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Choice of Measurement Sets in Qubit Tomography
Mark D de Burgh,1 Nathan K. Langford,1 Andrew C. Doherty,1 and Alexei Gilchrist1
1School of Physical Sciences, University of Queensland, St Lucia, Queensland 4072, Australia
(Dated: February 1, 2008)
Optimal generalized measurements for state estimation are well understood. However, practical
quantum state tomography is typically performed using a fixed set of projective measurements and
the question of how to choose these measurements has been largely unexplored in the literature. In
this work we develop theoretical asymptotic bounds for the average fidelity of pure qubit tomography
using measurement sets whose axes correspond to vertices of Platonic solids. We also present
complete simulations of maximum likelihood tomography for mixed qubit states using the Platonic
solid measurements. We show that overcomplete measurement sets can be used to improve the
accuracy of tomographic reconstructions.
I. INTRODUCTION AND BACKGROUND
Quantum tomography [1], the practical estimation of
quantum states through the measurement of large num-
bers of copies, is of fundamental importance in the study
of quantum mechanics. With the emergence of quantum
information science, the tomographic reconstruction of
finite dimensional systems [2] has also become an essen-
tial technology for characterizing the experimental per-
formance of practical quantum gates and state prepara-
tion. Examples include tomography of the polarization
states of light [3, 4] and electronic states of trapped ions
[5, 6]. These experiments would benefit from a systematic
study of the optimal measurement and state estimation
strategy to use.
There has been much theoretical work in this area,
and optimal bounds on state estimation, and construc-
tions for measurements that achieve these bounds are
known [7, 8, 9, 10, 11]. These bounds require all copies
of the state to be collected and a combined measurement
performed across all the copies. While these collective
measurements are known to be more powerful than per-
forming measurements on each copy one at a time (local
measurements) [7], they are currently totally impractical.
Qubit tomography is currently done using fixed local
projective measurements [12, 13]. The important ques-
tion of which fixed local projective measurement sets to
use remains an open problem, although there are several
relevant theoretical discussions [14, 15, 16, 17, 18, 19].
In practice, measurement sets with the minimal num-
ber of measurements such as those described in [12] have
become popular, particularly in optical experiments (but
see [4]). This situation arises since in optical experiments
a significant amount of time can be spent on changing the
measurement settings. However, such choices are made
without much quantitative understanding of how the per-
formance of the tomographic reconstruction is affected by
restricting the measurement sets in this way.
In this paper we investigate how choice of measure-
ments affects the quality of tomographic reconstruction.
We follow Jones [14] and investigate a class of measure-
ment sets based on Platonic solids that provide excellent
performance for tomography using fixed local projective
0.98
0.985
0.99
0.995
0.975
1.00
ϑ (pi rad)
ϕ
(pi
ra
d)
0 0.5 1 1.5 2
−0.5
0
0.5
ϕ
(pi
ra
d)
0 0.5 1 1.5 2
−0.5
0
0.5
+/−
+i/−i
0/
1
−1
0
1−1 0 1
−1
−0.5
0
0.5
1
+/−
+i/−i
0/
1
−1
0
1−1 0 1
−1
−0.5
0
0.5
1
FIG. 1: Performance of tomographically reconstructed states
on the Bloch sphere for two measurement sets. The top set
is a popular set of measurements (James4) used in the liter-
ature [12], the bottom set is a more isotropic set composed
of measurements along the three spatial axes (a cube mea-
surement set), see text for details. For each target state the
average fidelity with 400 reconstructed states is plotted by
color. The ensemble of reconstructed states was generated by
adding Posssonian noise to a simulated experiment with a to-
tal of 4000 counts, and then performing maximum-likelihood
tomography.
measurements. This class of measurement sets allows one
to trade off performance against the number of measure-
ment settings, with more overcomplete measurement sets
performing better. For example in Fig. 1 we show that
both the average and worst case fidelities of the tomo-
graphic reconstruction are greatly improved by replacing
the popular minimal measurement set of [12] with the
overcomplete cube Platonic measurement set.
Platonic solid measurements were first proposed for to-
mography in [14] for the special case of pure states and
using mutual information as a figure of merit. However,
this figure of merit makes it hard to compare these re-
sults to more recent work which typically uses the average
fidelity. We derive new asymptotic bounds for the perfor-
mance of Platonic solid measurements for pure states us-
ing average fidelity as a figure of merit. We then present
full numerical simulations of both one and two-qubit to-
mography performance as a function of the number of
copies of the system, using these measurement sets. This
provides a detailed study of how information is acquired

