# Morphophoric POVMs, generalised qplexes,
and 2-designs

Wojciech Słomczyński<EMAIL_ADDRESS>Anna Szymusiak
<EMAIL_ADDRESS>Institute of Mathematics, Jagiellonian University,
Łojasiewicza 6, 30-348 Kraków, Poland

###### Abstract

We study the class of quantum measurements with the property that the image of
the set of quantum states under the measurement map transforming states into
probability distributions is similar to this set and call such measurements
morphophoric. This leads to the generalisation of the notion of a qplex, where
SIC-POVMs are replaced by the elements of the much larger class of
morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace)
POVMs. The intrinsic geometry of a generalised qplex is the same as that of
the set of quantum states, so we explore its external geometry, investigating,
inter alia, the algebraic and geometric form of the inner (basis) and the
outer (primal) polytopes between which the generalised qplex is sandwiched. In
particular, we examine generalised qplexes generated by MUB-like 2-design
POVMs utilising their graph-theoretical properties. Moreover, we show how to
extend the primal equation of QBism designed for SIC-POVMs to the morphophoric
case.

##  1 Introduction and preliminaries

Over the last ten years in a series of papers [30, 31, 32, 2, 3, 68, 49]
Fuchs, Schack, Appleby, Stacey, Zhu and others have introduced first the idea
of QBism (formerly quantum Bayesianism) and then its probabilistic embodiment
- the (Hilbert) qplex, both based on the notion of symmetric informationally
complete positive operator-valued measure (SIC-POVM), representing the
standard (or reference) measurement of a quantum system. In spite of the fact
that SIC-POVMs do indeed exhibit some unique properties that distinguish them
among other IC-POVMs [24, 25, 52], we believe that similar (or, in a sense,
even richer from a purely mathematical point of view) structures, say,
generalised qplexes, can be successfully used to faithfully represent quantum
states as probability distributions if we replace SIC-POVMs with the broader
class of morphophoric POVMs, containing in particular rank-$1$ measurements
generated by complex projective $2$-designs. As in the QBism interpretation of
quantum mechanics, the result is a new formalism for quantum theory, which
involves only probabilities.

The rank-$1$ equal-trace case111In fact, Fuchs and Schack considered such a
generalisation in [30, pp. 22-23], but it seems that they rejected it. See
also Fuchs’ letter to Zauner [29, p. 1933], as well as [73]. is easier to cope
with than the general one, as it leads to geometry similar, though not
identical, to that for qplexes, as well as to an almost identical form of the
primal equation (‘Urgleichung’) that plays the central role in the QBism
approach to quantum mechanics. The general case, where the effects of the
measurement are not necessarily rank-$1$, seems to be more complex: while the
geometric picture remains in principle almost the same, the primal equation
takes a slightly different form.

Let us imagine that the system is in a given initial state. Then either we
apply a measurement ‘on the ground’ directly, or we first send the quantum
system through the measurement device ‘in the sky’. The original ‘Urgleichung’
allows us to express the probabilities of the outcomes of the measurement ‘on
the ground’ in terms of the probabilities of the results of the counterfactual
measurement ‘in the sky’ (which is a SIC-POVM in the QBism formulation) and
the conditional, also counterfactual, probabilities of obtaining a given
outcome of the former measurement conditioned on receiving a given result of
the latter, provided the state of the system after performing the latter is
given by the Lüders instrument. Note, however, that in the general (not
rank-$1$) case these conditional probabilities may depend on the initial state
of the system [30, p. 23] and, therefore, the primal equation cannot be easily
extended to this case. That is why we propose another equation connecting
those probabilities which is equivalent to the primal equation in the rank-$1$
equal-trace case, but employs the conditional probabilities defined for the
maximally mixed initial state rather than those for the original initial state
for which the primal equation is formulated. Despite this change, the new
primal equation is also defined in purely probabilistic terms, although its
form does not resemble any of the known laws of classical probability.

To see what path leads to this generalisation, let us start from several
technicalities. In standard (finite) $d$-dimensional quantum mechanics the
states of a system are identified with the set of positive trace-one operators
$\mathcal{S}(\mathbb{C}^{d})$ (also known as density operators or density
matrices) and the pure states with the extreme boundary of this set,
$\mathcal{P}(\mathbb{C}^{d})$, i.e. the rank-1 orthogonal projections or,
equivalently, with the elements of the projective complex vector space
$\mathbb{CP}^{d-1}$. The quantum states can be also described as the elements
of the ($d^{2}-1$)-dimensional real Hilbert space
$\mathcal{L}_{s}^{0}\left(\mathbb{C}^{d}\right)$ of Hermitian traceless
operators on $\mathbb{C}^{d}$ endowed with the Hilbert-Schmidt product.
Namely, the map
$\mathcal{S}\left(\mathcal{H}\right)\ni\rho\rightarrow\rho-I/d\in\mathcal{L}_{s}^{0}\left(\mathbb{C}^{d}\right)$
gives an affine embedding of the set of states (resp. pure states) into the
(outer) ($d^{2}-1$)-dimensional ball (resp. sphere) in
$\mathcal{L}_{s}^{0}\left(\mathbb{C}^{d}\right)$ centred at $0$ of radius
$\sqrt{1-1/d}$. The image of this map is called the Bloch body (resp. the
generalised Bloch sphere). Only for $d=2$ the map is onto, and for $d>2$ its
image is a convex subset of the ball of full dimension and contains the
maximal (inner) ball of radius $1/\sqrt{d\left(d-1\right)}$ [6].

A measurement with a finite number $n$ of possible outcomes can be described
by a positive operator-valued measure (POVM), i.e. an ensemble of positive
non-zero operators $\Pi=(\Pi_{j})_{j=1}^{n}$ on $\mathbb{C}^{d}$ that sum to
the identity operator, i.e. $\sum\nolimits_{j=1}^{n}\Pi_{j}=I$. According to
the Born rule, the affine measurement map $p_{\Pi}$ sends an input quantum
state $\rho\in\mathcal{S}(\mathbb{C}^{d})$ into the vector of probabilities of
the respective outcomes
$(\operatorname{tr}\left(\rho\Pi_{j}\right))_{j=1}^{n}$ that belongs to the
standard (probability) simplex
$\Delta_{n}:=\\{x\in\left(\mathbb{R}^{+}\right)^{n}:\sum\nolimits_{j=1}^{n}x_{j}=1\\}$.
If this map is one-to-one, we call $\Pi$ informationally complete; then,
necessarily, $n\geq d^{2}$. In this situation the results of the measurement
uniquely determine the input state, and so quantum tomography, i.e. the task
of reconstructing the quantum state from the outcomes of the measurement, can
be concluded. In this case
$\mathcal{Q}_{\Pi}:=p_{\Pi}(\mathcal{S}(\mathbb{C}^{d}))$ (the probability
range of $\Pi$) is an affine ($d^{2}-1$)-dimensional image of
$\mathcal{S}(\mathbb{C}^{d})$ and so, clearly, the set of extreme points of
the closed convex set $\mathcal{Q}_{\Pi}$, i.e. its extreme boundary
$\textnormal{ext}\mathcal{Q}_{\Pi}$, is equal to
$p_{\Pi}(\mathcal{P}(\mathbb{C}^{d}))$. We shall analyse the structure of the
image of $p_{\Pi}$ in Sections 2 and 3. If $\mathcal{S}(\mathbb{C}^{d})$ and
$\mathcal{Q}_{\Pi}$ are of the same shape (geometrically similar), we call
such $\Pi$ morphophoric (or form-bearing, from Greek $\mu
o\rho\phi\acute{\eta}$ \- shape, form and $\phi\acute{o}\varrho o\sigma$ \-
bearing). In this case we call $\mathcal{Q}_{\Pi}$ a generalised (Hilbert)
qplex. It seems that morphophoric POVMs share with SIC-POVMs many properties
crucial for developing natural generalisations of QBism and qplexes.

The main result of the present paper, Theorem 4, provides a necessary and
sufficient condition for morphophoricity: the orthogonal projections onto
$\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$ of the effects constituting the POVM in
question need to form a tight operator frame in this space. In consequence, we
can identify the set of quantum states with a convex subset of $\Delta_{n}$.
In Section 3 we take a closer look at the structure of the class of
morphophoric POVMs. In particular, we show that this notion coincides with
that of tight IC-POVMs introduced by Scott in [70] for POVMs consisting of
effects of equal trace (Corollary 8), but in general these classes are
different (Example 2).

In this paper we pay special attention to POVMs $\Pi=(\Pi_{j})_{j=1}^{n}$ such
that the effects $\Pi_{j}$ ($j=1,\ldots,n$) are not only of equal trace but
also one-dimensional, and so proportional to certain pure states $\rho_{j}$
($j=1,\ldots,n$). We show that in this case $\Pi$ is morphophoric if and only
if these states constitute a complex projective $2$-design (Corollary 9), or,
equivalently, their image in the Bloch body under the natural embedding is a
tight frame in $\mathcal{L}_{s}^{0}\left(\mathbb{C}^{d}\right)$. Note that the
class of $2$-designs contains, besides SIC-POVMs [33], also complete sets of
MUBs (mutually unbiased bases) [50], MUB-like designs [47], orbits of the
multiqubit Clifford group [83] (in fact, the latter are $3$-designs), group
designs [38], and many other discrete objects representing quantum
measurements [6, 7, 9]. Let us now take a closer look at the geometry of
$2$-design POVMs.

The main difference between (Hilbert) qplexes and their generalisation lies
not in their intrinsic geometry, but rather in their external geometry (i.e.
the location in the underlying real vector space), which we describe in detail
in Section 7. Both a (Hilbert) qplex and a (Hilbert) generalised qplex are
isomorphic (similar) through the measurement map to the space of quantum
states (or, to put it another way, the Bloch body). The former, however, is a
subset of the standard $(d^{2}-1)$-dimensional simplex $\Delta_{d^{2}}$ lying
in a $d^{2}$-dimensional real vector space, whilst the latter is a subset of a
$(d^{2}-1)$-dimensional primal polytope $\Delta$, which is a
$(d^{2}-1)$-section of the ‘larger’ simplex $\Delta_{n}$. We show that the
section in question has to be central, i.e. it passes through the centre of
the simplex $c_{n}$, and, what is more, medial, i.e. the vertices of the
simplex are equidistant from the primal affine space $\mathcal{A}$ generated
by $\Delta$. (The term ‘medial’ in the general case, i.e. not restricted to
the hyperplane sections, has been introduced in [78].) The distance equal to
$1-d^{2}/n$ measures the deviation from minimality: for $n=d^{2}$ we get
necessarily a SIC-POVM. Moreover, we consider another polytope, namely a basis
polytope $D$ generated by the elements of a $2$-design transformed by
$p_{\Pi}$. This polytope is the image under the homothety (with centre at
$c_{n}$ and ratio $1/(d+1)$) of the orthogonal projection of $\Delta_{n}$ on
$\mathcal{A}$.

The generalised qplex $\mathcal{Q}_{\Pi}$ is a convex set sandwiched between
these two polytopes $D$ and $\Delta$. The situation is somewhat analogous to
the quantum (Bell) correlation picture, where the quantum convex body lies in-
between the local polytope [64] and the non-signalling polytope [67], see also
[14, 28, 13, 35, 66, 36], or to the related idea of applying graph theoretical
concepts [39, 51] to the analysis of quantum contextuality, where the
probabilities of the results of an experiment in classical, quantum, and more
general probabilistic theories are described by, respectively, stable set
polytope $\subset$ theta body $\subset$ fractional stable set polytope (or, in
other words, clique-constrained stable set polytope) of the graph related to
the experiment [16, 17, 1]. In fact, the analogy is even deeper since also in
our situation $D$ and $\Delta$ represent, in a sense, classical and ‘beyond-
quantum’ behaviour, respectively.

The polytopes $D$ and $\Delta$ are dual in $\mathcal{A}$, see [81], namely $D$
and $\iota(\Delta)$, where $\iota$ is the inversion through $c_{n}$, are polar
with respect to the central sphere of radius $m:=1/\sqrt{n(d+1)}$. Moreover,
$\mathcal{Q}_{\Pi}$ is self-dual in this sense, every pair of elements
$p,q\in\mathcal{Q}_{\Pi}$ fulfils the fundamental inequalities:
$dm^{2}\leq\langle p,q\rangle\leq 2dm^{2}$, and $\mathcal{Q}_{\Pi}$ lies in-
between two central balls: the inscribed ball of $\Delta$ of radius
$r:=m/\sqrt{d-1}$ and the circumscribed ball of $D$ of radius
$R:=m\sqrt{d-1}$. Note that $m=\sqrt{rR}$. Thus, the overall geometric picture
is similar to the (Hilbert) qplex case represented schematically in [3, Figs.
2 and 3], but more complicated as our polytopes are not necessarily simplices,
see Figs. 4 and 5. These geometric properties, especially the mediality of the
constitutive cross-section space $\mathcal{A}$, can be chosen as a starting
point for an abstract definition of generalised qplexes (in preparation). Note
that most of these properties hold true for every morphophoric POVM, as we see
at the end of Section 7.

This geometric approach can be supplemented by an algebraic one we present in
Section 4, namely, for the extreme boundary
$\textnormal{ext}\mathcal{Q}_{\Pi}$ of the qplex, we show that it can be
described by a set of polynomial equations (Theorem 10): $n$ linear, one
quadratic, and one cubic (for $d>2$). The system of $n$ linear equations
defining the affine space $\mathcal{A}$ is just an analogue of the famous
primal equation (‘Urgleichung’) known from the QBism theory where $\Pi$ plays
the role of both the ‘ground’ and the ‘sky’ measurement. Note, however, that
for SIC-POVMs these equations do not impose any constraints on probabilities
as they are always satisfied (Corollary 11.(ii)). For low-dimensional systems
($d=2,3$) we provide also the description of the whole generalised qplex
$\mathcal{Q}_{\Pi}$ with the help of a set of polynomial equations and
inequalities (Corollary 12).

We provide a detailed analysis of several examples of $2$-designs and the
corresponding generalised qplexes: cubical, cuboctahedral and icosahedral
POVMs in dimension $2$, as well as SIC-POVMs in dimension $3$ in Section 5.
The linear equations defining primal affine space $\mathcal{A}$ take an
unexpectedly simple form for complete sets of MUBs and MUB-like 2-designs
(i.e. two-distance 2-designs with one of inner products equal to $0$), see
Section 6. Apart from complete sets of MUBs, we know four sporadic objects of
this type, one in each of dimensions 4, 5, 6 and 28, and they all generate the
orthogonality graph on the elements of the POVM that is a strongly regular
graph with some special properties. In this case the equations describing
$\mathcal{A}$ can be replaced by the requirement that the sum of the
probabilities over each orthonormal basis is constant (Theorem 13).

