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The Wigner function and quantum state tomography
Boris Braverman
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Dated: May 16, 2012)
We present a pedagogical introduction to the subjects of Wigner functions and optical homodyne
tomography. The basic formalism of Wigner functions is given and its connection to tomographic
reconstruction of the quantum state of light discussed. The experimental signatures of nontrivial
quantum states such as Schrödinger cats are presented, along with a survey of notable experimental
measurements of Wigner functions. A complete simulated quantum tomography experiment is given,
highlighting both the robustness and limitations of the procedure.
INTRODUCTION
WIGNER FUNCTION
Quantum mechanics is the most complete theory of
Nature currently known, but it often conflicts with our
classical intuition. A notable deviation between quantum
and classical theories is the manner in which the states of
systems are specified. Quantum systems are described as
vectors (wavefunctions) |ψi within a Hilbert space, while
classical systems are properly characterized as a point
in phase space. A powerful use of the classical phase
space formalism is the Gibbs ensemble commonly used
in statistical mechanics, in which the states of a fictitious ensemble of systems are described as a probability
distribution in phase space [1]. Time evolution in this
picture is given by Hamilton’s equations, and the expectation values of any physical observables can be obtained
by averaging over the phase space.
Such an ensemble picture is often used to introduce the
inherently uncertain and statistical nature of quantum
mechanics and quantum measurement. Despite all its
strengths, a Gibbs ensemble cannot replicate all the predictions of quantum mechanics, in particular ones where
interference between different components of the ensemble can occur. In addition, while a classical system can
be perfectly localized, the state of a quantum mechanical
system must satisfy Heisenberg’s uncertainty principle.
Nonetheless, representations of quantum states within
a phase space can be made, of which the Wigner function
[2] is the most notable. The Wigner function provides an
accurate description for certain measurements performed
on quantum states, but only at the cost of being potentially negative. Therefore, the Wigner function cannot
be interpreted as a regular probability distribution for a
random variable, and is usually referred to as a “quasiprobability distribution”. While the possibility of negative probabilities having physical meaning seems bizzare,
this possibility has been studied by notable physicists,
such as Dirac [3] and Feynman [4]. If one comes to terms
with the possibility of having a negative-valued quasiprobability distribution for the states of a system, then
the entirety of quantum mechanics can be reformulated
in terms of the dynamics of the Wigner function within
phase space [5, 6].
Consider a system that exists in a continuous 1dimensional Hilbert space. Any state |ψi in the space
can
R be represented as a complex wavefunction: |ψi =
|qi ψ(q)dq. Similarly, a Rgeneral mixed state is expressed
as a density matrix ρ̂ = ρ(q1 , q2 ) |q1 i hq2 |. The Wigner
function W (q, p) provides an equivalent representation of
any quantum state in the quadrature phase space (q, p).
In the general case of a mixed state, the Wigner function
equals [2, 7]:
Z x
x ipx/h̄
1
ρ q − ,q +
e
dx, (1)
W (q, p) ≡
2πh̄
2
2
while for the special case of a pure quantum state, the
expression is evidently
Z
1
x x ipx/h̄
ψ∗ q −
W (q, p) ≡
ψ q+
e
dx.
2πh̄
2
2
Since this relationship is nothing but an unusual Fourier
transform, one can straighforwardly invert it to obtain
the density matrix from the Wigner function:
Z
x1 + x2
ρ(x1 , x2 ) = h̄ W
, p e−ip(x2 −x1 )/h̄ dp. (2)
2
Note that while the Wigner function is most commonly used in discussions of the quantum states of a
1-dimensional harmonic oscillator, such as a single mode
of the electromagnetic field, this representation is quite
general and can be used to represent any system. The
only thing that changes then is the time evolution of the
Wigner function, governed by the Moyal equation [5]. In
subsequent discussion, we assume that the Hamiltonian
of the system is that of a harmonic oscillator,
Ĥ =
P̂ 2
mω 2 Q̂2
+
,
2m
2
with energy eigenstates
1
Ĥ |ni = h̄ω n +
|ni ,
2
2
n = 0, 1, 2, . . ., and using the coherent states, defined as
|αi ≡
∞
X
αn
√ |ni .
n!
n=0
The most important feature of the Wigner function
is that it provides a faithful prediction for measurements of quadratures of the quantum mechanical state.