2during a tomography experiment. To our knowledge such
a study has not been performed in experimental work
but would determine whether tomographic reconstruc-
tions are limited by the expected statistics or by other
experimental imperfections, such as drifting sources.
The average fidelity is the most commonly used mea-
sure of the performance of state reconstruction conse-
quently we make use of this figure of merit. Much of
the original interest in fidelity arises from the fact that it
bounds the distinguishability of quantum states, see [20].
Recently a true quantum Chernoff bound on the distin-
guishability of states has become available [21] and this
quantity is a tighter bound than the fidelity. As a result
we use this quantity as figure of merit also, although the
results do not depend greatly on this choice.
Very recently, Roy and Scott [19] identified a class of
measurements, including all the Platonic solid measure-
ments, that give the optimal performance for another fig-
ure of merit, the mean squared Hilbert-Schmidt distance.
Nevertheless, it will turn out that when average fidelity
is used as the figure of merit, there is a range of perfor-
mance within the class, with the higher order Platonic
solids performing better. This indicates that the choice
of figure of merit can certainly have a qualitative effect
on the comparison between tomographic procedures.
The structure of the paper is as follows: We will in-
troduce the tomography problem, and discuss figures of
merit for tomography schemes in section II. In section III
we define the Platonic solid measurements. In section IV
we describe some other measurement sets that have been
studied in the literature for comparison with the Pla-
tonic solid measurements. We present theoretical pure
state asymptotic bounds on the average fidelity of the to-
mographic reconstruction for the different platonic solid
measurements in section V. We then give some intuition
for mixed state tomography in section VI. After present-
ing details of our mixed state simulations for atomic and
photonic qubit systems in section VII we give our results
for one and two qubit systems in section VIII. Finally
we will investigate the effects of using different figures of
merit in sections IX and X.
II. QUBIT TOMOGRAPHY
A d dimensional quantum state ρ is represented by a
d×d positive semidefinite density matrix, with trace one.
There are d2−1 real parameters to be estimated. In the
case of optical experiments such as [3, 4] the flux must
also be estimated giving d2 real parameters to estimate.
Any setting l of an experimental apparatus designed
to measure the quantum state may be described by a
positive operator value measure (POVM). Each of the k
outcomes of a measurement setting is represented by a
positive semidefinite operator Olk. The operators satisfy
∑
k Olk = I. The probability of observing outcome k is
given by the Born rule plk = tr(ρOlk). In state tomogra-
phy each of the l measurements are performed on a large
number of copies and estimates obtained for each of the
plk. We denote these estimates pˆlk. When the number of
linearly independent Olk equals or exceeds the number of
parameters to estimate, our measurements are known as
informationally complete [22]. If these probability esti-
mates were perfect, so that pˆlk = plk, it would be possible
to reconstruct the state exactly. In this case there is a
set of operators Rlk, known as a dual basis or dual frame,
such that ρ = ∑lk plkRlk. (When the number of linearly
independent Olk exceeds the number of parameters to be
estimated the dual frame is not unique.) Since our prob-
ability estimates are not perfect due to the finite sample
size, they may well be inconsistent with any such recon-
struction. In this case it is common practise to resort to
a maximum likelihood estimate of the state [12, 13]. We
look at maximum likelihood reconstruction in detail in
section VIIC
To judge the quality of reconstruction we need a mea-
sure of similarity between the true state ρ and the re-
constructed state ρˆ. In this paper we will concentrate on
two measures. The first is the commonly used fidelity:
f(ρ, ρˆ) =
[
tr
√
√
ρˆρ
√
ρˆ
]2
. (1)
where 0 ≤ f(ρ, ρˆ) ≤ 1 and f = 1 implies ρ = ρˆ. The sec-
ond is the quantity that determines the recently derived
quantum Chernoff bound [21]:
λcb(ρ, ρˆ) = min
0≤s≤1
tr[ρsρˆ1−s]. (2)
where as for the fidelity, 0 ≤ λcb(ρ, ρˆ) ≤ 1 and λcb = 1
also implies ρ = ρˆ.
The quantum Chernoff bound has a clear physical in-
terpretation that can be understood by considering the
following situation. Suppose an experimentalist wishes to
determine if her state preparation device creates the state
ρ or the state σ. Time is limited and she only has N iden-
tically prepared copies of the state to work with. She is
assumed to be able to perform any measurement, includ-
ing a collective measurement, in her efforts to distinguish
the two states. The result of [21] is that the probability of
her making an error is asymptotically Pe ≈ eN lnλcb(ρ,σ)
where λcb is called the quantum Chernoff bound. When
one of the states is pure, the quantum Chernoff bound
coincides with the fidelity. Also the square root of the fi-
delity is always an upper bound on the quantum Chernoff
bound. [20, 21]. Somewhat confusingly this quantity has
also been termed the fidelity in other works and serves
as an alternative measure of distinguishability. We have
chosen the definition in (1) for it’s appealing interpre-
tation when one of the states is pure, and because all
of the theoretical results we mention are derived using
this definition. As the quantum Chernoff bound has a
clear physical interpretation even when both states are
mixed, we consider it a better motivated measure of the
distinguishability of ρ and σ.
Next we define quantities which rate the performance
of a tomography scheme for a particular true state ρ,

3using the fidelity and quantum Chernoff bound. The fig-
ure of merit we adopt is based on how distinguishable the
true state is from the reconstructed state, when averaged
over measurement outcomes and an appropriate ensem-
ble of true states. (An interesting alternative approach
was advocated by Blume-Kohout and Hayden [23].) A
tomography experiment produces a list of outcomes of
measurements performed on N copies of the true state
ρ. These outcomes are used to obtain an estimate of the
state ρˆ, and the fidelity (or quantum Chernoff bound)
between the true and estimated states is calculated. Av-
eraging over the outcomes of many such tomography ex-
periments we obtain the point fidelity
F (ρ,N) =
∑
ρˆ
pρ(ρˆ)f(ρ, ρˆ) (3)
where pρ(ρˆ) is the probability of estimating state ρˆ given
the true state ρ. Or point Chernoff bound :
λpt(ρ,N) =
∑
ρˆ
pρ(ρˆ)λcb(ρ, ρˆ). (4)
Finally averaging the point fidelity (or point Chernoff
bound) over all possible true states, we obtain the average
fidelity:
F =
∫
dρ F (ρ,N). (5)
or the average Chernoff bound :
λav =
∫
dρ λpt(ρ,N). (6)
These are the figures of merit we will use to rate a to-
mography scheme.
To perform the averages over true states in equa-
tions (5) and (6), we must choose a prior probability
distribution over true states. For pure states there is one
clear choice — the Haar measure. For mixed states the
choice is not so clear, see for example [24]. We will make
use of two priors, one based on the fidelity, and the other
on the quantum Chernoff bound.
The first, known as the Bures prior, is the density of
states induced by defining a volume element using the
distance 1/2 arccosf(ρ, ρ + dρ). It is a natural choice
when using the fidelity as our measure of similarity be-
tween states. It has also been argued that the Bures prior
corresponds to maximal randomness of the input states
[25]. The Bures prior is unitarily invariant, and it gives
a radial probability density on the Bloch sphere:
pB(r) =
4
π
r2√
1− r2
(7)
When the quantum Chernoff bound is used as the mea-
sure of similarity, it is natural to use a prior over states
based on this. The natural metric induced by the quan-
tum Chernoff bound was derived in [21]. Some of the
authors of [21] have also derived the corresponding vol-
ume element in the space of states which can be used as
a prior on mixed states [26]. In Appendix B we present
our general derivation for states of arbitrary dimension.
The resulting Chernoff prior is unitarily invariant and the
probability density over eigenvalues is given by:
p(λ1..λM ) =
C√
λ1..λM
∏
k<j
(λk − λj)2
(
√
λj +
√
λk)2
(8)
where the eigenvalues are also constrained to satisfy
∑M
j=1 λj = 1, and C is a normalisation constant. For
single qubit states this corresponds to a radial probabil-
ity density on the Bloch sphere given by:
pCB(r) =
2
π − 2
1−
√
1− r2√
1− r2
. (9)
A comparison of the density of the Bures and Chernoff
priors is given in Fig. 2. Both these priors have increasing
density with state purity. The Chernoff prior is slightly
more skewed towards pure states, than the Bures prior.
We will present all of our mixed state results in terms
of average fidelity with states drawn according to the
Bures prior first, as this has become the popular way of
assessing mixed state tomography. We will then present
our detailed results using the average Chernoff bound
figure of merit with states drawn according to the Cher-
noff prior. We believe that this figure of merit is better
motivated and thus it is important check that our quali-
tative conclusions do not depend on this choice of figure
of merit.
In other works the mean square Hilbert-Schmidt dis-
tance and the mean squared Euclidean distance between
Bloch vectors have been used as a figures of merit for to-
mography [18, 27]. For single qubits these measures are
equivalent up to constant factors. The Hilbert-Schmidt
distance is defined as Dhs(ρ, σ) =
√
tr(A†A) ,where
A = ρ − σ. It is again related to the probability of er-
ror when an experimentalist must determine if she has a
state ρ or state σ, however in this case the experimentalist
has only a single copy to measure. Under the assump-
tion that both states are equally likely before the mea-
surement, the probability of error in distinguishing single
qubit states ρ and σ is Pe = 12 − 12√2Dhs(ρ, σ). In order
to get an analytically tractable measure of distinguisha-
bility [18, 27] square Dhs before averaging this quantity
over measurement outcomes and states as we have done
for fidelity and the quantum Chernoff bound. This figure
of merit makes it possible to find the optimal measure-
ment sets analytically but after squaring and averaging
this quantity is not as well motivated. We will examine
the effect of using this figure of merit in section X and
show that it can lead to qualitatively different behaviour.