In the last section we show that a POVM is morphophoric if and only if it
obeys an equation which is a natural generalisation of the quantum law of
total probability of QBism (Theorem 18 and Corollary 19). Namely, for a
morphophoric measurement $\Pi=(\Pi_{j})_{j=1}^{n}$ ‘in the sky’ and an
arbitrary measurement $\Xi=(\Xi_{k})_{k=1}^{N}$ ‘on the ground’ we define the
deviations of the probability distributions of the measurements results for
the pre-measurement state $\rho$ from those for $\rho_{\ast}$, where
$\rho_{\ast}:=I/d$ is the maximally mixed state:
$\delta_{\Pi}(\rho):=p_{\Pi}(\rho)-p_{\Pi}(\rho_{\ast})$ and
$\delta_{\Xi}(\rho):=p_{\Xi}(\rho)-p_{\Xi}(\rho_{\ast})$. Moreover, we
consider the transposed covariance matrix $C$ of two consecutive measurements
$\Pi$ and $\Xi$ given by
$C_{kj}:=p_{jk}^{\Pi\Xi}(\rho_{\ast})-p_{j}^{\Pi}(\rho_{\ast})p_{k}^{\Xi}(\rho_{\ast})$,
for $j=1,\ldots,n$ and $k=1,\ldots,N$, and the constant
$\zeta:=(d^{2}-1)^{-1}\sum_{j=1}^{n}(p_{jj}^{\Pi\Pi}(\rho_{\ast})-(p_{j}^{\Pi}(\rho_{\ast}))^{2})$.
Then the following simple generalisation of the primal equation

$\zeta\delta_{\Xi}=C\delta_{\Pi}$

links the probabilities of both measurements.

##  2 The probability range of POVMs

Let us consider a quantum system represented by $\mathbb{C}^{d}$ and a general
discrete quantum measurement on it described by a POVM (positive operator-
valued measure) $\Pi=(\Pi_{j})_{j=1}^{n}$. In particular, if $\Pi_{j}$ are
orthogonal projections, we call $\Pi$ a PVM (projection-valued measure). A
POVM is called informationally complete (IC-POVM) if its statistics determine
uniquely the pre-measurement state, i.e. the conditions
$\textnormal{tr}(\rho\Pi_{j})=\textnormal{tr}(\sigma\Pi_{j})$ ($j=1,\ldots,n$)
imply $\rho=\sigma$ for every input states
$\rho,\sigma\in\mathcal{S}(\mathbb{C}^{d})$. Let us denote by
$\mathcal{Q}_{\Pi}$ the set of ‘allowed’ probabilities, i.e. the set of all
possible probability distributions of the measurement outcomes over all
quantum states. It is known also as the _probability range_ of $\Pi$ [40].
More formally, if we denote by $\mathcal{L}_{s}(\mathbb{C}^{d})$ the
$d^{2}$-dimensional real ordered vector space of Hermitian operators on
$\mathbb{C}^{d}$, then

$\mathcal{Q}_{\Pi}:=p_{\Pi}(\mathcal{S}(\mathbb{C}^{d})),$

where

$p_{\Pi}:\mathcal{L}_{s}(\mathbb{C}^{d})\ni
A\mapsto(\textnormal{tr}(A\Pi_{1}),\ldots,\textnormal{tr}(A\Pi_{n}))\in\mathbb{R}^{n}$

is a positive linear map. We introduce also $\mathcal{Q}_{\Pi}^{1}$ for the
probability distributions attained for the pure states, i.e.

$\mathcal{Q}_{\Pi}^{1}:=p_{\Pi}(\mathcal{P}(\mathbb{C}^{d})).$

As a straightforward consequence, we get the following properties of
$\mathcal{Q}_{\Pi}$ and $\mathcal{Q}_{\Pi}^{1}$:

######  Fact 1.

$\mathcal{Q}_{\Pi}=\textnormal{conv}(\mathcal{Q}_{\Pi}^{1})$ and
$\textnormal{ext}(\mathcal{Q}_{\Pi})\subset\mathcal{Q}_{\Pi}^{1}$.

Let us recall that the _joint numerical range_ of Hermitian matrices
$A_{1},A_{2},\ldots,A_{n}$ is defined by

$\mathcal{F}(A_{1},A_{2},\ldots,A_{n})=\\{(\langle z|A_{1}|z\rangle,\langle
z|A_{2}|z\rangle,\ldots,\langle
z|A_{n}|z\rangle)^{T}|z\in\mathbb{C}^{d},\|z\|=1\\}\subset\mathbb{R}^{n}.$

As any pure state $\rho$ is necessarily the orthogonal projection onto some
unit vector $z\in\mathbb{C}^{d}$, we have
$\textnormal{tr}(\rho\Pi_{j})=\langle z|\Pi_{j}|z\rangle$, and so
$\mathcal{Q}_{\Pi}^{1}$ can be described also as the joint numerical range of
$\Pi_{1},\Pi_{2},\ldots,\Pi_{n}$.

We are particularly interested in the shape of $\mathcal{Q}_{\Pi}$. For $d=2$,
$\mathcal{Q}_{\Pi}$ is an ellipsoid, possibly degenerated, as a linear image
of 3-dimensional ball. Some examples of these ellipsoids are presented below.

######  Example 1.

Let us consider a one-parameter family of rank-1 normalised POVMs on
$\mathbb{C}^{2}$ consisting of $n=4$ effects:

$\Pi_{1}:=\frac{1}{4}\begin{pmatrix}1+\sin\alpha&\cos\alpha\\\
\cos\alpha&1-\sin\alpha\end{pmatrix},\
\Pi_{2}:=\frac{1}{4}\begin{pmatrix}1+\sin\alpha&-\cos\alpha\\\
-\cos\alpha&1-\sin\alpha\end{pmatrix},$
$\Pi_{3}:=\frac{1}{4}\begin{pmatrix}1-\sin\alpha&-i\cos\alpha\\\
i\cos\alpha&1+\sin\alpha\end{pmatrix},\
\Pi_{4}:=\frac{1}{4}\begin{pmatrix}1-\sin\alpha&i\cos\alpha\\\
-i\cos\alpha&1+\sin\alpha\end{pmatrix},$

where $\alpha\in[0,\pi/2]$. The pure states $\rho_{j}=2\Pi_{j}$ for
$j=1,\ldots,4$ are represented on the Bloch sphere by the vertices of certain
tetragonal disphenoid, i.e. a tetrahedron whose four faces are congruent
isosceles triangles. Such a tetrahedron can be uniquely characterised by the
length of the base (or legs) of that triangle. Taking into account possible
degenerations, we get five cases. For $\alpha=\frac{\pi}{2}$ the disphenoid
degenerates to a segment (length of the base $=0$) and the measurement
effectively acts as the projective von Neumann measurement. For
$\alpha\in(0,\pi/2)$ there is no degeneracy and the POVM is informationally
complete. In particular, for
$\alpha\in(\arcsin\frac{1}{\sqrt{3}},\frac{\pi}{2})$ the length of the base of
the face triangle is less than the length of its legs. For
$\alpha=\arcsin\frac{1}{\sqrt{3}}$ we get the tetrahedron (faces being
equilateral triangles) representing SIC-POVM, the only case of 2-design here.
For $\alpha\in(0,\arcsin\frac{1}{\sqrt{3}})$ the disphenoid is such that the
length of the base of the face triangle is greater than the length of its
legs. Finally, for $\alpha=0$ the disphenoid degenerates to a square (length
of the legs $=0$) representing a non-informationally complete highly symmetric
POVM. Its probability range in these cases is presented in Fig. 1 and evolves
as follows: for $\alpha=\frac{\pi}{2}$ we obtain a segment, then for
$\alpha\in(\arcsin\frac{1}{\sqrt{3}},\pi/2)$ – an elongated spheroid, for
$\alpha=\arcsin\frac{1}{\sqrt{3}}$ – a ball, for
$\alpha\in(0,\arcsin\frac{1}{\sqrt{3}})$ – a flattened spheroid and, finally,
for $\alpha=0$ – a disk.

Figure 1: $\mathcal{Q}_{\Pi}$ in the cases: $\alpha=\frac{\pi}{2}$ – a
segment; $\alpha\in(\arcsin\frac{1}{\sqrt{3}},\pi/2)$ – an elongated spheroid;
$\alpha=\arcsin\frac{1}{\sqrt{3}}$ – a ball;
$\alpha\in(0,\arcsin\frac{1}{\sqrt{3}})$ – a flattened spheroid; $\alpha=0$ –
a disk. All solids are tangent to $\Delta_{4}$.

It is much more difficult to describe the shape of $\mathcal{Q}_{\Pi}$ in
dimensions higher than 2. However, it seems that there are two cases which are
in some sense ‘nice’. The first one is when $\mathcal{Q}_{\Pi}$ is the full
simplex. The second one is when it is of the same shape as
$\mathcal{S}(\mathbb{C}^{d})$. In what follows we deal with the sufficient and
necessary conditions for these two to appear. As for the full simplex case,
the following characterisation may be categorised as quantum information
folklore [40, p. 4], see below. The second case will be thoroughly analysed in
the next section.

######  Definition 1.

We say that a POVM $\Pi$ has the norm-1-property if $\|\Pi_{j}\|=1$ for
$j=1,\ldots,n$ (here $\|A\|:=\max_{\|x\|_{2}=1}\|Ax\|_{2}$).

######  Theorem 1.

$\mathcal{Q}_{\Pi}=\Delta_{n}$ if and only if $\Pi$ has the norm-1-property.
Moreover, in such case, $\mathcal{Q}_{\Pi}^{1}=\Delta_{n}$, $n\leq d$, and
$n=d$ if and only if $\Pi$ is a rank-1 PVM.

##  3 The probability range similar to quantum state space

In order to provide a characterisation of the second case we need to make some
preparations. From now on for $A,B\in\mathcal{L}_{s}(\mathbb{C}^{d})$ we will
denote their Hilbert-Schmidt inner product by $(A|B):=\textnormal{tr}(AB)$ and
the induced norm of $A$ by $\|A\|_{HS}$. We also consider
$\mathcal{L}_{s}^{0}(\mathbb{C}^{d}):=\\{A\in\mathcal{L}_{s}(\mathbb{C}^{d})|\textnormal{tr}A=0\\}$,
i.e. the ($d^{2}-1$)-dimensional subspace of traceless Hermitian operators. It
is easy to see that

$\pi_{0}:\mathcal{L}_{s}(\mathbb{C}^{d})\ni A\mapsto
A-(\textnormal{tr}A/d)I\in\mathcal{L}_{s}(\mathbb{C}^{d})$ (1)

is the orthogonal projection onto $\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$.

######  Proposition 2.

The following conditions are equivalent:

  1. i.

$\Pi=(\Pi_{j})_{j=1}^{n}$ is an IC-POVM;

  2. ii.

$\textnormal{lin}(\Pi):=\textnormal{lin}\\{\Pi_{j}:j=1,\ldots,n\\}=\mathcal{L}_{s}(\mathbb{C}^{d})$;

  3. iii.

$\textnormal{lin}(\pi_{0}(\Pi)):=\textnormal{lin}\\{\pi_{0}(\Pi_{j}):j=1,\ldots,n\\}=\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$.

###### Proof.

The equivalence of (i) and (ii) can be found in [41, Prop. 3.51]. In order to
see the equivalence of (ii) and (iii), let us first observe that
$(\textnormal{lin}(\Pi))^{\perp}\subset\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$
since $I\in\textnormal{lin}(\Pi)$. Thus

$\displaystyle\mathcal{L}_{s}^{0}\cap(\textnormal{lin}(\pi_{0}(\Pi)))^{\perp}$
$\displaystyle=\\{A\in\mathcal{L}_{s}^{0}:(A|\Pi_{j})-\textnormal{tr}A\textnormal{tr}\Pi_{j}/d=0,j=1,\ldots,n\\}$
$\displaystyle=\\{A\in\mathcal{L}_{s}^{0}:(A|\Pi_{j})=0,j=1,\ldots,n\\}=\mathcal{L}_{s}^{0}\cap(\textnormal{lin}(\Pi))^{\perp}=\mathcal{L}_{s}\cap(\textnormal{lin}(\Pi))^{\perp}.\qed$

The following definition gives a precise description of what we mean by ‘the
same shape’ in terms of similarity. Throughout the paper we use
$\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ to denote the standard inner
product and the induced norm on $\mathbb{R}^{n}$.

######  Definition 2.

We say that $\mathcal{Q}_{\Pi}$ is similar to quantum state space
$\mathcal{S}(\mathbb{C}^{d})$ if there exists $\alpha>0$ such that

$\|p_{\Pi}(\rho)-p_{\Pi}(\sigma)\|^{2}=\alpha\|\rho-\sigma\|_{HS}^{2}\quad\textnormal{for
all }\rho,\sigma\in\mathcal{S}(\mathbb{C}^{d}).$ (2)

######  Remark 1.

The above definition can be expressed equivalently as

$\langle
p_{\Pi}(\rho)-c_{\Pi},p_{\Pi}(\sigma)-c_{\Pi}\rangle=\alpha(\rho-\rho_{\ast}|\sigma-\rho_{\ast})$
(3)

for all $\rho,\sigma\in\mathcal{S}(\mathbb{C}^{d})$, where
$c_{\Pi}:=p_{\Pi}(\rho_{\ast})=(\textnormal{tr}\Pi_{1}/d,\ldots,\textnormal{tr}\Pi_{n}/d)$.
(Note that $\rho-\rho_{\ast}=\pi_{0}(\rho)$ and
$p_{\Pi}(\rho)-c_{\Pi}=p_{\Pi}(\pi_{0}(\rho))$.) In particular,
$p_{\Pi}|_{\mathcal{L}_{s}^{0}(\mathbb{C}^{d})}$ is a similarity with the
similarity ratio $s=\sqrt{\alpha}$.

######  Definition 3.

We say that a POVM $\Pi$ is _morphophoric_ if $\mathcal{Q}_{\Pi}$ is similar
to the quantum state space $\mathcal{S}(\mathbb{C}^{d})$.