That is, if we measure the quadrature operator M̂ (θ) ≡
Q̂ cos θ + P̂ sin θ, then the statistical distribution of the
measurement outcomes m will equal the Radon transform of the Wigner function, given by
Z∞
W (m cos θ − p sin θ, m sin θ + p cos θ)dp.
p(m, θ) =
−∞
(3)
where S(Q̂, P̂ ) is summed over all possible orderings of Q̂
and P̂ . For example, if m = 2, n = 2, then S(Q̂2 , P̂ 2 ) =
(Q̂2 P̂ 2 + P̂ 2 Q̂2 + P̂ Q̂P̂ Q̂ + Q̂P̂ Q̂P̂ + P̂ Q̂2 P̂ + Q̂P̂ 2 Q̂)/6.
Because of these limitations in the intuitive predictive
power of the Wigner function, it is most of all useful
as an intermediate tomographic stage en route to the
reconstruction of the quantum state, which can then of
course predict the outcome of any measurement.
Finally, we note that the Wigner function is by no
means a unique way of representing a quantum state in
phase space. Two other most commonly used representations of quantum states in phase space are the Q and
P functions. The Q function equals the convolution of
the Wigner function of a state with the Wigner function of the vacuum, which turns out to be the same as
the overlap of the quantum state with different coherent
states:
Q(Re(α), Im(α)) =
1
hα| ρ̂ |αi .
2π
(5)
On the other hand, the P function is the deconvolution of
the Wigner function of a state with the Wigner function
of the vacuum, which turns out to be the coefficients of
a decomposition of the state into coherent states:
Z
ρ̂ = P ((Re(α), Im(α)) |αi hα| dα.
(6)
FIG. 1: The Radon transform R(m, θ) of the function f (x, y)
is found by integrating the function along the line connecting
A and A0 .
The Radon transform is a familiar operation from tomography, finding uses in fields such as medical imaging
[8] and pattern recognition [9]. Its action is very simple,
projecting out a 2-dimensional image onto different axes,
as illustrated in Figure 1. For an example, compare the
function in Figure 4d and its Radon transform shown in
Figure 6.
Note that this integral is precisely the manner in which
one would predict the statistics of measurements performed on an ensemble of classical harmonic oscillators
based on a statistical distribution of their locations in
phase space. Beyond the potential negativity of the
Wigner function, this quantum-classical analogy is limited in that one can predict the outcomes of only a subset
of all possible quantum mechanical measuremnts in such
a straightforward manner. In particular, one can predict
the outcomes of measuring symmetrized operators:
h
m
tr ρ̂S Q̂ , P̂
n
i
Z∞ Z∞
=
−∞ −∞
W (q, p)q m pn dqdp,
(4)
These representations are sometimes useful, although
mostly in a complementary role to the Wigner function.
Especially notable is the fact that the P function can be
used to define a rigorous classicality criterion: A state is
classical if its P function is non-negative and is not more
singular than a Dirac δ function anywhere. An equivalent set of nonclassicality criteria exist for the Wigner
function. A state is nonclassical if its Wigner function is
non-negative anywhere, which immediately implies that
it cannot be expressed as a mixture of classical coherent states, or if the variance of any quadrature operator
M̂ (θ) is smaller than its value in the vacuum state. These
nonclassicality criteria for the Wigner function are useful
because a deconvolution operation is necessary to find P ,
making it extremely difficult to reconstruct accurately in
experiments.
Wigner function examples
The Wigner functions for the Fock states |0i, |1i, |2i,
|3i are shown in Figure 2. Several common features
are immediately apparent between these functions. The
Wigner function of |ni resembles the wavefunction for |ni
in having n zero-crossings; the functions are all radially
symmetric; the even n states have a peak at the origin,
while the odd n states have a dip at the origin. The
magnitudes of the peak and dip are both equal to 1/π.
3
(a) Vacuum state |0i
(b) Fock state |1i
(c) Fock state |2i
(d) Fock state |3i
FIG. 2: Wigner functions of the first four Fock states. Each
Wigner function is shown as a contour plot and as a 3D surface
map, with red corresponding to positive values and blue to
negative ones. Contours are spaced at 0.02.