40 0.2 0.4 0.6 0.8 10
2
4
6
8
10
Bloch radius
Pr
ob
ab
ilit
y
de
ns
ity


Bures Prior
Chernoff prior
FIG. 2: Plot of probability density against Bloch radius for
the Bures and Chernoff priors over states. Both distributions
have densities which increase with state purity. The Chernoff
prior is slightly more skewed towards the pure states than the
Bures prior.
III. PLATONIC SOLID MEASUREMENTS
The state density matrix of a qubit can be written as:
ρ = I + ~r.~σ
2
(10)
where ~r is a real three dimensional vector known as the
Bloch vector, and ~σ is a vector of the Pauli matrices
σx, σy and σz. In the special case of pure states, the
Bloch vector is unit length. A two-outcome local projec-
tive measurement on the lth measurement setting can be
represented by two projectors:
O(± ~ml) =
I ± ~ml.~σ
2
(11)
where ~ml is a real three dimensional unit Bloch vector.
The orthogonal outcomes correspond to opposite direc-
tions on the Bloch sphere.
As states are isotropically arranged in the Bloch ball,
one intuitively expects tomography performance to de-
pend on how isotropically spaced our measurement vec-
tors are on the Bloch sphere. This intuition was sup-
ported (and made precise) by the asymptotic pure state
results of [14], and is demonstrated in Fig. 1. In this pa-
per we will concentrate on the measurement sets whose
Bloch vectors form Platonic solids. These are the five
convex regular polyhedra: tetrahedron, cube, octahe-
dron, dodecahedron and icosahedron. Such shapes are
highly isotropic and thus would be expected to give ex-
cellent tomographic performance.
Mathematically, the idea of isotropically spacing points
on a sphere is also captured by the concept of a spher-
ical t-design. A description of t-designs is beyond the
scope of this paper but note that the properties of t-
designs mean that they are excellent sets of points for per-
forming numerical estimation of integrals over the sphere
with the quality of the approximation being expected
to improve as t is increased. This suggests that good
measurement sets for tomography may well be associ-
ated with t-designs. In fact, popular measurement sets
for tomography of general quantum systems include the
so-called symmetric informationally complete POVMs
SicPOVMs [28] which are spherical 2-designs. We note
that the tetrahedron measurement set described below
is the SicPOVM for a single qubit, and all the Platonic
solids are also 2-designs. In fact it is well known [29] that
the cube and octagon are 3-designs and the dodecahedron
and icosahedron are 5-designs. (Note that a t-design is
also a t′-design for all t′<t). This further motivates us to
concentrate on the performance of the Platonic solids.
There are two possible ways of defining Platonic mea-
surement sets. We define a Platonic solid measurement
set to be a set of L different measurement settings whose
2L Bloch vectors ± ~m1 · · · ± ~mL match the 2L centres
of the faces of a Platonic solid. We use this definition so
that an octagon has eight measurement directions. An al-
ternative definition would be to define the Platonic solid
measurement as the set of Bloch vectors matching the
vertices of a Platonic solid. This definition would result
in the same set of shapes because the centres of the faces
of one Platonic solid, match the vertices of another Pla-
tonic solid. Such solids are said to be duals of each other.
The dual pairs are the tetrahedron with itself, the cube
with the octagon and the dodecahedron with the icosa-
hedron. We find the first definition more appealing and
use it throughout the paper.
The experiments we will be modelling are described
in detail in section VII. In the atomic qubit experiment,
and the dual-detector photonic experiment each measure-
ment has two orthogonal outcomes. These correspond to
opposite directions on the Bloch sphere. Thus two oppo-
site directions are measured simultaneously by a single
measurement. All the Platonic solids except the tetrahe-
dron have a face opposite every face and thus these Pla-
tonic solid measurements can be measured in this way
while the tetrahedron cannot. (Attempting to measure
the tetrahedron directions would result in us measuring
the octagon instead). For the single-detector photonic
qubit experimental model, we will require L measure-
ment settings to obtain the L outcomes, as the orthog-
onal outcome is not detected. This enables a tetrahe-
dron measurement to be meaningfully measured in this
case. Direct measurement of the tetrahedron is possible
in principle and schemes have been proposed [30, 31],
that would however require significantly greater experi-
mental resources so we will not consider this possibility
in the following.
We will also be interested in two qubit states. The ex-
perimentally simplest measurements are projective mea-
surements performed locally on both subsystems (record-
ing joint outcome probabilities). While we would expect