Let us recall some basic facts about finite tight frames. Let $V$ be a finite-
dimensional Hilbert space with inner product $\langle\cdot|\cdot\rangle$ and
let $F:=\\{f_{1},\ldots,f_{m}\\}\subset V$.

######  Definition 4.

$F$ is a tight frame if there exists $\alpha>0$ such that

$\alpha\|v\|^{2}=\sum_{i=1}^{m}|\langle
v|f_{i}\rangle|^{2}\quad\textnormal{for all }v\in V.$

In such situation $\alpha$ is referred to as the frame bound. The operator
$S:=\sum_{i=1}^{m}|f_{i}\rangle\langle f_{i}|$ is called the frame operator.

The next theorem is a well-known fact in frame theory (see, e.g. [80]) that
provides us with some other handful equalities characterising tight frames.

######  Theorem 3.

The following conditions are equivalent:

  1. i.

$F$ is a tight frame with frame constant $\alpha$;

  2. ii.

$\sum_{i=1}^{m}\langle u|f_{i}\rangle\langle f_{i}|v\rangle=\alpha\langle
u|v\rangle$ for all $u,v\in V$;

  3. iii.

$S=\alpha I$.

Now we can proceed to the characterisation of morphophoric POVMs.

######  Theorem 4.

Let $\Pi$ be a POVM. Then $\mathcal{Q}_{\Pi}$ is similar to the quantum state
space if and only if $\pi_{0}(\Pi)$ is a tight (operator) frame in
$\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$. Moreover, the frame bound and the
square of the similarity ratio coincide and are equal to

$\alpha=\frac{1}{d^{2}-1}\left(\sum_{j=1}^{n}\textnormal{tr}\Pi_{j}^{2}-\frac{1}{d}\sum_{j=1}^{n}(\textnormal{tr}\Pi_{j})^{2}\right).$
(4)

###### Proof.

Since
$\textnormal{span}\\{\rho-\rho_{\ast}:\rho\in\mathcal{S}(\mathbb{C}^{d})\\}=\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$,
it follows that (3) holds for all quantum states if and only if it holds for
all $\rho,\sigma\in\mathcal{L}_{s}(\mathbb{C}^{d})$ such that
$\textnormal{tr}\rho=\textnormal{tr}\sigma=1$. Thus, by linearity, we can
extend this condition to arbitrary Hermitian operators by writing it
equivalently as

$\langle
p_{\Pi}(A)-\textnormal{tr}Ac_{\Pi},p_{\Pi}(B)-\textnormal{tr}Bc_{\Pi}\rangle=\alpha(A-(\textnormal{tr}A/d)I|B-(\textnormal{tr}B/d)I)\quad\textnormal{for
all }A,B\in\mathcal{L}_{s}(\mathbb{C}^{d}).$ (5)

Transforming the lhs, we get

$\displaystyle\langle p_{\Pi}(A)$
$\displaystyle-\textnormal{tr}Ac_{\Pi},p_{\Pi}(B)-\textnormal{tr}Bc_{\Pi}\rangle=\sum_{j=1}^{n}(\textnormal{tr}(A\Pi_{j})-\textnormal{tr}A\textnormal{tr}\Pi_{j}/d)(\textnormal{tr}(B\Pi_{j})-\textnormal{tr}B\textnormal{tr}\Pi_{j}/d)$
$\displaystyle=\sum_{j=1}^{n}\textnormal{tr}[(A-(\textnormal{tr}A/d)I)(\Pi_{j}-(\textnormal{tr}\Pi_{j}/d)I)]\textnormal{tr}[(B-(\textnormal{tr}B/d)I)(\Pi_{j}-(\textnormal{tr}\Pi_{j}/d)I)]$
$\displaystyle=\sum_{j=1}^{n}(A-(\textnormal{tr}A/d)I|\pi_{0}(\Pi_{j}))(\pi_{0}(\Pi_{j})|B-(\textnormal{tr}B/d)I)$
$\displaystyle=(A-(\textnormal{tr}A/d)I|S|B-(\textnormal{tr}B/d)I),$

where $S:=\sum_{j=1}^{n}|\pi_{0}(\Pi_{j}))(\pi_{0}(\Pi_{j})|$. Thus, (5) is
equivalent to

$(A|S|B)=\alpha(A|B)\quad\text{for all
}A,B\in\mathcal{L}_{s}^{0}(\mathbb{C}^{d}),$ (6)

which is equivalent to $\pi_{0}(\Pi)$ being a tight operator frame in
$\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$ with the frame bound $\alpha$. In order
to calculate $\alpha$, let us take the trace on both sides of the equation

$\sum_{j=1}^{n}|\pi_{0}(\Pi_{j}))(\pi_{0}(\Pi_{j})|=\alpha\mathcal{I}_{0},$
(7)

where $\mathcal{I}_{0}$ denotes the identity superoperator on
$\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$. On the rhs we get $\alpha(d^{2}-1)$,
since $\dim\mathcal{L}_{s}^{0}(\mathbb{C}^{d})=d^{2}-1$. On the lhs we get

$\displaystyle\sum_{j=1}^{n}(\pi_{0}(\Pi_{j})|\pi_{0}(\Pi_{j}))$
$\displaystyle=\sum_{j=1}^{n}\left[(\Pi_{j}|\Pi_{j})-\frac{\textnormal{tr}\Pi_{j}}{d}(I|\Pi_{j})-\frac{\textnormal{tr}\Pi_{j}}{d}(\Pi_{j}|I)+\frac{(\textnormal{tr}\Pi_{j})^{2}}{d^{2}}(I|I)\right]$
$\displaystyle=\sum_{j=1}^{n}\textnormal{tr}\Pi_{j}^{2}-\frac{1}{d}\sum_{j=1}^{n}(\textnormal{tr}\Pi_{j})^{2},$

which completes the proof. ∎

It is easy to see that if $\Pi^{1},\Pi^{2},\ldots,\Pi^{m}$ are morphophoric
POVMs with squares of the similarity ratios equal to
$\alpha_{1},\alpha_{2},\ldots,\alpha_{m}$, then also
$\Pi:=(t_{1}\Pi^{1})\cup(t_{2}\Pi^{2})\cup\ldots\cup(t_{m}\Pi^{m})$ is a
morphophoric POVM for any $t_{1},t_{2},\ldots,t_{m}\geq 0$ such that
$t_{1}+t_{2}+\ldots+t_{m}=1$. In such case the square of the similarity ratio
for $\Pi$ is equal to
$\alpha=t_{1}\alpha_{1}+t_{2}\alpha_{2}+\ldots+t_{m}\alpha_{m}$.

The following propositions not only give us an insight into the structure of
the set of morphophoric POVMs on $\mathbb{C}^{d}$ with a fixed number of
elements $n$ but also provide us with examples of such POVMs. The first one
states that an addition of classical noise $q\mathbb{I}=(q_{j}I)_{j=1}^{n}$,
where $q\in\Delta_{n}$, preserves the morphophoricity.

######  Proposition 5.

Let $\Pi=(\Pi_{j})_{j=1}^{n}$ be a morphophoric POVM and let $q\in\Delta_{n}$.
Then $\Pi_{\lambda,q}:=\lambda\Pi+(1-\lambda)q\mathbb{I}$ is also a
morphophoric POVM for $\lambda\in(0,1]$. In such case, the square of the
similarity ratio for $\Pi_{\lambda,q}$ is equal to $\lambda^{2}\alpha$, where
$\alpha$ is the square of the similarity ratio for $\Pi$.

###### Proof.

It is enough to observe that
$\mathcal{Q}_{\lambda\Pi+(1-\lambda)q\mathbb{I}}=\lambda\mathcal{Q}_{\Pi}+(1-\lambda)q$,
which is a homothety (therefore a similarity) with the centre at $q$ and the
ratio equal to $\lambda$. ∎

######  Definition 5.

We say that a POVM $\Pi$ is _boundary_ if $\Pi_{j}$ has at least one
eigenvalue equal to zero for every $j\in\\{1,\ldots,n\\}$.

The next proposition states that every morphophoric POVM is a unique convex
combination of a boundary morphophoric POVM with some classical noise.

######  Proposition 6.

Let $\Pi$ be a morphophoric POVM which is not boundary. Then there exists a
unique boundary morphophoric POVM $E=(E_{j})_{j=1}^{n}$ such that $\Pi=\lambda
E+(1-\lambda)q\mathbb{I}$ for some $\lambda\in(0,1)$ and $q\in\Delta_{n}$.

###### Proof.

Let $0\leq\lambda_{j,1}\leq...\leq\lambda_{j,d}$ be the eigenvalues of
$\Pi_{j}$. As $\Pi$ is not boundary, $\sum_{j=1}^{n}\lambda_{j,1}>0$.
Moreover, from the fact that
$\sum_{j=1}^{n}\sum_{k=1}^{d}\lambda_{j,k}=\sum_{j=1}^{n}\textnormal{tr}\Pi_{j}=d$
and $\Pi\neq(I/n)_{j=1}^{n}$ we obtain $\sum_{j=1}^{n}\lambda_{j,1}<1$. Let
$q_{j}:=\lambda_{j,1}/(\sum_{j=1}^{n}\lambda_{j,1})$ and
$\lambda:=1-\sum_{j=1}^{n}\lambda_{j,1}$. Then $q\in\Delta_{n}$ and
$\lambda\in(0,1)$. Put $E_{j}:=(\Pi_{j}+(\lambda-1)q_{j}I)/\lambda$. Then the
eigenvalues of $E_{j}$ are of the form
$\mu_{j,k}:=(\lambda_{j,k}+(\lambda-1)q_{j})/\lambda=(\lambda_{j,k}-\lambda_{j,1})/\lambda\geq
0$ and so $\mu_{j,1}=0$. Clearly, $\sum_{j=1}^{n}E_{j}=I$ and
$\pi_{0}(E_{j})=\pi_{0}(\Pi_{j})/\lambda$. Thus, $E=(E_{j})_{j=1}^{n}$ is a
boundary morphophoric POVM.

To prove uniqueness, let us assume that there exist a boundary morphophoric
POVM $E^{\prime}$, $\lambda^{\prime}\in(0,1)$ and $q^{\prime}\in\Delta_{n}$
such that
$\Pi=\lambda^{\prime}E^{\prime}+(1-\lambda^{\prime})q^{\prime}\mathbb{I}$. Let
$\mu_{j,k}^{\prime}:=(\lambda_{j,k}+(\lambda^{\prime}-1)q^{\prime}_{j})/\lambda^{\prime}$
denote the $k$-th eigenvalue of $E_{j}^{\prime}$. Then
$\mu_{j,1}^{\prime}=0=\mu_{j,1}$ and, in consequence,
$(\lambda^{\prime}-1)q_{j}^{\prime}=(\lambda-1)q_{j}$ for $j=1,\ldots,n$.
Summing over $j$, we obtain $\lambda^{\prime}=\lambda$, and so
$q_{j}^{\prime}=q_{j}$ for $j=1,\ldots,n$. ∎

A similar reasoning provides us with a _dual_ boundary morphophoric POVM:

######  Proposition 7.

Let $\Pi$ be a morphophoric POVM. Then there exists a unique boundary
morphophoric POVM $\tilde{E}$ such that
$\lambda\Pi+(1-\lambda)\widetilde{E}=q\mathbb{I}$ for some $\lambda\in(0,1)$
and $q\in\Delta_{n}$.

###### Proof.

Let $0\leq\lambda_{j,1}\leq\ldots\leq\lambda_{j,d}$ be the eigenvalues of
$\Pi_{j}$. Let $q_{j}:=\lambda_{j,d}/(\sum_{j=1}^{n}\lambda_{j,d})$ and
$\lambda:=1/(\sum_{j=1}^{n}\lambda_{j,d})$. Then $q\in\Delta_{n}$ and
$\lambda\in(0,1)$ since $\sum_{j=1}^{n}\lambda_{j,d}\geq 1$. Put
$\widetilde{E_{j}}:=(1-\lambda)^{-1}(-\lambda\Pi_{j}+q_{j}I)$. The eigenvalues
of $\widetilde{E_{j}}$ are of the form
$\widetilde{\lambda_{j,k}}:=(-\lambda\lambda_{j,k}+q_{j})/(1-\lambda)=(\lambda/(1-\lambda))(\lambda_{j,d}-\lambda_{j,k})\geq
0$ and $\widetilde{\lambda_{j,d}}=0$. Obviously,
$\sum_{j=1}^{n}\widetilde{E_{j}}=I$ and
$\pi_{0}(\widetilde{E_{j}})=\frac{-\lambda}{1-\lambda}\pi_{0}(E_{j})$. Thus,
$\widetilde{E}=(\widetilde{E_{j}})_{j=1}^{n}$ is a boundary morphophoric POVM.

To prove uniqueness, we proceed in the same way as in the previous
proposition. Let us assume that there exist a boundary morphophoric POVM
$\widetilde{E^{\prime}}$, $\lambda^{\prime}\in(0,1)$ and
$q^{\prime}\in\Delta_{n}$ such that
$\lambda^{\prime}\Pi+(1-\lambda^{\prime})\widetilde{E^{\prime}}=q^{\prime}\mathbb{I}$.
Let
$\widetilde{\lambda_{j,k}^{\prime}}:=(-\lambda^{\prime}\lambda_{j,k}+q^{\prime}_{j})/(1-\lambda^{\prime})$
denote the $k$-th eigenvalue of $\widetilde{E_{j}^{\prime}}$. Then
$\widetilde{\lambda_{j,d}^{\prime}}=0=\widetilde{\lambda_{j,d}}$ and, in
consequence, $q_{j}^{\prime}/\lambda^{\prime}=q_{j}/\lambda$ for
$j=1,\ldots,n$. Summing over $j$, we obtain $\lambda^{\prime}=\lambda$, and so
$q_{j}^{\prime}=q_{j}$ for $j=1,\ldots,n$. ∎

The condition of $\pi_{0}(\Pi)$ being a tight operator frame in
$\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$ brings to mind the definition of a tight
IC-POVM given by Scott in [70]. Let us introduce the following notation:
$P_{j}:=\Pi_{j}/\textnormal{tr}\Pi_{j}$, $P:=(P_{j})_{j=1}^{n}$ and
$\mathcal{F}:=\sum_{j=1}^{n}\textnormal{tr}\Pi_{j}|P_{j})(P_{j}|=\sum_{j=1}^{n}\frac{1}{\textnormal{tr}\Pi_{j}}|\Pi_{j})(\Pi_{j}|$.