In fact, this value bounds any possible Wigner function:
|W (q, p)| ≤ 1/π for any physical W (q, p).
The vacuum state |0i is the only state that is simultaneously a coherent state and a Fock state, and indeed
the Wigner function of any coherent state |αi is identical to that of the vacuum state, with a translation in
phase space to P = Re(α), Q = Im(α). This embodies
the quasi-classical, minimum-uncertainty nature of the
coherent states–their quasi-probability distributions are
as close to a δ function as quantum mechanics will allow.
The Wigner functions of these Fock state have been
successfully measured in many experiments. Some notable examples are the first reconstructions of a |1i
Wigner function for the motion of an atom [10] and for
an optical mode [11], as well as the |2i state in an optical
mode [12].
Notice also that the Wigner function is linear in the
density matrix of a quantum state. This means that the
Wigner function is additive for the incoherent mixture
of two states, but is not additive for a coherent superposition, which allows it to encode both phase and coherence of quantum states. For an example, compare
Figures 3a, 3b, and 3c. For the coherent superpositions
|0i + |1i and |0i + i |1i, the Wigner function gains an
evident directionality which encodes the phase between
the two states. On the other hand, the incoherent superposition ρ̂ = |0i h0| + |1i h1| has a Wigner function
that is radially symmetric, with no phase to encode, and
non-negative anywhere, indicating that the state is now
classical. This destruction of quantum behaviour due
to decoherence and state mixing is a familiar trend in
quantum mechanics, and can be readily visualized and
understood in the language of Wigner functions [13].
Finally, we consider some experimentally important
examples of nonclassical states. Figure 4a shows a
squeezed vacuum state, which in general is a state where
the variance of one of a pair of canonically conjugate
variables has been reduced below the vacuum noise level,
while the variance of the other variable has correspondingly increased to satisfy the Heisenberg uncertainty
principle. For the specific state in Figure 4a, the variance
of the momentum hP̂ 2 i has been reduced by a factor of
eζ ≈ 5, with a concommitant increase in the variance of
the position variable by the same factor. These states
can notably be generated using nonlinear optical processes [14], which makes them experimentally acessible,
and they function as the primary source of nonclassical
light in most current quantum optics experiment. Some
of the first reconstructions of Wigner functions were indeed those of squeezed states [15, 16].
The state shown in Figure 4b is a photon subtracted
squeezed vacuum state. This is a squeezed vacuum state
that has been projected onto the subspace {|1i , |2i , . . .},
and then all energy eigenstates are reduced by one:
|ni → |n − 1i. Again, the significance of this state comes
from its experimental accessibility; one can generate such
a state by taking a squeezed vacuum state and impinging
it on a beamsplitter with near-unity transmission. If a
photon is detected in the reflection mode from the beamsplitter, then this state will be generated in the transmission mode. These states have also been recently observed
[17, 18].
4
In Figure 4c, we show the Wigner function for a
Schrödinger kitten state, which is simply a superposition of the two coherent states |α = 1i and |α = −1i.
These two states are usually taken to represent the cat’s
macroscopically distinguishable states |alivei and |deadi
from Schrödinger’s famous thought experiment [19]. As
can be seen, the two states |α = 1i and |α = −1i are not
very distinguishable, and there are not two quasi-classical
humps corresponding to the two states. However, if we
increase the amplitude of the coherent states to 3, as
done in Figure 4d, then we see all the features typical
of Wigner functions of macroscopic superposition states:
there are two humps corresponding to the Wigner function of each state independently, and in between there
are rapid oscillations with large negative values that encode the coherence of the superposition of the two states.
If the state was mixed rather than coherent, these fringes
would not be present. The the generation and quantum
tomography of cat states has only been realized recently
[13, 20] because of the extreme difficulty of creating such
large nonclassical states.
(a) Coherent superposition state |0i + |1i
(b) Coherent superposition state |0i + i |1i
HOMODYNE DETECTION
Above, we mentioned the remarkable property of the
Wigner function to predict the statistics of the quadrature measurements of quantum light, which can be measured through the technique of balanced homodyne detection. Indeed, the popularity of the Wigner function
as a representation of quantum optical states can be directly connected to its usefulness in understanding this
type of experiments.