5measurement sets that include measurements entangled
across both subsystems to give improved results, these
measurements are difficult to perform. We will examine
the performance of Platonic solid measurements on each
subsystem. That is measurements of the form:
O(± ~mi)⊗O(± ~mj) (12)
where both i and j range over the L measurements of the
same Platonic solid.
IV. MEASUREMENT SETS FOR
COMPARISON
For simulations of one and two qubit optical exper-
iments (single-detector configuration) we will compare
our results to the popular measurement sets described
in [12]. The one qubit state measurement set consists
of measuring polarizations { H, V, D, R }, that is the
horizontal and vertical linear polarizations, the diagonal
linear polarization [|D〉=(|H〉+|V 〉)/
√
2], and the right
circular polarization [|R〉=(|H〉+i|V 〉)/
√
2]. We refer to
this measurement set as James4. The two qubit state
measurement set is { HH, HV, VV, VH, RH, RV, DV,
DH, DR, DD, RD, HD, VD, VL, HL, RL } where for ex-
ample the measurement setting HL means measuring hor-
izontal polarization on the first qubit subsystem and left
circular polarization [|L〉=(|H〉−i|V 〉)/
√
2] on the second
qubit subsystems [36]. We refer to this measurement set
as James16. (Notice that this measurement set does not
form complete POVMs. The implementation we have in
mind is explained fully in Section VII)
In the case of two qubits we also consider a modified
version of the SicPOVM of [28] for comparison. We call
this measurement a projective SicPOVM. The original
SicPOVM for two qubit states, is an entangled measure-
ment in which a single measurement produces one of 16
outcomes. Experimentally this measurement is difficult
to realize. Our projective SicPOVM is a measurement
set of 16 settings, in which each setting projects onto one
of the directions of the SicPOVM. While still entangled
this measurement should be simpler to implement exper-
imentally than the SicPOVM.
V. PURE STATE ESTIMATION OF QUBITS
WITH PLATONIC SOLIDS
The bound on average fidelity for collective measure-
ments is known to be: F≤(N+1)/(N+2) ≈ 1−1/N
where the approximation is good for large N . In fact
this bound is achievable with collective measurements [7].
When restricted to local measurements, measuring the
three orthogonal axes of the Bloch sphere (a cube mea-
surement) is known to give an average fidelity bounded
by F≤1−13/(12N) and again this bound is achievable
[32]. We have derived average fidelity bounds for the
other Platonic solids. The derivation only slightly gen-
eralizes Appendix B of [15]. We give an overview of our
derivation here, leaving the details to Appendix A.
These bounds on average tomography performance
are a consequence of the Crame´r-Rao bound of classical
statistics. The Crame´r-Rao bound concerns the problem
of sampling from a probability distribution, where the
probability distribution is determined by some parame-
ters. The goal is to estimate these parameters based on
the results of sampling the distribution. The Crame´r-Rao
bound states that the variance of an unbiassed estimator
is asymptotically lower bounded by O(1/N), and the co-
efficient of this scaling is given by the Fisher information
(the definition may be found in Appendix A).
In a tomography experiment, with a fixed set of mea-
surements, the outcome probability distribution is fixed
by the parameters of the quantum state we are trying
to estimate, and we can directly apply the Crame´r-Rao
bound. The procedure is as follows:
1. Consider a tomography experiment on a quantum
state with parameters η, and its fidelity to the
state estimated from the tomographic reconstruc-
tion with parameters ηˆ. In an asymptotic regime
our estimates will be close to the true state, so we
can expand the fidelity in a Taylor series about the
true state.
2. Calculating the expected value of the fidelity over
the different measurement outcomes, we obtain an
asymptotic expression for the point fidelity which is
a function of the covariance matrix of the estimated
parameters.
3. By applying the Crame´r-Rao bound to the covari-
ance matrix of the estimated parameters, we can
obtain an upper bound on the point fidelity which
is a function only of the number of each kind of
measurement we have made, and the state param-
eters.
4. Finally by averaging this point fidelity expres-
sion uniformly over the state parameters (using
the Haar measure) we obtain the average fidelity
bound.
While our expressions are analytic up to the final step,
we use numerical integration to average over the state
parameters using the Haar measure and obtain our final
asymptotic results.
We find F≤1−y/N where y is 1.083 for the cube (in
agreement with the value of 13/12 in [32]), 1.049 for the
octahedron, 1.018 for the dodecahedron, and 1.008 for
the icosahedron. It is well known in classical statistics
that the Crame´r-Rao bound can be achieved using a max-
imum likelihood estimator. Hence the above bounds are
also tight. These results show that for pure states the
performance of the Platonic solids measurements in to-
mography is close to the collective measurement bound
of y=1.

6Recall that the quantum Chernoff bound is equal to
the fidelity if one of the states involved is pure. When,
as here, both states are pure the HilbertSchmidt distance
is also an equivalent measure as, D2hs=(1−f)/2. So that
the choice of figure of merit does not affect the results of
this section at all.
VI. MIXED STATE QUBIT TOMOGRAPHY
When the state to be reconstructed can be mixed an
analytical treatment is more complicated since the re-
quirement that the density matrix be positive semidefi-
nite leads to constraints on the allowed set of parameters.
The situation is also complicated by the way in which the
fidelity function depends on the mixedness of the state to
be reconstructed. We partially address both these issues
in this section, with the main aim to develop intuition.
For quantitative results we will resort to simulation.
For mixed state qubit tomography, the state Bloch vec-
tor is no longer unit length. The state vector is now:
~r = [rx, ry , rz]T (13)
where rx=r cos(φ) sin(θ), ry=r sin(φ) sin(θ), rz=r cos(θ)
and r satisfies r≤1. This constraint on the length of the
Bloch vector reflects the requirement that the density
matrix is positive semidefinite.
Mixed state tomography introduces two new complica-
tions which we discuss in this section. Both concern be-
haviour close to the boundary of the Bloch sphere. The
first is due solely to the definition of the mixed state fi-
delity function:
f(~r, ~ˆr) = 1 + ~r.
~ˆr +
√
1− r2
√
1− rˆ2
2
. (14)
where ~ˆr is the estimated state vector. It can be un-
derstood by considering only errors in the Bloch vec-
tor length r [17]. Suppose r=1−δ and our estimate is
rˆ=r−ǫ. Then provided ǫ≪δ and both are small we find:
F≈1−ǫ2/8. From the statistical arguments of the previ-
ous section we would expect ǫ to scale asymptotically like
1/
√
N , so we expect to see all point fidelities (except for
perfectly pure states) to scale like F≈1−1/8N once N
is large enough that our errors are well inside the Bloch
ball. Alternatively if δ≪ǫ, we find that the point fidelities
scale like F≈1−ǫ/2. Thus the general behaviour of point
fidelities is a transition from 1/
√
N scaling at low N to
a scaling of 1/N at large N . The transition takes place
when the error associated with the state reconstruction ǫ
becomes small enough that rˆ+ǫ lies inside the Bloch ball.
Thus the location of the transition happens at higher N
for higher purity states. This is illustrated in Fig. 3 where
numerical simulations of point fidelities for states with
different radial parameters are shown. The Bures prior
has a radial distribution that is strongly peaked at r = 1
and it turns out that when point fidelities are averaged
over this prior the 1/
√
N dependence of the point fidelity
10−4 10−3 10−2 10−1
101
102
103
104
105
1 − F
N