######  Definition 6.

A POVM $\Pi$ is a tight IC-POVM if $\pi_{0}(P)$ is a tight operator frame with
respect to the trace measure $\tau$ ($\tau(j)=\textnormal{tr}\Pi_{j}$,
$j=1,\ldots,n$) in $\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$, i.e. if there exists
$\beta>0$ such that

$\sum_{j=1}^{n}\textnormal{tr}\Pi_{j}|\pi_{0}(P_{j}))(\pi_{0}(P_{j})|=\beta\mathcal{I}_{0}.$

######  Remark 2.

The definition of a tight IC-POVM can be expressed equivalently in the
following way:

$\mathcal{F}=\beta\mathcal{I}+\frac{1-\beta}{d}|I)(I|,$

where $\mathcal{I}$ stands for the identity superoperator on
$\mathcal{L}_{s}(\mathbb{C}^{d})$. Note that $\beta$ can be calculated by
taking the trace on both sides of one of the above equations:

$\beta=\frac{1}{d^{2}-1}\left(\sum_{j=1}^{n}\frac{\textnormal{tr}\Pi_{j}^{2}}{\textnormal{tr}\Pi_{j}}-1\right).$

Applying Theorem 4, we get the following statement as a simple observation.

######  Corollary 8.

Let $\Pi$ be a POVM consisting of effects of equal trace. Then $\Delta_{\Pi}$
is similar to the quantum state space if and only if $\Pi$ is a tight IC-POVM.
The square of the similarity ratio $\alpha$ is then equal to
$\frac{1}{d^{2}-1}\left(\sum_{j=1}^{n}\textnormal{tr}\Pi_{j}^{2}-\frac{d}{n}\right)$.

Observe, however, that the equal-trace assumption cannot be dropped.

######  Example 2 (A morphophoric POVM which is not a tight IC-POVM).

Let $X$, $Y$ and $Z$ denote the Pauli matrices and let
$a:=\frac{\sqrt{3}-1}{4}$ and $b:=\frac{3-\sqrt{3}}{6}$. Put

$\Pi_{1}:=aZ+aI,\quad\Pi_{2}:=-aZ+aI,\quad\Pi_{3}:=bX+bI,\quad\Pi_{4}:=-\frac{b}{2}X+aY+bI,\quad\Pi_{5}:=-\frac{b}{2}X-aY+bI.$

Then $(\Pi_{j})_{j=1}^{5}$ is a rank-1 POVM which is morphophoric but not
tight. Indeed, since $\\{X,Y,Z,I\\}$ forms an (equal-norm) orthogonal basis of
$\mathcal{L}_{s}(\mathbb{C}^{2})$, it is easy to see that the projections of
$\Pi_{j}$, $j=1,\ldots,5$, onto the 3-dimensional subspace of traceless
operators can be identified as vertices of a triangular bipyramid: $(0,0,a)$,
$(0,0,-a)$, $(b,0,0)$, $(-b/2,a,0)$, $(-b/2,-a,0)$, constituting a tight
frame, see Fig. 2. On the other hand, non-tightness of IC-POVM can be easily
verified by direct computation.

Figure 2: A triangular bipyramid with vertices at $(0,0,a)$, $(0,0,-a)$,
$(b,0,0)$, $(-b/2,a,0)$, $(-b/2,-a,0)$. The set of vertices forms a tight
frame and therefore represents projections of effects constituting a
morphophoric POVM.

##  4 The probability range of 2-design POVMs

######  Definition 7.

We say that $\\{\rho_{j}\\}_{j=1}^{n}\subset\mathcal{P}(\mathbb{C}^{d})$ is a
$t$-design if

$\frac{1}{n^{2}}\sum_{j,k=1}^{n}f(\textnormal{tr}(\rho_{j}\rho_{k}))=\iint_{(\mathcal{P}(\mathbb{C}^{d}))^{2}}f(\textnormal{tr}(\rho\sigma))d\mu(\rho)d\mu(\sigma)$

holds for every $f:\mathbb{R}\to\mathbb{R}$ polynomial of degree $t$ or less,
where by $\mu$ we mean the unique unitarily invariant measure on
$\mathcal{P}(\mathbb{C}^{d})$.

Obviously, any $t$-design is also an $s$-design for $s<t$. Note that
$\\{\rho_{j}\\}_{j=1}^{n}\subset\mathcal{P}(\mathbb{C}^{d})$ is a 1-design if
and only if $(\frac{d}{n}\rho_{j})_{j=1}^{n}$ is a POVM. Moreover, it is a
2-design if and only if $(\frac{d}{n}\rho_{j})_{j=1}^{n}$ is a tight IC-POVM
[70, Prop. 13]. We will refer to such rank-1 POVM as a 2-design POVM. That
property allows us to rephrase the definition of a 2-design in a more
comprehensible way, namely
$\\{\rho_{j}\\}_{j=1}^{n}\subset\mathcal{P}(\mathbb{C}^{d})$ is a 2-design if
and only if

$\tau=(d+1)\sum_{j=1}^{n}\frac{d}{n}\textnormal{tr}(\tau\rho_{j})\rho_{j}-I$
(8)

for every $\tau\in\mathcal{L}_{s}(\mathbb{C}^{d})$ such that
$\textnormal{tr}\tau=1$.

######  Corollary 9.

Let $\Pi$ be a rank-1 POVM consisting of effects of equal trace. Then
$\mathcal{Q}_{\Pi}$ is similar to the quantum state space if and only if $\Pi$
is a 2-design POVM. The square of the similarity ratio $\alpha$ is then equal
to $\frac{d}{n(d+1)}$.

From now on we assume that $\Pi=(\frac{d}{n}\rho_{j})_{j=1}^{n}$ is a 2-design
POVM. For $\tau\in\mathcal{S}(\mathbb{C}^{d})$,
$p_{j}(\tau):=(p_{\Pi}(\tau))_{j}=\frac{d}{n}\textnormal{tr}(\tau\rho_{j})$ is
the probability of obtaining the $j$-th outcome. The following equalities are
derived directly from (8):

$\displaystyle\sum_{j=1}^{n}p_{j}(\tau)$ $\displaystyle=1,$ (9)
$\displaystyle\sum_{j=1}^{n}(p_{j}(\tau))^{2}$
$\displaystyle=\frac{d(\textnormal{tr}(\tau^{2})+1)}{n(d+1)},$ (10)
$\displaystyle\sum_{j,k,l=1}^{n}p_{j}(\tau)p_{k}(\tau)p_{l}(\tau)\textnormal{tr}(\rho_{j}\rho_{k}\rho_{l})$
$\displaystyle=\frac{\textnormal{tr}(\tau+I)^{3}}{(d+1)^{3}}.$ (11)

In particular, for pure states we get

$\displaystyle\sum_{j=1}^{n}(p_{j}(\tau))^{2}$
$\displaystyle=\frac{2d}{n(d+1)},$ (12)
$\displaystyle\sum_{j,k,l=1}^{n}p_{j}(\tau)p_{k}(\tau)p_{l}(\tau)\textnormal{tr}(\rho_{j}\rho_{k}\rho_{l})$
$\displaystyle=\frac{d+7}{(d+1)^{3}}.$ (13)

######  Remark 3.

Note that the rhs of the last equation does not depend on $n$.

The proof of the following statement can be found in [48].

######  Fact 2.

A self-adjoint operator $\tau$ is a pure quantum state if and only if
$\textnormal{tr}\tau=1$, $\textnormal{tr}\tau^{2}=1$ and
$\textnormal{tr}\tau^{3}=1$.

This fact will be helpful in proving the next theorem that characterises
algebraically the probability distributions belonging to
$\mathcal{Q}_{\Pi}^{1}$. In spite of its apparent similarity to [37, Cor.
2.3.4.], the statement is more general and has different meaning since we do
not assume that the vector satisfying conditions (i)-(iii) below is a
probability distribution from $\mathcal{Q}_{\Pi}$. In fact, we do not even
assume that it is a probability distribution at all.

######  Theorem 10.

Let $\Pi=(\frac{d}{n}\rho_{j})_{j=1}^{n}$ be a 2-design POVM. Then
$(p_{1},\ldots,p_{n})\in\mathcal{Q}_{\Pi}^{1}$ if and only if

  1. i.

$\frac{n}{d}p_{l}=(d+1)\sum_{j=1}^{n}p_{j}\textnormal{tr}(\rho_{j}\rho_{l})-1$
for $l=1,\ldots,n$;

  2. ii.

$\sum_{j=1}^{n}p_{j}^{2}=\frac{2d}{n(d+1)}$;

  3. iii.

$\sum_{j,k,l=1}^{n}p_{j}p_{k}p_{l}\textnormal{tr}(\rho_{j}\rho_{k}\rho_{l})=\frac{d+7}{(d+1)^{3}}$.

###### Proof.

We already know that every probability distribution from
$\mathcal{Q}_{\Pi}^{1}$ satisfies all the equations above. Thus, to complete
the proof, it suffices to show that any vector
$(p_{1},\ldots,p_{n})\in\mathbb{R}^{n}$ fulfilling equalities (i)-(iii)
belongs to $\mathcal{Q}_{\Pi}^{1}$. Let us define
$\tau:=(d+1)\sum_{j=1}^{n}p_{j}\rho_{j}-I$. Then

$\textnormal{tr}(\tau\rho_{l})=(d+1)\sum_{j=1}^{n}p_{j}\textnormal{tr}(\rho_{j}\rho_{l})-\textnormal{tr}(\rho_{l})=\frac{n}{d}p_{l}$

and thus $p_{l}=\frac{d}{n}\textnormal{tr}(\tau\rho_{l})$. It is now enough to
prove that $\tau\in\mathcal{P}(\mathbb{C}^{d})$. Obviously, $\tau^{*}=\tau$.
Using (i), we show that the numbers $p_{1},p_{2},\ldots,p_{n}$ sum up to 1:

$\sum_{l=1}^{n}p_{l}=\frac{d}{n}\left((d+1)\sum_{j=1}^{n}p_{j}\textnormal{tr}\left(\rho_{j}\sum_{l=1}^{n}\rho_{l}\right)-n\right)=\frac{d}{n}(d+1)\frac{n}{d}\sum_{j=1}^{n}p_{j}-\frac{d}{n}n=(d+1)\sum_{j=1}^{n}p_{j}-d,$

which gives $\sum_{l=1}^{n}p_{l}=1$. In consequence,
$\textnormal{tr}\tau=(d+1)\sum_{j=1}^{n}p_{j}\textnormal{tr}(\rho_{j})-d=1$.
Since (10) and (11) hold true for any Hermitian operator with trace equal to
1, we can compare them with our assumptions concerning the sum of squares and
triple products, respectively, to obtain
$\textnormal{tr}\tau^{2}=\textnormal{tr}\tau^{3}=1$, which completes the
proof. ∎

Let us take a closer look at equations (i)-(iii) from Theorem 10. As we
already know from Corollary 9, they need to describe a $(2d-2)$-dimensional
submanifold of the $(d^{2}-2)$-dimensional sphere. From the proof above we
deduce that $p\in\mathbb{R}^{n}$ fulfils the system of $n$ linear equations
(i) if and only if $p$ lies in the affine span of $\mathcal{Q}_{\Pi}$.

######  Definition 8.

By the _primal affine space_ we mean the affine span of $\mathcal{Q}_{\Pi}$
and denote it by $\mathcal{A}$.

Note that $\mathcal{A}\subset\\{x\in\mathbb{R}^{n}:x_{1}+\ldots+x_{n}=1\\}$
and $\dim\mathcal{A}=d^{2}-1$, so the number of equations can be reduced to
$n-(d^{2}-1)$. In particular, if $n=d^{2}$, this system of equations reduces
to the single equation $\sum_{j=1}^{d^{2}}p_{j}=1$. The only 2-design POVMs
with $d^{2}$ elements are SIC-POVMs, i.e. those with
$\textnormal{tr}(\rho_{j}\rho_{k})=\frac{1}{d+1}$ for $j\neq k$. The quadratic
form (ii) is obviously the equation of the sphere in $\mathbb{R}^{n}$ centred
at 0 and of radius $\sqrt{\frac{2d}{n(d+1)}}$. The intersection of this sphere
with the affine subspace $\mathcal{A}$ gives us the $(d^{2}-2)$-dimensional
sphere we are looking for. Finally, the cubic form (iii) is responsible for
cutting the $(2d-2)$-dimensional submanifold from the sphere in question. We
can now state the following:

######  Corollary 11.

  1. i.

The set $\mathcal{Q}_{\Pi}^{1}$ is fully described by equations (i)-(ii) if
and only if $d=2$.

  2. ii.

The linear equations (i) can be reduced to the single equation
$\sum_{j=1}^{n}p_{j}=1$ if and only if $\Pi$ is a SIC-POVM.

A natural question arises whether we can provide a similar characterisation of
$\mathcal{Q}_{\Pi}$ as we did for $\mathcal{Q}_{\Pi}^{1}$ in Theorem 10. As we
can see, the proof of the latter is based on the fact that the pure states are
fully described by
$\textnormal{tr}\tau=\textnormal{tr}\tau^{2}=\textnormal{tr}\tau^{3}=1$. The
mixed states obviously satisfy $\textnormal{tr}\tau=1$ and
$\textnormal{tr}\tau^{2}\leq 1$ and it is tempting to write down the third
inequality in a similar manner. However, it turns out that in order to provide
a complete characterisation of $\tau\in\mathcal{S}(\mathbb{C}^{d})$ in terms
of $\textnormal{tr}\tau^{k}$, we need additional $d-2$ inequalities. Put
together, they take the form $S_{k}\geq 0$ ($k\in\\{2,\ldots,d\\}$), where
$S_{k}$ can be defined recursively by
$S_{k}=\frac{1}{k}\sum_{j=1}^{k}(-1)^{j-1}\textnormal{tr}(\tau^{j})S_{k-j}$
and $S_{0}=S_{1}=1$ [15]. For example, the condition on
$\textnormal{tr}\tau^{3}$ is as follows:
$3\textnormal{tr}\tau^{2}-2\textnormal{tr}\tau^{3}\leq 1$. Thus, such a
characterisation is possible but it becomes more and more complicated with the
growth of $d$. However, for $d=2,3$ we can characterise $\mathcal{Q}_{\Pi}$ in
a relatively simple way.

