0 
(see p. 16 of <cit.>) has been applied. In summary, we have verified that both sides
of Eq. (<ref>) agree, which completes the proof of Result <ref>.





§ PROOF OF RESULT <REF>


In this appendix, we demonstrate the tomography formula 
as given in Result <ref> for an operator A by 
applying the reconstruction formula of Result <ref>. The proof relies on mapping
A to (a set of) spherical functions
f^(ℓ)(θ,ϕ) as detailed in Sec. <ref>.


By substituting g(θ,ϕ) with  f^(ℓ)(θ,ϕ) 
in Result <ref>
for each label ℓ separately, one obtains that
f_j^(ℓ)(β, α)    =    s_j ⟨ R_αβ Y_j0(θ,ϕ) | f^(ℓ)(θ,ϕ)⟩_L^2.
Note that ⟨ A^(ℓ) | B^(ℓ)⟩
=⟨ f_A^(ℓ) | f_B^(ℓ)⟩_L^2 for the 
spherical functions f_A^(ℓ) and f_B^(ℓ) corresponding to the 
operators A and B, which can easily be  verified by expanding the 
arguments into tensor operators and spherical harmonics and
applying their orthonormality relations. Moreover,
the correspondence between operators and spherical functions
is covariant under rotations (see Proposition 1(d) in <cit.>), i.e.,
the operator ℛ_αβ T_j0^(ℓ) is mapped to
R_αβ Y_j0(θ,ϕ) 
[
This relation can also be established using the theory of Wigner-D matrices <cit.>
where 
D^j_m'm(α,β,γ) 
= ⟨ Y_jm'(θ,ϕ) | R_αβγ 
Y_jm(θ,ϕ) ⟩_L^2 (see p. 41 and 276 of <cit.>)
and D_m'm^j(α,β,γ) = 
⟨ T_jm'^(ℓ) | ℛ_αβγ T_jm^(ℓ)⟩
(see p. 45 of <cit.>).
Here, 
ℛ_αβγ C := R_αβγ C  R^-1_αβγ
with 
R_αβγ:=
e^- i α F_z e^- i β F_y e^- i γ F_z, 
cf. Sec. <ref>
where 
ℛ_αβ = ℛ_αβ 0.].
The last two statements imply that f_j^(ℓ)(β, α)    =    s_j ⟨ℛ_αβ T_j0^(ℓ) | A^(ℓ)⟩
=    s_j ⟨ℛ_αβ T_j0^(ℓ) | A⟩, where the last step follows as ⟨ A^(ℓ) | A^(ℓ')⟩ = 0 if ℓ≠ℓ'
(which is a consequence of the orthonormality of the tensor operators T_jm^(ℓ)). This 
completes the proof of Eq. (<ref>) in Results <ref>, and 
Eq. (<ref>) is then a direct consequence
due to the fact that ℛ_αβ T_j0^(ℓ)=
[ℛ_αβ T_j0^(ℓ)]^† is Hermitian.








§ DERIVATION OF EQ. (<REF>)