r=0.5
r=0.95
r=0.98
r=0.999
r=0.5 Theory
1/2√N
FIG. 3: Numerical simulations of atomic point fidelities for a
state rotated 45 degrees about the x-axis from the +ve z-axis
are plotted against radius on the Bloch sphere. The cube mea-
surement set was used for these simulations. Observe mixed
states always have a 1/N scaling. States that are close to pure
start (after we get to a sufficiently large N to obtain a sensi-
ble state estimate) with a 1/
√
N dependence and transition
to 1/N scaling with larger N . This knee occurs at larger N
for purer states. Also the statistical theory of equation (15)
is plotted for r=0.5 giving excellent agreement
persists long enough that the dependence of the average
fidelity on N is not N−1 but rather N−3/4 [33].
The second complication arises since our estimated
physical state has to be positive semidefinite. For qubits
this is equivalent to rˆ≤1. An estimator that predicts
states lying outside the boundary of the Bloch ball can
be improved by mapping these states onto the boundary
of the Bloch ball which always moves the estimate closer
to the true state. Maximum likelihood estimation is a
specific example of an estimator that does exactly this.
This procedure creates a biassed estimator that neverthe-
less performs better than any unbiassed one. The na¨ıve
Crame´r-Rao estimation is therefore not appropriate.
For states well away from the boundary, we can ignore
this positivity constraint, and use the unbiased statis-
tical methods of the previous section extended to three
parameters to calculate the cube measurement point fi-
delity:
F (~r,N) ≤ 1− 3
4N
3− 3r2 + 2(r2yr2x + r2zr2x + r2zr2y)
1− r2 .
(15)
This value is included in Fig. 3 for the case when r=0.5
and found to give excellent agreement with the numerical
simulations.
VII. SIMULATION DETAILS
The main result of this paper is the numerical simula-
tion of tomography using Platonic solid measurements.

7The simulations are performed in the following way.
1. Choose Ns states ρi at random according to the
Bures (or Chernoff) prior. For our simulations Ns
was chosen to be 10,000.
2. For each ρi, simulate measurements on Nc copies
of the state (divided equally between the measure-
ment sets) generating Nc measurement outcomes
χji . For our simulations we chose Nc=100,000.
3. Perform maximum-likelihood state estimation
based on increasing numbers N of these χ1i ..χNi ob-
taining state estimates ρˆi(N).
4. Calculate the fidelity f(ρˆi(N), ρi) [or Chernoff
bound λcb(ρˆi(N), ρi)] between each state estimate
and true state.
5. Calculate the average fidelity
F (N)=∑Nsi=1 f(ρˆi(N), ρi)/Ns [or the average Cher-
noff bound: λav(N)=
∑Ns
i=1 λcb(ρˆi(N), ρi)/Ns].
The Monte-Carlo averaging over states and measure-
ment outcomes is performed simultaneously.
We present our results on a log-log plot of number of
copies required against 1−F (N). To find the number of
copies required to achieve a target average fidelity, one
can select the target average fidelity on the horizontal
axis, and tracing vertically upward, find the number of
copies required to achieve that target average fidelity.
The inserts on the graphs are enlargements of the largest
N section of the graph. The width of the lines on these
inserts are approximately equal to the one standard devi-
ation statistical error in average fidelity due to our choice
of Ns.
We simulate two main scenarios, tomography on
atomic qubits, and tomography on photonic qubits.
A. Atomic Qubits
The kind of experiment we model here is an atomic
qubit as in experiments on trapped ions, such as [5, 6]
represented by two metastable energy levels |0〉 and |1〉,
and an auxiliary level |r〉 used for performing the mea-
surement with a laser driving the transition |0〉→|r〉 lead-
ing to fluorescence if the state was initially in |0〉 [34].
Arbitrary single qubit projective measurements are made
available by a first laser pulse tuned to the |0〉→|1〉 tran-
sition followed by turning a laser on the |0〉→|r〉 tran-
sition and looking for this fluorescence. We assume the
dominant source of error is statistical fluctuation due to
the finite number of copies of the state, and neglect all
other sources of error. Many of these could be taken into
account by replacing our projective measurements with
suitable POVMs. However the effects such as drifts in
either state preparation or measurement settings would
necessitate a different approach.
QWP
HWP
Polarizing
Beam Splitter
FIG. 4: The single photonic qubit experimental configuration
consists of a quarter wave plate and a half wave plate to set
the basis, and a polarizing beam splitter followed by a photon
detector to analyse in that basis. We call this a single-detector
configuration. An additional detector can be placed on the
reflected port in a dual-detector configuration, in which case
the statistics of the counts can be easily reduced to the atomic
case by waiting until a fixed number of counts are obtained
for each basis setting.
In this limit these single qubit system measurements
are well described by Eqns. (10), (11) and counts for
each outcome k of a measurement setting l drawn from
a multinomial distribution:
p(nl1..nlK) =
Nl
nl1..nlK
pnl1l1 p
nl2
l1 ...p
nlK
lK (16)
where Nl copies of the system are measured with mea-
surement setting l, plk=tr(ρOlk) and K is number of pos-
sible outcomes for the measurement (2 for qubit systems
and 4 for two-qubit systems).
For two-qubit systems, the measurement operators Olk
take the form of Eq. (12) and now the index l ranges over
all combinations of single qubit measurement settings on
the subsystems. Experimentally this corresponds to not-
ing the joint probability of the four possible outcomes of
fluorescence and no fluorescence in each subsystem, for
all combinations of measurement settings on the subsys-
tems.
B. Photonic qubits
Our model for experiments on single photonic qubits is
shown in Fig. 4. In this model our qubits are represented
by the polarization state of photons issuing from a source
with Poissonian arrival times. There are two configura-
tions to consider: a single-detector configuration and a
dual-detector configuration.
In the first configuration, the source is directed through
a quarter wave plate, a half wave plate and polarizing
beam splitter (or vertical polarizer) and finally a single
photon counter [12]. The angles of the fast axes of both
of the wave plates can be rotated, allowing the projection