######  Corollary 12.

Let $\Pi=(\frac{d}{n}\rho_{j})_{j=1}^{n}$ be a 2-design POVM. If
$(p_{1},\ldots,p_{n})\in\mathcal{Q}_{\Pi}$, then

  1. i.

$\frac{n}{d}p_{l}=(d+1)\sum_{j=1}^{n}p_{j}\textnormal{tr}(\rho_{j}\rho_{l})-1$
for $l=1,\ldots,n$;

  2. ii.

$\sum_{j=1}^{n}p_{j}^{2}\leq\frac{2d}{n(d+1)}$;

  3. iii.

$\sum_{j,k,l=1}^{n}p_{j}p_{k}p_{l}\textnormal{tr}(\rho_{j}\rho_{k}\rho_{l})\geq\frac{9n}{2d(d+1)^{2}}\sum_{j=1}^{n}p_{j}^{2}+\frac{d-2}{(d+1)^{3}}$.

Moreover, conditions (i)-(iii) are sufficient for $d=3$ and (i)-(ii) are
sufficient for $d=2$.

##  5 Examples

In the following examples we provide some explicit formulae for the
$(d^{2}-1)$-dimensional affine subspaces $\mathcal{A}$ defined by the system
of linear equations (i) in Theorem 10.

######  Example 3 (Cubical POVM).

Let us consider a cubical POVM $\Pi=(\frac{1}{4}\rho_{j})_{j=1}^{8}$, i.e. the
states $\rho_{1},\ldots,\rho_{8}$ are represented on the Bloch sphere by the
vertices of a cube, e.g. $\frac{1}{\sqrt{6}}(\pm 1,\pm 1,\pm 1)$. Let us label
these vertices in the following way: the vertices of the bottom face are
labelled in sequence with 1-4 and if $i$ and $j$ are the labels of the
opposite vertices, then $j=i+4$ ($mod\ 8$). The eight linear equations (i)
from Theorem 10 reduce to the following $8-(2^{2}-1)=5$ linearly independent
ones:

$\begin{cases}p_{1}+p_{5}=\frac{1}{4}\\\ p_{2}+p_{6}=\frac{1}{4}\\\
p_{3}+p_{7}=\frac{1}{4}\\\ p_{4}+p_{8}=\frac{1}{4}\\\
p_{1}+p_{3}+p_{6}+p_{8}=\frac{1}{2}.\end{cases}$

It is easy to see that the first four of them guarantee that
$p_{1}+\ldots+p_{8}=1$. Obviously, the 3-dimensional affine subspace of
$\mathbb{R}^{8}$ defined by these equations is label-dependent. The number of
possible labellings is $8!=40320$. However, two of them produce the same set
of equations, if the permutation transforming one to the other corresponds to
an isometric operation on the cube. The isometries of the cube are described
by the elements of the octahedral group $O_{h}$. Thus the number of different
subspaces is $\frac{8!}{|O_{h}|}=\frac{40320}{48}=840$.

######  Example 4 (Cuboctahedral and icosahedral POVMs).

The polyhedra defining these POVMs (cuboctahedron and icosahedron) have the
same number of vertices: $n=12$, which can be represented on the Bloch sphere
by $\frac{1}{\sqrt{2(s^{2}+1)}}(\pm s,\pm 1,0)$,
$\frac{1}{\sqrt{2(s^{2}+1)}}(0,\pm s,\pm 1)$ and
$\frac{1}{\sqrt{2(s^{2}+1)}}(\pm 1,0,\pm s)$, where $s=1$ for cuboctahedron
and $s=(1+\sqrt{5})/2$ (_golden ratio_ , usually denoted by $\varphi$) for
icosahedron. If we label these vertices as follows: $v_{1}=(+,+,0)$,
$v_{3}=(+,-,0)$, $v_{5}=(0,+,+)$, $v_{7}=(0,+,-)$, $v_{9}=(+,0,+)$,
$v_{11}=(-,0,+)$ and $v_{2j}=-v_{2j-1}$ for $j\in\\{1,\ldots,6\\}$, then the
different values of the parameter $s$ result in slightly different affine
subspaces defined by the following systems of 9 linear equations:

$\textnormal{cuboctahedron:}\begin{cases}p_{1}+p_{2}=\frac{1}{6}\\\
p_{3}+p_{4}=\frac{1}{6}\\\ p_{5}+p_{6}=\frac{1}{6}\\\
p_{7}+p_{8}=\frac{1}{6}\\\ p_{9}+p_{10}=\frac{1}{6}\\\
p_{11}+p_{12}=\frac{1}{6}\\\ p_{1}-p_{3}+p_{6}+p_{8}=\frac{1}{6}\\\
p_{5}-p_{7}+p_{10}+p_{12}=\frac{1}{6}\\\
p_{9}-p_{11}+p_{2}+p_{4}=\frac{1}{6}\end{cases},\quad\textnormal{icosahedron:}\begin{cases}p_{1}+p_{2}=\frac{1}{6}\\\
p_{3}+p_{4}=\frac{1}{6}\\\ p_{5}+p_{6}=\frac{1}{6}\\\
p_{7}+p_{8}=\frac{1}{6}\\\ p_{9}+p_{10}=\frac{1}{6}\\\
p_{11}+p_{12}=\frac{1}{6}\\\ \varphi(p_{1}-p_{3})+p_{6}+p_{8}=\frac{1}{6}\\\
\varphi(p_{5}-p_{7})+p_{10}+p_{12}=\frac{1}{6}\\\
\varphi(p_{9}-p_{11})+p_{2}+p_{4}=\frac{1}{6}\end{cases}.$

As in the previous case, the definitions of the subspaces are label-dependent.
The number of possible labellings is $12!$, but taking into account the
symmetry groups of cuboctahedron and icosahedron, i.e. the octahedral group
$O_{h}$ and the icosahedral group $I_{h}$, we get the numbers of different
subspaces are $\frac{12!}{|O_{h}|}=\frac{12!}{48}=9979200$ and
$\frac{12!}{|I_{h}|}=\frac{12!}{120}=3991680$.

Starting from dimension $d=3$, the equation of third degree plays a crucial
role in defining $\mathcal{Q}_{\Pi}^{1}$. In the following example we take a
closer look at the 1-parameter family of 2-design POVMs for which the
equations of first and second degree coincide, but not these of third degree.
But first let us observe that the lhs of (iii) in Theorem 10 can be written in
a slightly different way if we assume that both (i) and (ii) from the same
theorem hold. We start with:

$\sum_{j,k,l=1}^{n}p_{j}p_{k}p_{l}\textnormal{tr}(\rho_{j}\rho_{k}\rho_{l})=\sum_{j=1}^{n}p_{j}^{3}\textnormal{tr}(\rho_{j}^{3})+3\sum_{j\neq
k}p_{j}^{2}p_{k}\textnormal{tr}(\rho_{j}^{2}\rho_{k})+\sum_{j\neq k\neq l\neq
j}p_{j}p_{k}p_{l}\textnormal{tr}(\rho_{j}\rho_{k}\rho_{l}).$

But $\rho_{j}^{2}=\rho_{j}$ and $\textnormal{tr}(\rho_{j}^{3})=1$. Moreover,

$\displaystyle\sum_{j\neq k}p_{j}^{2}p_{k}\textnormal{tr}(\rho_{j}\rho_{k})$
$\displaystyle=\sum_{j=1}^{n}p_{j}^{2}\left(\sum_{k=1}^{n}p_{k}\textnormal{tr}(\rho_{j}\rho_{k})-p_{j}\textnormal{tr}(\rho_{j}^{2})\right)=\sum_{j=1}^{n}p_{j}^{2}\left(\frac{1}{d+1}\left(\frac{n}{d}p_{j}+1\right)-p_{j}\right)$
$\displaystyle=\left(\frac{n}{d(d+1)}-1\right)\sum_{j=1}^{n}p_{j}^{3}+\frac{1}{d+1}\sum_{j=1}^{n}p_{j}^{2}=\left(\frac{n}{d(d+1)}-1\right)\sum_{j=1}^{n}p_{j}^{3}+\frac{2d}{n(d+1)^{2}}.$

Thus, the third degree equation now takes the form:

$\left(\frac{3n}{d(d+1)}-2\right)\sum_{j=1}^{n}p_{j}^{3}+\sum_{j\neq k\neq
l\neq
j}p_{j}p_{k}p_{l}\textnormal{tr}(\rho_{j}\rho_{k}\rho_{l})=\frac{n(d+7)-6d(d+1)}{n(d+1)^{3}}$
(14)

######  Example 5 (SIC-POVMs in dimension 3).

Let

$v_{0,j}^{t}=\frac{1}{\sqrt{2}}(-e^{it}\eta^{j},0,1),\quad
v_{1,j}^{t}=\frac{1}{\sqrt{2}}(1,-e^{it}\eta^{j},0),\quad
v_{2,j}^{t}=\frac{1}{\sqrt{2}}(0,1,-e^{it}\eta^{j}),$

where $\eta=e^{2\pi i/3}$, $j\in\\{0,1,2\\}$ and $t\in[0,\pi/3)$. Then
$\Pi^{t}:=(\frac{1}{3}\rho_{m,j}^{t})_{m,j=0}^{2}:=(\frac{1}{3}|v_{m,j}^{t}\rangle\langle
v_{m,j}^{t}|)_{m,j=0}^{2}$ is a SIC-POVM and for $t\neq t^{\prime}$ the sets
$\Pi^{t}$ and $\Pi^{t^{\prime}}$ are not unitarily equivalent. It is easy to
see that the equations of first and second degree are the same for every $t$,
but that is not the case for the third degree equation. Let us introduce the
following notation:

$\displaystyle J$
$\displaystyle:=\\{(\alpha,\beta,\gamma)\in(\mathbb{Z}_{3}\times\mathbb{Z}_{3})^{3}|\alpha\neq\beta\neq\gamma\neq\alpha\\}$
$\displaystyle J_{k}$
$\displaystyle:=\\{(\alpha,\beta,\gamma)\in(\mathbb{Z}_{3}\times\mathbb{Z}_{3})^{3}|\alpha_{1}\neq\beta_{1}\neq\gamma_{1}\neq\alpha_{1},\alpha_{2}+\beta_{2}+\gamma_{2}=k\
(\textnormal{mod }3)\\},\quad k=0,1,2$ $\displaystyle J_{3}$
$\displaystyle:=\\{(\alpha,\beta,\gamma)\in(\mathbb{Z}_{3}\times\mathbb{Z}_{3})^{3}|\alpha_{1}=\beta_{1}=\gamma_{1},\alpha_{2}\neq\beta_{2}\neq\gamma_{2}\neq\alpha_{2}\\}$
$\displaystyle J^{\prime}$ $\displaystyle:=J\setminus(J_{0}\cup J_{1}\cup
J_{2}\cup J_{3}).$

Now, using (14), we get (for the sake of clarity we omit the label $t$)

$\displaystyle\frac{1}{32}$
$\displaystyle=\frac{1}{4}\sum_{\alpha\in\mathbb{Z}_{3}\times\mathbb{Z}_{3}}p_{\alpha}^{3}+\sum_{(\alpha,\beta,\gamma)\in
J}p_{\alpha}p_{\beta}p_{\gamma}\textnormal{tr}(\rho_{\alpha}\rho_{\beta}\rho_{\gamma})$
$\displaystyle=\frac{1}{4}\sum_{\alpha\in\mathbb{Z}_{3}\times\mathbb{Z}_{3}}p_{\alpha}^{3}-\frac{1}{8}\cos(3t)\sum_{(\alpha,\beta,\gamma)\in
J_{0}}p_{\alpha}p_{\beta}p_{\gamma}+\frac{1}{16}(\cos(3t)+\sqrt{3}\sin(3t))\sum_{(\alpha,\beta,\gamma)\in
J_{1}}p_{\alpha}p_{\beta}p_{\gamma}$
$\displaystyle+\frac{1}{16}(\cos(3t)-\sqrt{3}\sin(3t))\sum_{(\alpha,\beta,\gamma)\in
J_{2}}p_{\alpha}p_{\beta}p_{\gamma}-\frac{1}{8}\sum_{(\alpha,\beta,\gamma)\in
J_{3}}p_{\alpha}p_{\beta}p_{\gamma}+\frac{1}{16}\sum_{(\alpha,\beta,\gamma)\in
J^{\prime}}p_{\alpha}p_{\beta}p_{\gamma}$

For $t=0$ we get the Hesse configuration and the above equation takes a
particularly nice form [77]:

$\sum_{\alpha\in\mathbb{Z}_{3}\times\mathbb{Z}_{3}}p_{\alpha}^{3}-\frac{1}{2}\sum_{(\alpha,\beta,\gamma)\in
J_{0}\cup J_{3}}p_{\alpha}p_{\beta}p_{\gamma}=0.$

##  6 MUB-like POVMs

###  6.1 MUBs

Another remarkable example of a $2$-design POVM is a complete set of mutually
unbiased bases (MUBs) in $\mathbb{C}^{d}$, where $d$ is a prime power. (The
question whether complete sets of MUBs exist in other dimensions remains
open). Such POVM $\Pi$ consists of $d(d+1)$ effects of the form
$\frac{1}{d+1}\rho_{j}$, where
$\rho_{j}\in\mathcal{P}\left(\mathbb{C}^{d}\right)$ for $j=1,2,\ldots,d(d+1)$,
and

$\operatorname{tr}(\rho_{kd+l}\rho_{k^{\prime}d+l^{\prime}})=\left\\{\begin{tabular}[c]{ll}$1$&$\text{\text{for
}}k=k^{\prime},l=l^{\prime}$\\\ $0$&$\text{\text{for }}k=k^{\prime},l\neq
l^{\prime}$\\\ $\frac{1}{d}$&for\text{ }$k\neq
k^{\prime}$\end{tabular}\right.$

for $k,k^{\prime}\in\\{0,\ldots,d\\}$ and $l,l^{\prime}\in\\{1,\ldots,d\\}$.

The fact that a complete set of MUBs constitutes a $2$-design POVM was first
shown explicitly in [50], however, this also follows easily [5] from the fact
that in this case $\Pi$ fulfils both the Welch bound [82] and the Levenshtein
bound [54] for the angles between $2$-design lines.

Now, the system of linear equations defining the primal affine space
$\mathcal{A}$ takes a particularly nice form since, instead of the initial
$n=d(d+1)$ equations, it suffices to consider the following $d+1$ equations
expressing the simple fact that the probabilities over any basis
($k=0,\ldots,d$) need to sum up to $\frac{1}{d+1}$:

$\sum_{l=1}^{d}p_{kd+l}=\frac{1}{d+1}\text{.}$

We shall show that this result is valid even in more general settings, see
Theorem 13 below.