The basic setup of a balanced homodyne tomography
experiment is shown in Figure 5. The quantum state
under study ρ̂ is interfered on a 50% transmission beamsplitter with a strong coherent state |αi = ||α|eiθ i, called
the local oscillator (i.e. |α| nm , where hn| ρ̂ |ni is negligible for n > nm ). The phase θ of the local oscillator
is varied, and the statistics of the difference in the currents I1 and I2 through the photodiodes PD1 and PD2
are recorded for different θ. If the current is proportional
to the number of photons hitting each detector, then we
have
I2 − I1 ∝ n̂21 = n̂2 − n̂1 = â†2 â2 − â†1 â1 ,
where a1 and a2 are the annihilation operators for the
light modes hitting the two photodetectors. Using the
beamsplitter Hamiltonian, we have the operators for the
photodiode modes
1
â1 = √ (â − α)
2
1
â2 = √ (â + α)
2
(c) Incoherent mixture |0i h0| + |1i h1|
FIG. 3: The Wigner function encodes information about the
amplitudes, phases, and coherences of quantum states.
where we are treating the coherent state classically, so its
operator is replaced by its classical amplitude. √
Recalling
that α =√ |α|(cos θ + i sin θ), Q̂ = (â + ↠)/ 2, P̂ =
(â − ↠)/ 2, we find
√
√
(7)
n̂21 = 2|α| Q̂ cos θ + P̂ sin θ = 2|α|M̂ ,
where M̂ is the quadrature operator defined earlier. Notice that in addition to allowing measurement of the operator M̂ , the method also amplifies the weak photonic
signal by a factor of |α|. This is a hugely important experimental consideration, as it is a technical challenge
to detect single photons, especially over a background of
experimental noise. The addition of this enhancement
factor allows the signal to noise ratio in current experiments to greatly exceed the signal to noise ratio possible
with single photon detectors. Physically, this enhancement comes about because the amplitudes of the quantum state ρ̂ and the coherent state α that are added on
5
(a) Squeezed vacuum state with squeezing parameter ζ = 1.6
FIG. 5: Configuration of a balanced homodyne tomography
experiment.
(b) Photon subtracted squeezed vacuum, ζ = 1.6
qubits encoded in atomic systems. Thus, if one encodes
a qubit in the spin of a nucleus, it is readily possible
to perform arbitrary rotations of the state and perform
measurements in any basis on it. On the other hand, for
a photonic qubit encoded in the different Fock states, the
equivalent operations are extremely challenging.
(c) Schrödinger kitten |α = 1i + |α = −1i
(a)
(b)
FIG. 6: (a) Radon transform of the Wigner function of the
cat state |ψc i in Fig 4d, which equals the ideal experimental
distribution of quadrature measurements. The vertical axis is
normalized to 1. (b) The same, after the addition of simulated
experimental noise.
(d) Schrödinger cat |α = 3i + |α = −3i
FIG. 4: Examples of Wigner functions for some typical nonclassical states.
the beamsplitter, and when the amplitude is squared to
obtain an intensity, the inteference term is proportional
to the magnitude of both ρ̂ and α.
Another notable reason for the popularity of homodyne tomography for the measurement of the quantum
states of light is the fact that there are no robust ways to
transform the quantum state of light, unlike for example
FIG. 7: Reconstructed Wigner function of the cat state |ψc i.
6
(a)
(a)
(b)
(b)
FIG. 8: (a) Real and imaginary parts of the density matrix
ρ(x1, x2) of the cat state |ψc i. (b) Same, for the reconstructed
state obtained from the Wigner function in Figure 7.
RECONSTRUCTION EXAMPLE
After the discussion of the basics of Wigner functions
and homodyne tomography, we present a simulated quantum tomography experiment to illustrate the whole procedure. In this experiment, we simulate the measurement
of the Schrodinger cat state |ψc i = |α = 3i + |α = −3i.
The Wigner function of this state is shown in Figure 4d,
while the Radon transform of the Wigner function, which
is equal to the statistical distribution of quadrature measurements, is shown in Figure 6a. Note the narrow interference features near θ = π/2, which encode the phase
coherence between the two coherent states that make up
the Schrödinger cat state.