8onto vertical polarization fixed by the final polarizer to be
rotated into any polarization state that the experimenter
requires. Photon counts are accumulated over a time
t for each measurement setting ml. The total photons
incident on the measurement apparatus during this time
is drawn from the Poisson distribution:
p(Nl) =
e−Nf t(Nf t)Nl
Nl!
(17)
where Nf is the mean photon flux. The number of these
photons reaching the detector nl is then drawn from a
binomial distribution with probability pl = tr(ρOl). The
flux is estimated from a set of r measurements satisfying
r
∑
i=1
Oi = I (18)
via
Nf =
r
∑
i=1
ni/t. (19)
For the Platonic solid measurements, the four measure-
ments of the tetrahedron satisfy (18) and for higher order
Platonic solids each axis has two measurements satisfy-
ing (18). In the later case an average is made over all the
axes to obtain an improved flux estimate.
Adding a detector on the reflected port of the polariz-
ing beam splitter gives a dual-detector configuration and
potentially better flux estimation. In this configuration
both orthogonal directions are measured simultaneously,
and the total flux is estimated by summing the two detec-
tor counts. By waiting until a set number of total counts
are obtained for each measurement set, the statistics of
the counts will reduce to the atomic case covered above.
We also simulate two photon state tomography. A
source produces photon pairs whose arrivals are again
Poisson distributed according (17). The single photon
configuration (single- and dual-detectors) are duplicated
(giving two and four detectors). Now only coincidences
are counted between the two single photon configura-
tions. The number of co-incidences are drawn from the
multinomial distribution of (16). For the single-detector
configuration for each qubit we estimate the flux via
Nf =
∑r
j=1
∑r
i=1 nij
r2t/4 . (20)
C. Maximum likelihood state reconstruction
After simulating measurements on all copies of the
state we have nlk detections corresponding to each mea-
surement operator Olk. Maximum likelihood reconstruc-
tion is the problem of finding the state ρ which max-
imizes the likelihood p(...nlk...|ρ) or equivalently min-
imizes − log p(...nlk...|ρ). To make our inversions effi-
ciently solvable, we convert (an approximate version of)
this problem into a semidefinite program.
For atomic (single and multiple) qubit systems, and
for each measurement setting l, the outcomes follow the
multinomial distribution of (16).
We will approximate these statistics with a Gaussian
distribution with mean µlk=plk(ρ)Nl and a covariance
matrix Vl with diagonal entries (Vl)ii=Nlpeli(1−peli), and
off-diagonal entries (Vl)ij=−Nlpelipelj where pelk=nlk/Nl.
Since the outcomes of different measurement settings are
statistically independent, the combined likelihood func-
tion is p(...nlk...|ρ)=
∏
l p(nl1..nlK). Defining ~nl as the
vector with kth element nlk, and ~µl as the vector with
kth element µlk, our approximate problem is to find a
state ρ which minimises
− log p(...nlk...|ρ) ∝
∑
l
(~nl − ~µl)TV −1l (~nl − ~µl) (21)
We have assumed that the term in the exponential of the
Gaussian dominates the likelihood and used experimen-
tal probabilities pelk instead of plk(ρ) in the covariance
matrices Vl to make the problem convex. This enables
the problem to be converted into a semidefinite program
and solved with the SeDuMi semidefinite program solver
software [35].
For the dual-detector simulations of single photon and
two photon experiments the maximum likelihood recon-
struction is identical to the reconstruction for the atomic
case (if we wait for a fixed number of counts).
For single-detector simulations of single photon and
two photon experiments, the counts are Poisson dis-
tributed and as we only measure one outcome per mea-
surement setting, all outcomes are statistically inde-
pendent. We make a Gaussian approximation with
the expected number of counts for measurement out-
come Ol equal to the variance in those counts i.e.
µl=σ2l =Nl=plNf t where pl=tr(ρOl). Our approximate
problem is to find the state ρ which minimise the log
likelihood
− log[p(n1, n2, ...|ρ)] =
∑
l
(nl −Nl)2
2Nl
=
Nf t
2
∑
l
(pel − pl(ρ))2
pl(ρ)
,
where pel = nl/(Nf t). In this case we were able to use
Nl for mean and variance without losing the convexity of
the problem.
VIII. RESULTS
A. Single atomic qubit results
Simulation results for atomic qubit states are shown
in Fig. 5(a). The tetrahedron is not plotted because
for atomic measurements orthogonal outcomes are al-
ways measured simultaneously so, as noted above, the
tetrahedron measurement does not arise without more

9(a) 1 atomic qubit (c) 1 photonic qubit
10−4 10−3 10−2 10−1
101
102
103
104
105
1 − F
N


Cube
Octa
Dodeca
Icosa
10−4 10−3
104
105
10−4 10−3 10−2 10−1
101
102
103
104
105
1 − F
N


James4
Tetra
Cube
Octa
Dodeca
Icosa
10−4 10−3
104
105
(b) 2 atomic qubits (d) 2 photonic qubits
10−4 10−3 10−2 10−1
102
103
104
105
106
1 − F
N


Cube
Octa
Dodeca
10−4 10−3
105
10−4 10−3 10−2 10−1 100
102
103
104
105
106
1 − F
N


James16
SicPOVM
Cube
Octa
Dodeca
10−3 10−2
105
FIG. 5: Tomography simulation results with a Bures prior. The plot is the number of copies of the state against 1−F where F
is the average fidelity. To read, select the desired average fidelity on the horizontal axis, and trace vertically to find the number
of copies of the state required to achieve that fidelity. Each plot shows the performance of various Platonic solids. Tomography
for: (a) single atomic qubit states. The tetrahedron can not be measured in this configuration. (b) two atomic qubit states.
(c) single photonic qubit with single-detector configuration. (d) two photonic qubit with single-detector configuration.
experimental effort. Observe that, for sufficiently large
N , increasing the order of the Platonic solid improves to-
mography performance. Recall that N is the total num-
ber of systems measured so this result holds despite the
fact that each outcome probability is estimated from a
smaller number of measurements. The improvement of
the icosahedron over the dodecahedron is only marginal
but the difference in performance from the other mea-
surements is statistically significant.
B. Two atomic qubit state results
The simulation results for two qubit atomic states are
shown in Fig. 5(b). The performance is effectively the
same for all the Platonic solids and experimentally one
should choose the cube tensor measurement to minimize
the time consumed by changes in measurement settings.
C. Single photonic qubit results
The single photonic qubit results using a single pho-
ton counter are shown in Fig. 5(c). In addition to the
Platonic solid measurement sets, we have also included,
the popular measurement sets proposed by James et al.
in [12]. This set is labelled James4. The most striking
aspect of these results is how poorly the James4 mea-
surement set performs compared to the Platonic solids.
It is a strong recommendation that experimentalists us-
ing this measurement set should switch to Platonic solid
measurements. The performance of dual Platonic solids
is similar (the cube and octagon, and the dodecahedron
and the icosahedron), so we recommend using the cube
or dodecahedron measurements.
D. Two photonic qubit state results
The photonic two-qubit simulation results with single-
detectors (one for each qubit subsystem) are shown in
Fig. 5(d). In addition to the Platonic solid measurement
sets, we have also included, the popular measurement
sets proposed by James et al. in [12] labelled James16,
and the projective SicPOVM measurements.
Again the most striking result is how poorly the
James16 measurement set performs compared to tensor
products of Platonic solids. It is our strong recommen-
dation that experimentalists using these settings change
to a measurement consisting of tensor products of cube
measurements or if measurement setting changes con-
sume too much time, at least to tensors of the tetrahedral
measurements.
We also observe that while the projective SicPOVM

10
(a) 1 atomic qubit (c) 1 photonic qubit
10−4 10−3 10−2 10−1
101
102
103
104
105
1 − λ
av
N