###  6.2 MUB-like 2-designs: definition and the orthogonality graph

Let us consider a $2$-design POVM $\Pi=(\frac{d}{n}\rho_{j})_{j=1}^{n}$, where
$\rho_{j}\in\mathcal{P}\left(\mathbb{C}^{d}\right)$ for $j=1,\ldots,n$. We
call $\Pi$ MUB-like [47] if there exists $a>0$ such that
$\left\\{\operatorname{tr}\left(\rho_{j}\rho_{k}\right):j,k=1,\ldots,n,j\neq
k\right\\}=\left\\{0,a\right\\}$. The orthogonality graph for $\Pi$ is the
graph $\Gamma$ with the vertex set $\left\\{\rho_{j}\right\\}_{j=1}^{n}$ and
the adjacency relation defined by $\rho\sim\sigma$ iff $\rho\perp\sigma$ iff
$\operatorname{tr}\left(\rho\sigma\right)=0$. Now, we can apply the results of
Hoggar, who, in even more general setting, examined the properties of $\Gamma$
[44, Theorem 5.2][46, Sec. 3], employing the unpublished notes of Neumaier
[57]. In particular, he showed that $\Gamma$ is a distance regular graph with
diameter $2$, and so strongly regular, see [11]. We now specify Hoggar’s
results for MUB-like $2$-design POVMs. (Note, however, that Hoggar analysed
the graph complementary to $\Gamma$.) Denote by:

  * •

$\varkappa$ \- the number of vertices of $\Gamma$ adjacent (orthogonal) to a
given vertex;

  * •

$\lambda$ (resp. $\mu$) - the number of vertices of $\Gamma$ adjacent to a
pair of adjacent (resp. not adjacent) vertices.

Obviously, $\Gamma$ is non-complete and connected unless $\mu=0$. The
adjacency matrix of $\Gamma$ has three eigenvalues $\varkappa$, $r$, and $-q$,
where $r,q\in\mathbb{N}$, $1\leq q\leq r^{2}$,
$\psi:=\frac{r(q-1)}{r+q}\in\mathbb{N}$, with the multiplicities $1$, $f$, and
$g$, respectively. Note that the inequality $q\leq r^{2}$ follows from the
Delsarte-Goethals-Seidel bound [26], see also [53, Theorem 7.2], for the
cardinality of sets of lines in $\mathbb{C}^{d}$ having a prescribed number of
angles. Moreover, $a=\frac{1}{r+1}$, and

$\displaystyle d$ $\displaystyle=r+\psi+1\text{, \ \ }n=d(r+2q)\text{,}$
$\displaystyle\varkappa$ $\displaystyle=(d-1)q\text{, \ \ }\mu=\psi q\text{, \
\ }\lambda=\mu+r-q\text{,}$ $\displaystyle f$ $\displaystyle=n-d^{2}\text{, \
\ }g=d^{2}-1\text{.}$

Clearly, $\Pi$ saturates both the Welch bound and the Levenshtein bound, which
are in this case equivalent as

$a=\frac{2n-d(d+1)}{(n-d)(d+1)}\text{.}$ (15)

In fact, as observed in [27, Proposition 3], if $\Pi$ is a rank-one POVM
consisting of effects of equal trace such that
$\left\\{0,a\right\\}=\left\\{\operatorname{tr}\left(\rho_{j}\rho_{k}\right):j,k=1,\ldots,n,j\neq
k\right\\}$, then $\Pi$ forms a $2$-design if and only if (15) holds.

Clearly, a maximal clique (full subgraph) in $\Gamma$, i.e. a maximal
orthogonal set, has at most $d$ elements, which is just the Hoffman-Delsarte
bound for $\Gamma$ [72], as $1+\varkappa/q=d$. Let us recall that a clique $C$
is called regular if every point not in $C$ is adjacent to the same number
(the nexus) of points in $C$. It follows from [58, Theorem 1.1, Corollary 1.2]
that all regular cliques in $\Gamma$, if exist, have $d$ elements (and so they
are so called Delsarte cliques) with the nexus equal to $\psi$. From [72,
Theorem 3.1.iv] we deduce that every clique in $\Gamma$ of size $d$ is
regular. Hence, $\Gamma$ is a Delsarte clique graph (for definition see [4])
if and only if for every $j,k=1,\ldots,n$, $j\neq k$ such that
$\rho_{j}\perp\rho_{k}$, $\left\\{\rho_{j},\rho_{k}\right\\}$ is contained in
the same number (say $c$) of bases. More precisely, $\Gamma$ is then a
Delsarte clique graph with parameters $\left(\varkappa,d-1,c\right)$.
Moreover, double counting argument shows that the number of maximal cliques in
every local graph is $cq$, and the number of maximal cliques in $\Gamma$ is
$\left(r+2q\right)cq$. Note that if $\Gamma$ is an edge transitive graph and
there is a Delsarte clique in $\Gamma$, then $\Gamma$ is a Delsarte clique
graph. In particular, this is true for rank $3$ graphs, i.e. strongly regular
graphs generated by a rank $3$ permutation group of even order acting on a
finite set $X$ (i.e. a group with exactly three orbits on $X\times X$), as
they are edge transitive [42].

###  6.3 Examples of MUB-like 2-designs

Let us now consider two extreme cases: $q=1$ and $q=r^{2}$, where $n$ is,
resp., minimal and maximal, for a given $d$.

  * •

$q=1$ (iff $\psi=0$)

In this case $a=\frac{1}{d}$, $n=d(d+1)$, $\varkappa=d-1$, $\lambda=d-2$ and
$\mu=0$. In consequence, $\Pi$ must be a complete set of mutually unbiased
bases (MUBs).

  * •

$q=r^{2}$ (iff $\psi=r-1$)

Then $d=2r$, $a=\frac{1}{r+1}$,
$n=2r^{2}\left(2r+1\right)=d^{2}\left(d+1\right)/2$,
$\varkappa=\left(2r-1\right)r^{2}$, $\lambda=\left(r-1\right)^{2}r$ and
$\mu=\left(r-1\right)r^{2}$. Hoggar showed [46, Theorem 3.7] that this is also
a sufficient and necessary condition for $\Pi$ to be a (minimal or tight)
$3$-design. Note that $\Pi$ cannot be a $4$-design [44, Theorem 5.2].

Note that the case $q=r=1$, i.e. the POVM consisting of $3$ MUBs in dimension
$d=2$ that form the regular octahedron inscribed in the Bloch sphere, lies at
the intersection of these two extremes. Besides complete sets of MUBs, which
exist for every prime power $d$, but not necessarily for other dimensions, we
know four other MUB-like POVMs, one in each of dimensions $4$, $5$, $6$ and
$28$, see also [27]. The parameters of these POVMs and their orthogonality
graphs are collected in Tab. 1.

$n$ | $\varkappa$ | $\lambda$ | $\mu$ | $d$ | $r$ | $q$ | $\psi$ | $c$
---|---|---|---|---|---|---|---|---
$d(d+1)$ | $d-1$ | $d-2$ | $0$ | $d$ | $d-1$ | $1$ | $0$ | $1$
$40$ | $12$ | $2$ | $4$ | $4$ | $2$ | $4$ | $1$ | $1$
$45$ | $12$ | $3$ | $3$ | $5$ | $3$ | $3$ | $1$ | $1$
$126$ | $45$ | $12$ | $18$ | $6$ | $3$ | $9$ | $2$ | $3$
$4060$ | $1755$ | $730$ | $780$ | $28$ | $15$ | $65$ | $12$ | $45$
Table 1: Parameters of the orthogonality graphs generated by MUB-like POVMs.

Note that all these graphs are Delsarte clique graphs:

  * •

For MUBs $\Gamma$ is the disjoint union of maximal cliques and, in
consequence, a Delsarte clique graph with $\psi=0$ and $c=1$.

  * •

$d=4$ [44, Ex. 7] This $3$-design was introduced in [62, Sec. 6] under the
name of Penrose dodecahedron, see also [84, 63, 55], and, independently, as
the diameters of the Witting polytope [19, 20]. Waegell and Aravind [79]
showed that these two constructions are in fact equivalent. Hoggar [45] proved
that in this case $\Gamma$ is the collinearity graph of one of two generalised
quadrangles of order $(3,3)$ [60][61, Secs. 3.1.1 & 6.2.1], namely $W(3)$
arising as the set of absolute points and lines of a symplectic polarity of
projective geometry $PG(3,3)$, see also [12, Ex. 10.10]. Hence $\Gamma$ is a
(rank $3$) Delsarte clique graph with $\psi=1$ and $c=1$.

  * •

$d=5$ [44, Ex. 18] This $2$-design was defined in [26] and it is equal to the
line system $\mathcal{K}_{5}$ [53, pp. 107 & 166]. We shall show below (Ex. 6)
that in this case $\Gamma$ is the collinearity graph of the unique generalised
quadrangle of order $(4,2)$ [43][12, Ex. 10.11]. In consequence, $\Gamma$ is a
(rank $3$) Delsarte clique graph with $\psi=1$ and $c=1$.

  * •

$d=6$ [44, Ex. 7] This $3$-design was introduced in [56] and it is just the
line system $\mathcal{K}_{6}$ [53, pp. 107 & 166]. The orthogonality graph for
this design is a unique rank 3 strongly regular graph with parameters
$(126,45,12,18)$ [8][12, Ex. 10.32], having the collinearity graph of unique
quadrangle of order $(4,2)$, see above, as its local graph [21, Sec. 5.2].
Hence, $\Gamma$ is a (rank $3$) Delsarte clique graph with $\psi=2$ and $c=3$.

  * •

$d=28$ [44, Ex. 20] This MUB-like $2$-design (with $a=1/16$) was defined in
[18] and it comes from a $28$-dimensional representation of the sporadic
simple Rudvalis group. This group is a rank $3$ permutation group of even
order [42] on a $4060$-element set and, as such, it generates an edge
transitive strongly regular (Rudvalis) graph, which is the orthogonality graph
for the $2$-design. Now, $\Gamma$ is a (rank $3$) Delsarte clique graph with
$\psi=12$ and $c=45$ [12, Ex. 10.65].

However, unfortunately, we do not know neither whether it is true that the
orthogonality graph of a MUB-like POVM is always a Delsarte clique graph, nor
even whether any other MUB-like POVMs apart from those discussed above exist.
The answer to this question seems to be of some importance, as we use this
property in the next section.

###  6.4 Primal affine space in MUB-like case

The next result gives us an alternative description of the primal affine space
(see Definition 8), and ipso facto an alternative form of the particular case
of the primal equation in the MUB-like case, assuming that the orthogonality
graph $\Gamma$ is a Delsarte clique graph. On the other hand, it shows that
the primal polytope (see Section 7)
$\mathcal{A\cap}\Delta_{n}=\frac{d}{n}\cdot\operatorname{QSTAB}(\Gamma)\mathcal{\cap}\Delta_{n}$,
where $\operatorname{QSTAB}(\Gamma)$ is the clique constrained stable set
polytope for $\Gamma$ [1].

######  Theorem 13.

Let $\Pi$ be a MUB-like POVM and let its orthogonality graph be a Delsarte
clique graph. Denote by $\mathcal{C}$ the set of orthogonal bases (maximal
cliques) contained in $\left\\{\rho_{j}\right\\}_{j=1}^{n}$. For
$C\in\mathcal{C}$ set $J_{C}:=\left\\{j=1,\ldots,n:\rho_{j}\in C\right\\}$.
Then the following conditions are equivalent for
$p\in\left(\mathbb{R}^{+}\right)^{n}$:

  1. i.

$p$ fulfils (i) from Theorem 10;

  2. ii.

$p\in\Delta_{n}$ and for every $C\in\mathcal{C}$

$\sum_{j\in J_{C}}p_{j}=\frac{d}{n}\text{;}$ (16)

  3. iii.

$p\in\Delta_{n}$ and for every clique $C$ in $\Gamma$

$\sum_{j\in J_{C}}p_{j}\leq\frac{d}{n}\text{.}$ (17)

###### Proof.

For $l=1,\ldots,n$ denote by $\mathcal{C}_{l}:=\\{C\in\mathcal{C:}\rho_{l}\in
C\\}$,
$A_{l}:=\\{j=1,\ldots,n:\operatorname{tr}\left(\rho_{j}\rho_{l}\right)=a\\}$
and
$O_{l}:=\\{j=1,\ldots,n:\operatorname{tr}\left(\rho_{j}\rho_{l}\right)=0\\}$.
Note that $\left|\mathcal{C}_{l}\right|=cq$.

The implication $(ii)\Rightarrow(iii)$ follows from the fact that each clique
is contained in a maximal one.

To show $(iii)\Rightarrow(i)$, set $l=1,\ldots,n$. Applying (17), we get

$\sum_{C\in\mathcal{C}_{l}}\sum_{j\in J_{C}}p_{j}\leq cq\frac{d}{n}\text{.}$

In consequence,

$cqp_{l}+c\sum_{j\in O_{l}}p_{j}=cqp_{l}+\sum_{C\in\mathcal{C}_{l}}(\sum_{j\in
J_{C}}p_{j}-p_{l})\leq cq\frac{d}{n}\text{,}$

and so

$qp_{l}+\sum_{j\in O_{l}}p_{j}\leq q\frac{d}{n}\text{.}$ (18)

Multiplying both sides of (18) by $(d+1)a$, we get

$\displaystyle\frac{q}{r+q}=(d+1)aq\frac{d}{n}$
$\displaystyle\geq(d+1)a(qp_{l}+\sum_{j\in O_{l}}p_{j})$
$\displaystyle=(d+1)aqp_{l}+(d+1)a(1-p_{l}-\sum_{j\in A_{l}}p_{j})$
$\displaystyle=(d+1)\left(aq+(1-a)\right)p_{l}-(d+1)(p_{l}+a\sum_{j\in
A_{l}}p_{j})+(d+1)a$ $\displaystyle=\frac{n}{d}p_{l}-(d+1)(p_{l}+a\sum_{j\in
A_{l}}p_{j})+\frac{r+2q}{r+q}.$

Thus,

$\frac{n}{d}p_{l}\leq(d+1)(p_{l}+a\sum_{j\in A_{l}}p_{j})-1\text{.}$ (19)

On the other hand, the sums of both sides of (19) over $l=1,\ldots,n$ are
equal as

$\sum_{l=1}^{n}((d+1)(p_{l}+a\sum_{j\in
A_{l}}p_{j})-1)=(d+1)(1+a\left(n-\varkappa-1\right))-n=\frac{n}{d}=\frac{n}{d}\sum_{l=1}^{n}p_{l}\text{.}$

Hence, $(i)$ follows.