To simulate an experimental reconstruction, we add
noise to the quadrature components and attempt to recover the original quantum state. First, we coarse-grain
the quadrature measurement to only 80 possible outcomes, such that the width of each Gaussian hump near
θ = 0 is approximately 10 quadrature bins, and each
sharp interference fringe near θ = π/2 is about 5 bins
wide. We add a constant background technical noise giving a signal to noise ratio of 10 for the Gaussian humps
near θ = 0, and a photon shot noise corresponding to having 3000 measurements per phase setting. The number of
phase settings used for reconstruction is 30, and we simulate flucuations in the phase by averaging over a π/20
FIG. 9: (a) Real and imaginary parts of the density matrix
of the cat state |ψc i expressed in the Fock state basis, i.e.
ρmn = hm| ρ̂ |ni . Note that only when both m and n are even
is the matrix element non-zero, because of the destructive
interference between the odd Fock components of the two coherent states making up the Schrödinger cat. (b) The same,
for the reconstructed state.
FIG. 10: Diagonal elements ρmm of the density matrices from
Figure 9, comparing the actual state with the reconstructed
one. In particular note that the reconstructed state is unphysical, since it has negative diagonal matrix elements.
7
angular interval around the phase setting, corresponding
to the experimental reality that while the experiment is
running, phase fluctuations are not recorded, but simply
averaged over. The resulting simulated quadrature measurement is shown in Figure 6b. Note that these noise
values are quite conservative, with state of the art experiments achieving much higher quality signals.
Next, we perform an inverse Radon transform on the
quadrature distribution to reconstruct the Wigner function, which is shown in Figure 7. As can be seen, the
reconstruction is very noisy, but recovers the key features
of the state: the two humps at P = 0, Q = ±4, and the
narrow interference fringes near the origin. Despite the
large amount of noise added to the simulated measurements, the nonclassicality of the state is still very evident
in the reconstruction. This illustrates the robustness of
quantum state tomography using the Wigner function –
even with a large level of noise, one can recover indicators
of interesting quantum states from the experiment.
We can then use (2) to recover the density matrix of
the quantum state, which is compared to the true density matrix in Figure 8. Again, the reconstructed density matrix maintains the key nonclassical features of the
original state (notably the humps at x1 = −4, x2 = 4 and
x1 = 4, x2 = −4 which indicate the coherence between
the two coherent states), while suffering from added noise
from the reconstruction procedure. The conventional
way of presenting these density matrices is in the basis of harmonic oscillator eigenstates, which is shown in
Figure 9. The RMS error of the elements of the density
matrix is 0.012, which is consistent with the level of noise
introduced into the homodyne measurements.
This straightforward method of recovering the quantum state by performing an inverse Radon transform on
the measured quadrature values suffers from two limitations. First, the reconstruction relies on physically arbitrary parameters in the filtering necessary to perform the
backprojection for the inverse Radon transform. More
importantly though, the reconstruction can lead to nonphysical final states. This can be seen even in our reconstruction above. A physical density matrix must have
all non-negative eigenvalues, making it positive semidefinite. In particular, this implies that the elements of the
density matrix on its diagonal must be positive in any
basis. However, as can be seen in Figure 10, this is not
the case, with ρ̂21,21 and ρ̂23,23 being negative.
The most popular method of dealing with these limitations is maximum likelihood estimation [21, 22]. The
basic idea of the method is to ask: “what is the physically
allowed state that most likely would have produced the
observed distribution of quadratures”? This approach
guarantees that the reconstructed state will be physically
meaningful. From the perspective of Bayesian statistics,
this approach is equivalent to starting with a uniform
prior distribution over the allowed density matrices, and
using Bayes’ theorem for each measurement result to up-
date the distribution. This correspondence to a Bayesian
approach also leads to another advantage of maximum
likelihood estimation over the inverse Radon transform –
that one can find a reliable estimate of the uncertainties
in the reconstructed state.
CONCLUSION
The field of quantum state tomography has blossomed
in the last 20 years, driven by an increased understanding
of the quantum nature of light and the creation of highrate nonclassical light sources, primarily through spontaneous parametric downconversion. The Wigner function’s ability to visually display the behaviour of quantum states, as well as its straightforward experimental
interpretation have given it a central role in this new and
exciting field. Clearly, only a taste of the richness of the
field can be given here, but many excellent review papers and books have been written on the subject, such
as [7, 23, 24].
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