Cube
Octa
Dodeca
Icosa
10−4 10−3
104
105
10−4 10−3 10−2 10−1
101
102
103
104
105
1 − λ
av
N


James4
Tetra
Cube
Octa
Dodeca
Icosa
10−4 10−3
104
105
(b) 2 atomic qubits (d) 2 photonic qubits
10−4 10−3 10−2 10−1
102
103
104
105
106
1 − λ
av
N


Cube
Octa
Dodeca
10−4 10−3
105
10−4 10−3 10−2 10−1 100
102
103
104
105
106
1 − λ
av
N


James16
SicPOVM
Cube
Octa
Dodeca
10−3 10−2
105
FIG. 6: Tomography simulation results with the prior induced by the quantum Chernoff bound. The number of copies is
plotted against 1−λav, where λav is the average quantum Chernoff bound. Tomography for (a) a single atomic qubit, (b) two
atomic qubits, (c) a single photonic qubit with a single-detector configuration, (d) two photonic qubits using single-detector
configurations.
performed the best, its improvement was negligible over
the cube tensors. There seems to be negligible advan-
tage to going to Platonic solids any higher order than
the cube.
IX. USING THE QUANTUM CHERNOFF
BOUND AS A FIGURE OF MERIT
As discussed in section II we find the use of the aver-
age quantum Chernoff bound to be the most physically
appealing figure of merit. It is thus important to see if
our conclusions hold when the quantum Chernoff bound
is used instead of the fidelity.
We repeated our tomography simulations using the av-
erage quantum Chernoff bound as the figure of merit,
and the Chernoff bound prior over states. The results
for one and two atomic qubits are shown in Figs. 6(a)
and 6(b) respectively. The results for one and two
photonic qubits with single-detector configurations are
shown in Figs. 6(c) and 6(d) respectively.
In all cases the graphs are almost identical to those
using a Bures prior and average fidelity as a figure of
merit. It is clear that all of our conclusions from the
average fidelity are correct, and hold also when using the
quantum Chernoff figure of merit.
X. USING THE MEAN SQUARED
HILBERT-SCHMIDT DISTANCE AS A FIGURE
OF MERIT
While we have focussed on the average fidelity and
the average Quantum Chernoff bound as our figures of
merit for tomography schemes, there is one commonly
used figure of merit that does behave quite differently
and does affect our results — the mean squared Hilbert-
Schmidt distance.
Roy and Scott [19] found Mutually Unbiased Bases
(e.g. the cube measurement for a single qubit in our
terminology) were an optimal measurement set for sin-
gle qubit tomography using the mean squared Hilbert-
Schmidt distance as a figure of merit. (They also neglect
any effect of the positivity constraint on the density ma-
trix, but our simulations bear out the hope that this is
negligible.) These conclusions contrast to ours based on
average fidelity which show that higher order Platonic
solids perform better than cube measurements.
To explore this discrepancy we repeated the calculation
of (15) (which also neglects the positivity constraints)
replacing the fidelity with squared Hilbert-Schmidt dis-
tance to give a point squared Hilbert-Schmidt distance
of:
〈D2hs(r)〉 ≥
3
2N (3− r
2) (22)
where r is the radius of the state on the Bloch sphere.

11
This equation was derived in [18] by other means and
agrees with the results of [19]. Observe the point squared
Hilbert-Schmidt distance is independent of the angular
parameters of the state on the Bloch sphere, unlike the
point fidelity case which has an angular dependence, as
shown in Eqn. (15).
We have repeated our numerical simulations using
mean squared Hilbert-Schmidt distance as a figure of
merit, a Bures prior and full maximum likelihood estima-
tion. Our results show that for atomic qubits the Platonic
solids all performed equivalently. Indeed our results were
within statistical errors of the theoretical value obtained
by integrating (22) over a Bures prior DB ≥ 0.8438/N .
This suggests that the result of [19] that the cube mea-
surement is optimal for mean squared Hilbert-Schmidt
distance squared holds, even when the positivity con-
straint is included. We also note that for single-detector
configurations our conclusions still hold, the James4 mea-
surement set still performs badly and should be replaced
by a cube measurement.
XI. DISCUSSION AND CONCLUSIONS
A. Experimental Recommendations
The clear conclusion of this paper is that for single-
detector photonic qubit tomography, experimentalists
should immediately switch from the minimal four mea-
surement set of [12] to the overcomplete six measurement
cube set, or at the very least switch to the tetrahedron
measurement set if the number of measurement settings
is an issue. Moving to the dual-detector configuration
would further improve the results (subject to statistical
fluctuation being the major source of error).
There is also a consistent trend towards better asymp-
totic performance for higher order Platonic solids for sin-
gle qubit states. Experimentalists would have to judge
whether there is a net gain after time to switch mea-
surement settings is taken into account. We note that
measurements on trapped ions are generally performing
what we call the cube measurement set already.
We have found that this trend is maintained in two-
qubit experiments with tensor product Platonic solid
measurements performing very well even in comparison
to entangled measurement sets. This strongly suggests
that the use of these symmetrically arranged measure-
ments will be advantageous for experiments on systems
having larger numbers of qubits. In this case we note that
the number of measurement settings grows very rapidly
and the time to change between detector settings is likely
to become an issue. In this regard the tetrahedron mea-
surement should have much improved performance but
requires no more measurements than the generalisations
of James4 and James16.
We also note that while experiments in tomography
usually report an uncertainty in the reconstruction based
on a fixed number of measured systems, we are not aware
of studies of how this uncertainty is reduced over time
during the tomography. Such a study should contain
information about the noise sources affecting the experi-
ment. Specifically drifts in either the states or the mea-
surement over time might be expected to lead to a sat-
uration of the average fidelity with N . We have seen
that when the experiment is limited by statistical noise
then the quality of the tomography depends systemati-
cally on N and on the choice of measurement. This could
be tested experimentally and should give detailed infor-
mation about the true errors in real experiments. In the
case of optical experiments this simply requires rotating
regularly among the different measurement settings and
keeping track of the order in which counts arrive. State
inversions can then be done at various time intervals and
a graph of the number of qubits N against point fidelity
similar to our Fig. 3 could be straightforwardly obtained.
One should, for example, be able to observe the gen-
eral behaviour of a scaling like 1/
√
N at low numbers of
copies transitioning to a scaling of 1/N at large numbers
of copies. Likewise the improvement of the point fidelity
with the choice of measurement could be checked. Com-
parison to detailed point fidelity simulations should aid
characterization of experimental error sources.
B. Future work
One could certainly consider measurements made from
polyhedra other than the Platonic solids (for example the
semi-regular polyhedra). However this paper has shown
that the Platonic solids provide a good spread in the
number of measurements and performance, and going
to even higher order shapes appears to be of decreas-
ing value. We thus consider exploring other shapes to
be unfruitful with one likely exception. The SicPOVM
was designed to be a spherical t design with t = 2. It is
well known [29] that the cube and octagon are 3-designs
while the dodecahedron and icosahedron are 5-designs.
It is likely that this is the origin of the similarity in per-
formance of the dual Platonic solids and t is an indicator
of the quality of tomography performance. This being
the case it would be well worth investigating McLaren’s
“improved Snub Cube” [29] which is a 24 point 7-design.
Acknowledgements
We would like to thank Gerard Milburn for useful dis-
cussions, and the Australian Research Council for fund-
ing this work.