To show $(i)\Rightarrow(ii)$, we note first that summing the equalities in (i)
from Theorem 10 over $l=1,\ldots,n$, we get

$\sum_{l=1}^{n}p_{l}=\frac{n}{\left(d+1\right)\left(\left(n-\varkappa-1\right)a+1\right)-n/d}=1\text{.}$

Now fix $C\in\mathcal{C}$. Then

$\displaystyle\frac{n}{d}\sum_{l\in J_{C}}p_{l}$
$\displaystyle=(d+1)(\sum_{l\in J_{C}}p_{l}+a\sum_{l\in J_{C}}\sum_{j\in
A_{l}}p_{j})-d$ $\displaystyle=(d+1)\sum_{l\in
J_{C}}p_{l}+a(d+1)\left(d-\psi\right)\left(1-\sum_{l\in J_{C}}p_{l}\right)-d$
$\displaystyle=(d+1)\sum_{l\in J_{C}}p_{l}+(d+1)\left(1-\sum_{l\in
J_{C}}p_{l}\right)-d,$

and so (16) holds, which completes the proof. ∎

######  Example 6 (Two-distance $2$-design in dimension $5$).

This $2$-design consists of $45$ projections onto vectors $(1,0,0,0,0)$ and
$\frac{1}{4}(0,1,\pm\eta,\pm\eta,\pm 1)$ under all cyclic permutations of
their coordinates, where $\eta:=e^{2\pi i/3}$. The $2$-design POVM is then
formed by rescaling these projections by $\frac{1}{9}$. The inner products
between different states of this $2$-design take the values $0$ and
$\frac{1}{4}$. As a two-distance $2$-design, it carries a strongly regular
graph $\operatorname{srg}(45,12,3,3)$. It turns out that among $78$ non-
isomorphic strongly regular graphs with such parameters, our graph is the one
generated by the point graph of the generalised quadrangle of order $(4,2)$,
meaning that the $45$ vectors constituting the $2$-design can be arranged into
$27$ orthonormal bases in such a way that every vector belongs to $3$ bases,
every $2$ vectors belong to at most $1$ basis, and for every vector $v$ and
every base $B$ such that $v\notin B$ there exist a unique vector $v^{\prime}$
and a basis $B^{\prime}$ such that $v^{\prime}\in B^{\prime}$, $v\in
B^{\prime}$ and $v^{\prime}\in B$, see Fig. 3 (adapted from [65, p. 61] and
[69, Fig. 1]). Obviously, the sum of the probabilities over any basis is equal
to $\frac{1}{9}$. But there are $27$ bases and we need just $45-(25-1)=21$
linear equations to describe the affine space $\mathcal{A}$. An example of how
to get rid of $6$ equations (respective bases are marked in blue) in order to
obtain a system of $21$ linearly independent equations is presented in Fig. 3.

Figure 3: Two-distance 2-design in $\mathbb{C}^{5}$ as $\textnormal{GQ}(4,2)$.
Vertices represent vectors and curves (lines, circles or arcs of ellipses)
represent bases. Sums of probabilities over each of blue bases can be omitted
in the description of the primal affine subspace $\mathcal{A}$.

##  7 Geometric properties of generalised qplex

To understand a geometric picture hidden behind the algebraic equations
presented in the previous sections, let us consider a $2$-design POVM
$\Pi=(\frac{d}{n}\rho_{j})_{j=1}^{n}$, where
$\\{\rho_{j}\\}_{j=1}^{n}\subset\mathcal{P}(\mathbb{C}^{d})$. We show that the
geometry of generalised qplex generated by this design is much alike to that
of (Hilbert) qplex, in spite of the fact that now this object is not located
between two simplices but between two dual polytopes lying in a medial
($d^{2}-1$)-section of the probability simplex by an affine space. In our
reasoning we use similar steps as in [3], although some alterations related to
the fact that $\mathcal{Q}_{\Pi}$ is not a full-dimensional subset of
$\Delta_{n}$ are necessary. Finally, we show which geometric properties are
preserved if a general morphophoric POVM is considered.

Let us recall that from Corollary 9 it follows that the measurement map
$p_{\Pi}:\mathcal{S}\left(\mathbb{C}^{d}\right)\rightarrow\mathcal{Q}_{\Pi}$
is a similarity of ratio $m\sqrt{d}$, where $m:=1/\sqrt{n(d+1)}$. Clearly, the
uniform distribution
$c_{n}:=\left(1/n,\ldots,1/n\right)=p_{\Pi}\left(\rho_{\ast}\right)\in\mathcal{Q}_{\Pi}$.
We know that the maximal ball centred at $\rho_{\ast}$ and contained in
$\mathcal{S}\left(\mathbb{C}^{d}\right)$ has radius $1/\sqrt{d(d-1)}$. Hence,
the maximal ball centred at $c_{n}$ and contained in $\mathcal{Q}_{\Pi}$ has
radius $r:=m/\sqrt{d-1}$. On the other hand, $\mathcal{S}(\mathbb{C}^{d})$ is
contained in the ball with centre $\rho_{\ast}$ and radius $\sqrt{(d-1)/d}$
with the pure states contained in the corresponding sphere. Thus,
$\mathcal{Q}_{\Pi}$ is contained in the ball centred at $c_{n}$ with radius
$R:=m\sqrt{d-1}$ and $\mathcal{Q}_{\Pi}^{1}$ is contained in the corresponding
sphere. In short,

$B_{\mathcal{A}}(c_{n},r)\subset\mathcal{Q}_{\Pi}\subset
B_{\mathcal{A}}(c_{n},R)\quad\textnormal{and}\quad\mathcal{Q}_{\Pi}^{1}\subset\partial
B_{\mathcal{A}}(c_{n},R).$ (20)

For $j=1,\ldots,n$ introduce the basis distributions given by
$f_{j}:=p_{\Pi}\left(\rho_{j}\right)$ generating the basis polytope

$D:=\operatorname*{conv}\left\\{f_{j}:j=1,\ldots,n\right\\}\text{.}$

Note that $\left(f_{j}\right)_{k}=\left(f_{k}\right)_{j}$ for $j,k=1,\ldots,n$
and the uniform distribution
$c_{n}:=\left(1/n,\ldots,1/n\right)=p_{\Pi}\left(\rho_{\ast}\right)=p_{\Pi}\left(\frac{1}{n}\sum_{j=1}^{n}\rho_{j}\right)=\frac{1}{n}\sum_{j=1}^{n}f_{j}\in
D$.

######  Fact 3.

$B_{\mathcal{A}}\left(c_{n},R\right)$ is the circumscribed ball of $D$.

Now, the following statement is the result of simple computation.

######  Fact 4.

Let $p\in\mathbb{R}^{n}$. The following conditions are equivalent:

  1. i.

$p\in\mathcal{A}$;

  2. ii.

$p=\left(d+1\right)\sum_{j=1}^{n}p_{j}f_{j}-dc_{n}$;

  3. iii.

$p-c_{n}=\left(d+1\right)\sum_{j=1}^{n}p_{j}\left(f_{j}-c_{n}\right)$;

  4. iv.

$\frac{1}{d+1}\left\langle p,e_{k}\right\rangle=\left\langle
p,f_{k}\right\rangle-dm^{2}$ for every $k=1,\ldots,n$;

  5. v.

$p-c_{n}\perp e_{k}-\left(d+1\right)\left(f_{k}-c_{n}\right)$ for every
$k=1,\ldots,n$.

Note that (ii) is just the famous primal equation (‘Urgleichung’) introduced
by the founders of QBism and applied to the situation where $\Pi$ plays the
role of both the ‘ground’ and the ‘sky’ measurement.

Let us now consider the homothety in $\mathcal{A}$ with the centre at $c_{n}$
and the ratio equal to $1/(d+1)$, i.e. the map
$h:\mathcal{A}\rightarrow\mathcal{A}$ given by

$h\left(p\right):=c_{n}+\frac{p-c_{n}}{d+1}=\frac{1}{d+1}p+\frac{d}{d+1}c_{n}$

for $p\in\mathcal{A}$. Let
$P_{\mathcal{A}}:\mathbb{R}^{n}\rightarrow\mathcal{A}$ denote the orthogonal
projection on $\mathcal{A}$.

######  Fact 5.

In the above situation $h\left(P_{\mathcal{A}}e_{k}\right)=f_{k}$ for every
$k=1,\ldots,n$. In particular, $h\left(P_{\mathcal{A}}\Delta_{n}\right)=D$.

###### Proof.

Let $k=1,\ldots,n$. Then it follows from Fact 4.v that
$P_{\mathcal{A}}e_{k}=c_{n}+\left(d+1\right)\left(f_{k}-c_{n}\right)$. Hence,
$h\left(P_{\mathcal{A}}e_{k}\right)=f_{k}$, as desired. ∎

Define the primal polytope $\Delta$ as the ($d^{2}-1$)-section of $\Delta_{n}$
by $\mathcal{A}$, i.e.

$\Delta:=\mathcal{A}\cap\Delta_{n}\text{.}$

It follows from Theorem 10 that

$D\subset\mathcal{Q}_{\Pi}\subset\Delta\text{,}$

see Fig. 4. Thus

$D\cup
B_{\mathcal{A}}\left(c_{n},r\right)\subset\mathcal{Q}_{\Pi}\subset\Delta\cap
B_{\mathcal{A}}\left(c_{n},R\right)\text{.}$

######  Theorem 14.

The above ($d^{2}-1$)-section is medial, i.e. the vertices of $\Delta_{n}$ are
equidistant from $\mathcal{A}$. More precisely,

$\operatorname{dist}\left(e_{k},\mathcal{A}\right)=\sqrt{1-\frac{d^{2}}{n}}$

for every $k=1,\ldots,n$.

###### Proof.

Let $k=1,\ldots,n$. From the Pythagorean theorem, Fact 5, (20), and Theorem
10.ii we get

$\displaystyle\operatorname{dist}^{2}\left(e_{k},\mathcal{A}\right)$
$\displaystyle=\left\|P_{\mathcal{A}}e_{k}-e_{k}\right\|^{2}=\left\|e_{k}-c_{n}\right\|^{2}-\left\|P_{\mathcal{A}}e_{k}-c_{n}\right\|^{2}$
$\displaystyle=\frac{n-1}{n}-\left(d+1\right)^{2}\left\|f_{k}-c_{n}\right\|^{2}=\frac{n-1}{n}-\left(d+1\right)^{2}\left(d-1\right)m^{2}=1-\frac{d^{2}}{n}\text{.}\qed$

Figure 4: A central and medial 2-section of $\Delta_{4}$ by the primal affine
space $\mathcal{A}$. The points $f_{1},\ldots,f_{4}$ are images under a
homothety in $\mathcal{A}$ with the centre at $c_{4}$ of the orthogonal
projections of $e_{1},\ldots,e_{4}$ onto $\mathcal{A}$.

Note that $\mathcal{A}\subset\\{p\in\mathbb{R}^{n}:\sum_{j=1}^{n}p_{j}=1\\}$.
Hence $c_{n}\perp\mathcal{A}$, and so $\left\langle
p-c_{n},q-c_{n}\right\rangle=\left\langle p,q\right\rangle-\frac{1}{n}$ for
$p,q\in\mathcal{A}$. Let $\iota$ be the inversion in $\mathcal{A}$ through
$c_{n}$, given by $\iota\left(p\right):=2c_{n}-p$ for $p\in\mathcal{A}$. Now
we can characterise $\mathcal{A}$ more precisely. The following fact is the
result of direct computation.

######  Fact 6.

Let $p\in\mathcal{A}$. The following conditions are equivalent:

  1. i.

$p\in\Delta$;

  2. ii.

$p_{k}\geq 0$ for every $k=1,\ldots,n$;

  3. iii.

$\left\langle p,f_{k}\right\rangle\geq dm^{2}$ for every $k=1,\ldots,n$;

  4. iv.

$\left\langle p,q\right\rangle\geq dm^{2}$ for every $q\in D$;

  5. v.

$m^{2}\geq\left\langle p-c_{n},q-c_{n}\right\rangle$ for every
$q\in\iota\left(D\right)$.

######  Fact 7.

$B_{\mathcal{A}}(c_{n},r)$ is the inscribed (maximal central) ball contained
in $\Delta$.

###### Proof.

Taking $\rho_{k}^{\prime}:=(I-\rho_{k})/(d-1)$ and
$f_{k}^{\prime}:=p_{\Pi}(\rho_{k}^{\prime})$, from the similarity of
$\mathcal{S}(\mathbb{C}^{d})$ and $\mathcal{Q}_{\Pi}$ we get
$\left\|f_{k}^{\prime}-c_{n}\right\|^{2}=dm^{2}\|\rho_{k}^{\prime}-\rho_{\ast}\|_{HS}^{2}=\frac{m^{2}}{d-1}=r^{2}$
and $\langle
f_{k}^{\prime},f_{k}\rangle=dm^{2}(\textnormal{tr}(\rho_{k}^{\prime}\rho_{k})+1)=dm^{2}$.
Thus, from Fact 6 it follows that
$f_{k}^{\prime}\in\partial_{\mathcal{A}}\Delta\cap B(c_{n},r)$. ∎

Now we discuss the polarity of basis and primal polytopes.

######  Definition 9.

Let $C$ be a convex subset of $\mathcal{A}$, $s>0$. Define polar and dual of
$C$ in $\mathcal{A}$ with respect to the sphere with centre at $c$ of radius
$s$ by

$C^{\circ}:=\left\\{p\in\mathcal{A}:s^{2}\geq\left\langle
p-c,q-c\right\rangle\text{ for every }q\in
C\right\\}\quad\textnormal{and}\quad
C^{\star}:=\iota\left(C^{\circ}\right)\text{.}$

Note that if the sphere in the definition above is centred at $c_{n}$, we get

$C^{\star}=\left\\{p\in\mathcal{A}:\left\langle
p,q\right\rangle\geq\frac{1}{n}-s^{2}\text{ for every }q\in
C\right\\}\text{.}$

From the convex analysis we know that the following fact holds.

######  Fact 8.

In the above situation

  1. i.

$C^{\circ\circ}=C$;

  2. ii.