Here, we derive the formula of Eq. (<ref>)
given in Result <ref>
starting from Result <ref>. 
In standard NMR experiments, only the signatures of
Cartesian product operators <cit.> 
that contain a single transverse Cartesian operator I_ka with a∈{ x,  y}
(such as I_ka, 2I_kaI_lz, and 4I_kaI_lz I_mz)
can be measured 
directly, and hence the expectation values of axial operators T^(ℓ)_j0 are not directly 
accessible. 
Nevertheless, these expectation values can be measured indirectly
since the operators T^(ℓ)_j0=  ∑_n r^(ℓ,n)_j C^(ℓ,n)_j
can always be expressed as real linear combinations of (Hermitian) standard 
Cartesian product operators C^(ℓ,n)_j <cit.>. Thus, 
the tomography formula of Eq. (<ref>) can be rewritten as 
f^(ℓ)_j(β, α)=    s_j   ∑_n  r^(ℓ,n)_j⟨ R_αβ  
C^(ℓ,n)_j⟩_ρ. One obtains that
f^(ℓ)_j(β, α)=s_j   ∑_n  r^(ℓ,n)_j Tr{    C^(ℓ,n)_jρ̃} 
where ρ̃:=ℜ^†_αβρℜ_αβ
by exploiting the action 
R_αβ  C^(ℓ,n)_j = ℜ_αβ  C^(ℓ,n)_jℜ^†_αβ
via the operator ℜ_αβ:=exp(- i α∑_k=1^n I_kz) exp(- i β∑_k=1^n I_ky)
and the fact that the trace is invariant under cyclic permutations.
The operators C^(ℓ,n)_j can be transformed into measurable operators 
M^(ℓ,n)_j= U^(ℓ,n)_j C^(ℓ,n)_jU^(ℓ,n)†_j
with
unitary transformations
U^(ℓ,n)_j, which can be realized experimentally using radio-frequency pulses and 
coupling evolutions. Hence, the formula of Eq. (<ref>)
is given by f^(ℓ)_j(β, α) =  s_j   
∑_n r^(ℓ,n)_j Tr{  M^(ℓ,n)_jρ̃̃̃^(ℓ,j,n)}, where  
immaterial  cyclic permutations of the trace have again been applied and where
ρ̃̃̃^(ℓ,j,n):= U^(ℓ,n)_jρ̃ U^(ℓ,n)†_j.
Finally, the definition of the expectation value yields 
f^(ℓ)_j(β, α)= s_j ∑_n r^(ℓ,n)_j⟨ M^(ℓ,n)_j⟩_ρ̃̃̃^(ℓ,j,n).



§ PREPARATION AND DETECTION SEQUENCES 

We detail the explicit form of the preparation and detection sequences
used in the experiments in order to demonstrate our Wigner tomography.
We denote a pulse with flip angle β and phase α that is 
applied to spin k by [β]_α(I_k). Similarly,
[β]_α(I_k, I_l)
specifies two pulses both of flip angle β and phase α that are 
simultaneously applied to spins k and l.
We also use the notation 
𝒫^bil_x,y=[π/2]_y(I_2)-G-[π/2]_x,y(I_1)-t_a-
[π]_y,x(I_1, I_2)-t_a  which represents a pulse sequence
which is read from left to right. Here, G represents a pulsed magnetic field gradient 
that dephases all present transverse spin operators and t_a refers to a
time delay of length 1/(4J_12), where  J_12 is the coupling
constant between the first and second spin. The pulse sequences
𝒫^bil_x and 𝒫^bil_y
create from I_z the bilinear product operators 2 I_1xI_2z and 2 I_1yI_2z, respectively.
Moreover, the trilinear product operator  4I_xzz:=4I_1xI_2zI_3z is obtained from I_z
by applying the pulse sequence 
𝒫^tril_x=
[π/2]_y(I_2, I_3)-G-[π/2]_y(I_1)-t_b-[π]_y(I_1, I_3)-t_c-[π]_y(I_2)-t_d 
where the time delays are
t_b=1/(4J_13), t_c=1/(4J_13)-1/(4J_12), and t_d=1/(4 J_12).
Using these notations, the preparation sequences are given in 
Table <ref>.

For the detection, the Cartesian product operators C_j^(ℓ,n) 
have to be rotated into 
NMR-measurable operators M_j^(ℓ,n). The relevant pairs
of operators C_j^(ℓ,n) and M_j^(ℓ,n) are provided in
Table <ref>. The rotation pulse sequences 
are easily inferred, e.g., 
one uses the pulse [π/2]_y(I_k)
in order to rotate I_kz into I_kx. Similarly,
[π/2]_-y(I_k), [π/2]_x(I_k), and [π/2]_-x(I_k) rotate
respectively
I_kx, I_ky, and I_kz into I_kz, I_kz, and I_ky.
For example, 4I_zxy is rotated into 4I_xzz
using the pulse sequence [π/2]_x(I_3)-[π/2]_-y(I_2)-[π/2]_y(I_1).