12
APPENDIX A: DERIVATION OF PURE STATE
CRAME´R-RAO BOUNDS FOR SINGLE QUBIT
TOMOGRAPHY
Let η = [θ, φ]T be the state Bloch vector in spherical
polar coordinates, and let ηˆ be the state estimate. The
similarity between η and ηˆ is given by the fidelity fη(ηˆ) =
(1+r.rˆ)/2 where r = [cos(φ) sin(θ), sin(φ) sin(θ), cos(θ)]T
and similarly rˆ = [cos(φˆ) sin(θˆ), sin(φˆ) sin(θˆ), cos(θˆ)]T
Since we assume η and ηˆ are close we expand the fi-
delity in a Taylor series to second order giving:
fη(ηˆ) ≈ fη(η) +Dfη(η).(ηˆ − η) +
1
2
(ηˆ − η)H(η)(ηˆ − η)T
(A1)
where Dfη(η)i = ∂fη(ηˆ)∂ηi |ηˆ=η and H(η) is the Hessian ma-
trix defined by:
H(η)i,j =
∂2fη(ηˆ)
∂ηˆi∂ηˆj
∣
∣
ηˆ=η. (A2)
Now fη(η) = 1 and Dfη(η) = 0 so
fη(ηˆ) ≈ 1 +
1
2
∑
i,j
∂2fη(ηˆ)
∂ηi∂ηj
∣
∣
∣
∣
ηˆ=η
(ηˆi − ηi)(ηˆj − ηj). (A3)
During tomography, each of N identical copies of the
qubit state are each measured exactly once. Our mea-
surement set consists of l different measurement settings.
So N = N/l measurements are performed with each mea-
surement setting.
Each measurement setting is represented by Bloch vec-
tor ~mi, with outcomes χi= ± 1. One set of measure-
ments has 2l possible outcomes represented by the string
χ = χ1..χl. The probability of obtaining outcome χ is
given by:
pη(χ) =
∏
i
(1 + χi~r. ~mi)/2. (A4)
The point fidelity obtained by averaging over the pos-
sible outcomes χ of the set of measurements is given by
F (η) =
∑
χ
pη(χ)fη(ηˆ(χ)) (A5)
≈ 1 + 1
2
tr[H(η)V (η)] (A6)
where V (η) is the covariance matrix defined by:
V (η) =
∑
χ
p(χ)(ηˆ(χ)− η)(ηˆ(χ)− η)T . (A7)
The Crame´r-Rao bound for unbiased estimators states:
V (η) ≥ 1N I(η) (A8)
where I(η) is the Fisher Information matrix defined by:
Iij(η) =
∑
χ
pη(χ)
∂ ln pη(χ)
∂ηi
∂ ln pη(χ)
∂ηj
. (A9)
Noting that H(η) is negative definite, the point fidelity
bound is then given by:
F (η) ≤ 1 + 1
2N tr[H(η)I
−1(η)]. (A10)
Finally averaging over η uniformly according to the
Haar measure:
dρ = sin θ
4π dθdφ (A11)
gives average fidelity Crame´r-Rao Bound.
APPENDIX B: EIGENVALUE DISTRIBUTION
FOR THE QUANTUM CHERNOFF BOUND
METRIC
In [21] it was shown that the infinitesimal distance
between states ρ and ρ+dρ according to the Quantum
Chernoff bound is:
ds2 = 1
2
∑
jk
|〈j|dρ|k〉|2
(
√
λj +
√
λk)2
(B1)
where ρ = ∑j λj |j〉〈j| is the eigenvalue decomposition
of ρ. We will follow the same procedure as [25] in his
calculation of the Bures volume element to derive our
quantum Chernoff bound eigenvalue distribution.
First decompose this into an infinitesimal shift in
eigenvalues followed by an infinitesimal unitary rotation:
ρ+ δρ = (I + δU)(ρ + δΛ)(I + δU)† (B2)
= ρ + δΛ + [δU, ρ] (B3)
where we have ignored any terms which are a product of
multiple infinitesimal quantities and defined 〈j|δΛ|k〉 =
δjkdλj . In general we can rewrite the infinitesimal uni-
tary operator as:
δU =
∑
j≤k
(dxjk + idyjk)|j〉〈k|+ h.c. (B4)
Next we substitute Eq. B4 and Eq. B3 into Eq. B1 yield-
ing
ds2 = 1
8
∑
j
(dλj)2
λj
+
∑
j<k
(λk − λj)2
(
√
λj +
√
λk)2
(dx2jk + dy2jk).
We convert this into a differential volume element by mul-
tiplying together all the dsi formed by a shift in each pa-
rameter λi assuming the shift in the other parameters is
zero. Since the volume element will require normalization

13
we combine any constant factors. The final differential
volume element is given by:
dV = C′ dλ1..dλM√λ1..λM
∏
k<j
(λk − λj)2
(
√
λj +
√
λk)2
dxjkdyjk (B5)
where C′ is a normalisation constant.
Since the eigenvalue distribution is unitarily invariant
we can concern ourselves with the marginal probability
distribution over the eigenvalues of the density operators.
That is:
p(λ1..λM ) =
C√
λ1..λM
∏
k<j
(λk − λj)2
(
√
λj +
√
λk)2
(B6)
where the eigenvalues are also constrained to satisfy
∑M
j=1 λj = 1 and C is a normalisation constant. Next
we examine the single qubit case.
1. Single Qubit Case
For the qubit case we have:
p(λ1λ2) =
C√
λ1λ2
(λ1 − λ2)2
(
√
λ1 +
√
λ2)2
(B7)
The eigenvalues for the qubit case are related to r the
radius on the Bloch sphere according to: λ1=(1+r)/2
and λ2=(1−r)/2. This yields
p(r) = C
2
1−
√
1− r2√
1− r2
(B8)
By requiring
∫ 1
0 p(r)dr = 1 we calculate C=4/(π−2).
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