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@bib@innerbibempty

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[Note2()]Note2
  note Spherical harmonics Y_jm(θ ,ϕ )=r(θ ,ϕ )
   oexp[iη (θ ,ϕ )] (and spherical
  functions) are plotted throughout this paper by mapping their spherical
  coordinates θ and ϕ to the radial part r(θ ,ϕ ) and
  phase η (θ ,ϕ ).Stop
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[Note3()]Note3
  note Cartesian operators for single spins are I_x:=σ _x/2,
  I_y:=σ _y/2, and I_z:=σ _z/2, where the Pauli matrices are σ _x= ( [ 0 1; 1 0 ] ), σ _y= ( [  0 -i;  i  0 ] ), and σ _z= ( [  1  0;  0 -1 ] ). For n spins, one has the
  operators I_k η := ⊗@ @ _s=1^n I_a_s
  where a_s is equal to η for s=k and is zero otherwise; note
  I_0:=  ([ 1 0; 0 1 ]
  ).Stop
[Note4()]Note4
  note Hermitian operators lead to positive and negative values
  which are shown in red (dark gray) and green (light gray).Stop
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[Note5()]Note5
  note Even though our arguments do not rely on explicitly knowing
  the expansion coefficients c_jm'(α ,β ), it might be instructive
  to specify the expansion coefficients in terms of the widely used Wigner-D
  matrices D_m'm^j(α ,β ,γ ) <cit.>. Let us also
  assume that the rank-j parts g_j(θ ,ϕ ) are expanded into
  g_j(θ ,ϕ )@ =@ ∑@ @
  _m=-j^jtilde07Ec_jm Y_jm(θ ,ϕ )
  using certain coefficients tilde07Ec_jm. The
  theory of Wigner-D matrices <cit.> implies that R_αβ^-1 +.1667em@ Y_jm(θ ,ϕ ) @ =@ ∑@
  @ _m'=-j^j@ D_m'm^j(0,-β ,-α ) +.1667em@ Y_jm'(θ ,ϕ ). Hence,
  we obtain the formula R_αβ^-1 +.1667em g_j(θ ,ϕ ) =∑@ @
  _m=-j^j@ tilde07Ec_jm@ ×∑@ @ _m'=-j^j@
  D_m'm^j(0,-β ,-α ) +.1667em@ Y_jm'(θ ,ϕ ) which shows that c_jm'(α ,β
  ) = ∑@ @ _m=-j^jtilde07Ec_jm D_m'm^j(0,-β ,-α ).Stop
[Steinborn and Rueddenberg(1973)]SR73
  author author E. O. Steinborn and author K. Rueddenberg, 10.1016/S0065-3276(08)60558-4 journal journal Adv. Quantum Chem. volume 7, pages 1 (year 1973)NoStop
[Note6()]Note6
  note This relation can also be established using the theory of
  Wigner-D matrices <cit.> where D^j_m'm(α ,β
  ,γ ) =  "426830A  Y_jm'(θ ,ϕ ) | 
  R_αβγ Y_jm(θ ,ϕ ) 
  "526930B _L^2 (see p. 41 and 276 of <cit.>) and D_m'm^j(α
  ,β ,γ ) =  "426830A  T_jm'^(ℓ ) |
  ℛ_αβγ T_jm^(ℓ ) "526930B (see p. 45 of <cit.>). Here, ℛ_αβγ C := ℜ_αβγ C ℜ^-1_αβγ with ℜ_αβγ:= e^- i α F_z e^- i
  β F_y e^- i γ F_z (see Sec. <ref>) where ℛ_αβ = ℛ_αβ
  0.Stop