$C^{\star}=\left(\iota\left(C\right)\right)^{\circ}$;

  3. iii.

$C^{\star\star}=C$.

The last statement is a direct consequence of the preceding two. Two convex
sets $C$ and $G$ are called polar (resp. dual) in $\mathcal{A}$ with respect
to the central sphere of radius $s$ if $C^{\circ}=G$ (resp. $C^{\star}=G$).
From condition (v) in Fact 6 we deduce, see Fig. 5:

######  Theorem 15.

The polytopes $D$ and $\Delta$ are dual in $\mathcal{A}$ with respect to the
central sphere of radius $m=\sqrt{rR}$. The same is true for the balls
$B_{\mathcal{A}}\left(c_{n},r\right)$ and
$B_{\mathcal{A}}\left(c_{n},R\right)$.

Figure 5: Duality of the polytopes $D$ and $\Delta$ and of the balls $B_{r}$
and $B_{R}$. The qplex $\mathcal{Q}_{\Pi}$ is a convex set containing $D$ and
$B_{r}$ and contained in the intersection of $\Delta$ and $B_{R}$.

Now, we show that the elements of $\mathcal{Q}_{\Pi}$ fulfil the following
fundamental inequalities (or are consistent [3]).

######  Theorem 16.

For $p,q\in\mathcal{Q}_{\Pi}$ we have

$dm^{2}\leq\left\langle p,q\right\rangle\leq 2dm^{2}\text{.}$

###### Proof.

Take $\rho,\sigma\in\mathcal{S}\left(\mathbb{C}^{d}\right)$ such that
$p_{\Pi}\left(\rho\right)=p$ and $p_{\Pi}\left(\sigma\right)=q$. From the
similarity of $\mathcal{S}(\mathbb{C}^{d})$ and $\mathcal{Q}_{\Pi}$ it follows
that $\langle p,q\rangle=dm^{2}(\textnormal{tr}(\rho\sigma)+1)$. Now, it is
enough to apply the well-known inequality
$0\leq\operatorname*{tr}(\rho\sigma)\leq 1$. ∎

Finally, let us set again $s=m$ in the definition of dual sets. Then
$\mathcal{Q}_{\Pi}$ is a self-dual (and hence maximally consistent) set.

######  Theorem 17.

The probability range $\mathcal{Q}_{\Pi}$ is self-dual

$\mathcal{Q}_{\Pi}^{\star}=\mathcal{Q}_{\Pi}\text{.}$

###### Proof.

It follows from the first inequality in Theorem 16 that
$\mathcal{Q}_{\Pi}\subset\mathcal{Q}_{\Pi}^{\star}$. Let
$p\in\mathcal{Q}_{\Pi}^{\star}$ and $\tau\in\mathcal{L}_{s}(\mathbb{C}^{d})$
be such that $\textnormal{tr}\tau=1$ and $p=p_{\Pi}(\tau)$. Since (3) holds
for all $\rho,\sigma\in\mathcal{L}_{s}(\mathbb{C}^{d})$ such that
$\textnormal{tr}\rho=\textnormal{tr}\sigma=1$, from the definition of duality
we get $\textnormal{tr}(\tau\rho)\geq 0$ for every
$\rho\in\mathcal{S}(\mathbb{C}^{d})$. Hence $\tau\geq 0$ and, in consequence,
$\tau\in\mathcal{S}\left(\mathbb{C}^{d}\right)$ and $p\in\mathcal{Q}_{\Pi}$. ∎

It turns out that some of the geometric properties presented in this section
hold true for every morphophoric POVM. The reasoning behind these observations
is basically the same as in the case of 2-design POVMs, although some
adjustments are necessary. First of all, the balls that sandwich
$\mathcal{Q}_{\Pi}$ need to be centred at
$c_{\Pi}=(\textnormal{tr}\Pi_{1}/d,\ldots,\textnormal{tr}\Pi_{n}/d)$. The
radii of these balls are defined in the same way as previously, note, however,
that the parameter $m$ involved in the definition is strictly related to the
similarity ratio, i.e. $m=\sqrt{\alpha/d}$. Relations (20) now take the
following form:

$B_{\mathcal{A}}(c_{\Pi},r)\subset\mathcal{Q}_{\Pi}\subset
B_{\mathcal{A}}(c_{\Pi},R)\quad\textnormal{and}\quad\mathcal{Q}_{\Pi}^{1}\subset\partial
B_{\mathcal{A}}(c_{\Pi},R).$

We can also define the basis distributions $f_{j}=p_{\Pi}(\rho_{j})$
($j=1,\dots,n$) and the basis polytope
$D=\operatorname*{conv}\left\\{f_{j}:j=1,\ldots,n\right\\}$ setting
$\rho_{j}:=\Pi_{j}/\textnormal{tr}\Pi_{j}$. The states $\rho_{j}$ need not to
be pure this time, and so we do not have an analogue of Fact 3 but we still
have

$D\cup B_{\mathcal{A}}(c_{\Pi},r)\subset\mathcal{Q}_{\Pi}\subset\Delta\cap
B_{\mathcal{A}}(c_{\Pi},R).$

Finally, and most importantly, the appropriate dualities are preserved.
Indeed, for $\iota$ denoting now the inversion in $\mathcal{A}$ through
$c_{\Pi}$ (i.e. $\iota(p):=2c_{\Pi}-p$) we get the analogue of Fact 6 from
which we deduce the following properties:

  1. i.

The polytopes $D$ and $\Delta$ are dual in $\mathcal{A}$ with respect to the
sphere with centre at $c_{\Pi}$ of radius $m=\sqrt{\alpha/d}=\sqrt{rR}$. The
same is true for the balls $B_{\mathcal{A}}(c_{\Pi},r)$ and
$B_{\mathcal{A}}(c_{\Pi},R)$.

  2. ii.

For $p,q\in\mathcal{Q}_{\Pi}$ we have $-m^{2}\leq\langle
p-c_{\Pi},q-c_{\Pi}\rangle\leq R^{2}$.

  3. iii.

The probability range $\mathcal{Q}_{\Pi}$ is self-dual
($\mathcal{Q}_{\Pi}^{\star}=\mathcal{Q}_{\Pi}$) with respect to the sphere
with centre at $c_{\Pi}$ of radius $m$.

##  8 Morphophoric POVM and primal equation

Assume that a morphophoric POVM $\Pi=(\Pi_{j})_{j=1}^{n}$ is given and
$\rho\in\mathcal{S}\left(\mathbb{C}^{d}\right)$ is the state of the system
before the measurement. Then the probability that the result of the
measurement $\Pi$ is $j$ ($j=1,\ldots,n$) equals
$p_{j}^{\Pi}\left(\rho\right)=\operatorname{tr}(\Pi_{j}\rho)$. (In this
section we put the POVM in the superscript and the coordinates in the
subscript.) Assume also that the state of the system after the measurement is
described by the (generalised) Lüders instrument [41, p. 243], i.e. it is
given by
$\rho_{j}\left(\rho\right):=\Pi_{j}^{1/2}\rho\Pi_{j}^{1/2}/p_{j}^{\Pi}\left(\rho\right)$
provided the result was actually $j$ and $p_{j}^{\Pi}\left(\rho\right)\neq 0$.
Let us now consider another POVM $\Xi=(\Xi_{k})_{k=1}^{N}$. Fuchs et al [34]
called the former measurement in the sky and the latter on the ground, and
analysed two paths leading to the results of the second measurement. One can
either perform the measurement ‘on the ground’ directly or proceed first to
the (counterfactual) measurement ‘in the sky’ and subsequently to the
measurement ‘on the ground’. In the former case the probability that the
result of the measurement $\Xi$ is $k$ ($k=1,\ldots,N$) is given by
$p_{k}^{\Xi}\left(\rho\right)=\operatorname{tr}\left(\Xi_{k}\rho\right)$. In
the latter case, the probability that the results of the measurements $\Pi$
and $\Xi$ are subsequently $j$ and $k$ ($j=1,\ldots,n$, $k=1,\ldots,N$) is
given by
$p_{jk}^{\Pi\Xi}\left(\rho\right)=\operatorname{tr}\left(\Xi_{k}\rho_{j}\left(\rho\right)\right)p_{j}^{\Pi}\left(\rho\right)=\operatorname{tr}\left(\Xi_{k}\Pi_{j}^{1/2}\rho\Pi_{j}^{1/2}\right)$
and can be expressed via the conditional probability
$p_{k|j}^{\Xi|\Pi}\left(\rho\right):=\operatorname{tr}\left(\Xi_{k}\rho_{j}\left(\rho\right)\right)$
as
$p_{jk}^{\Pi\Xi}\left(\rho\right)=p_{k|j}^{\Xi|\Pi}\left(\rho\right)p_{j}^{\Pi}\left(\rho\right)$.

The central theme of QBism is answering the question of how the probabilities
$p_{k}^{\Xi}$ depend on $p_{j}^{\Pi}$ and $p_{k|j}^{\Xi|\Pi}$. Classically, if
the process of measurement does not alter the state, such quantities should be
linked by the law of total probability:
$p_{k}^{\Xi}=\sum_{j=1}^{n}p_{j}^{\Pi}p_{k|j}^{\Xi|\Pi}$; however, in the
quantum world the situation is quite different. In the case of rank-$1$ POVMs
the post-measurement states $\rho_{j}$ ($j=1,\ldots,n$) and, in consequence,
the conditional probabilities $p_{k|j}^{\Xi|\Pi}$ do not depend on the pre-
measurement state $\rho$. For SIC-POVMs Fuchs and Schack [30, p. 23] used this
fact to formulate the fundamental equation of QBism (called the ‘primal
equation’ or the ‘Urgleichung’):

$p_{k}^{\Xi}\left(\rho\right)=\sum_{j=1}^{n}p_{k|j}^{\Xi|\Pi}\left(\left(d+1\right)p_{j}^{\Pi}\left(\rho\right)-1/d\right)\text{.}$
(21)

Note, however, that the derivation of such an equation does not require $\Pi$
to be a SIC-POVM; in fact, any morphophoric rank-$1$ equal-trace POVM, i.e. a
$2$-design POVM (see Corollary 9), suits here, resulting in the equation of
the form

$p_{k}^{\Xi}\left(\rho\right)=\sum_{j=1}^{n}p_{k|j}^{\Xi|\Pi}\left(\left(d+1\right)p_{j}^{\Pi}\left(\rho\right)-d/n\right)\text{.}$
(22)

The proof is similar to the SIC-POVM case and follows directly from (8).

For an arbitrary morphophoric POVM $\Pi$ the situation could be much more
complicated, as, in general, the post-measurement states for $\Pi$ depend on
the initial state of the system. To derive an analogue of (22), we have to
introduce some notation. Recall that $c_{\Pi}=p_{\Pi}\left(\rho_{\ast}\right)$
and $c_{\Xi}=p_{\Xi}\left(\rho_{\ast}\right)$, where $\rho_{\ast}:=I/d$ is the
maximally mixed state. Consider the deviation of the probability distribution
of the measurement results for the pre-measurement state $\rho$ from that for
$\rho_{\ast}$ for both POVMs:

$\displaystyle\delta_{\Pi}\left(\rho\right)$
$\displaystyle:=p_{\Pi}\left(\rho\right)-c_{\Pi}\text{,}$
$\displaystyle\delta_{\Xi}\left(\rho\right)$
$\displaystyle:=p_{\Xi}\left(\rho\right)-c_{\Xi}\text{,}$

and the transposed covariance matrix $C$ of two consecutive measurements $\Pi$
and $\Xi$ given by:

$\displaystyle C_{kj}$
$\displaystyle:=p_{jk}^{\Pi\Xi}\left(\rho_{\ast}\right)-p_{j}^{\Pi}\left(\rho_{\ast}\right)p_{k}^{\Xi}\left(\rho_{\ast}\right)=\frac{1}{d}\left(\operatorname{tr}\left(\Xi_{k}\Pi_{j}\right)-\operatorname{tr}\left(\Xi_{k}\right)\operatorname{tr}\left(\Pi_{j}\right)/d\right)$
$\displaystyle=\operatorname{tr}\left(\pi_{0}\left(\Xi_{k}\right)\pi_{0}\left(\Pi_{j}\right)\right)/d$

for $j=1,\ldots,n$ and $k=1,\ldots,N$, where
$\pi_{0}\left(A\right)=A-\left(\operatorname{tr}A\right)\rho_{\ast}$ for
$A\in\mathcal{L}_{s}(\mathbb{C}^{d})$, see (1). Then the following simple
equation, which in the current context replaces (22), is fulfilled (see
Theorem 18):

$\delta_{\Xi}=\frac{d}{\alpha}C\delta_{\Pi}\text{,}$ (23)

where

$\alpha=\frac{d}{\dim(\mathcal{A})}\sum_{j=1}^{n}\left(p_{jj}^{\Pi\Pi}\left(\rho_{\ast}\right)-\left(p_{j}^{\Pi}\left(\rho_{\ast}\right)\right)^{2}\right)=\frac{1}{d^{2}-1}\sum_{j=1}^{n}\left(\operatorname{tr}\left(\Pi_{j}^{2}\right)-\frac{\left(\operatorname{tr}\left(\Pi_{j}\right)\right)^{2}}{d}\right)$
(24)

is the square of the similarity ratio for morphophoric POVM, see (4). Note
that, as in the original ‘Urgleichung’, all the quantities that appear in (23)
can be interpreted in probabilistic terms. In fact, the above equation fully
characterises morphophoric POVMs as the following theorem shows:

######  Theorem 18.

Assume that $\Pi=(\Pi_{j})_{j=1}^{n}$ is a POVM in $\mathbb{C}^{d}$. Then the
following conditions are equivalent:

  1. i.

$\Pi$ is morphophoric;

  2. ii.

for every POVM $\Xi=(\Xi_{k})_{k=1}^{N}$, (23) holds with $\alpha$ given by
(24);

  3. iii.

for some IC-POVM $\Xi=(\Xi_{k})_{k=1}^{N}$ and $\alpha>0$, (23) holds.

###### Proof.

The implication $(ii)\Rightarrow(iii)$ is obvious. We prove
$(iii)\Rightarrow(i)$. Let $k=1,\ldots,N$ and
$\rho\in\mathcal{S}\left(\mathbb{C}^{d}\right)$. Then

$\displaystyle\alpha\operatorname{tr}\left(\pi_{0}\left(\Xi_{k}\right)\left(\rho-\rho_{\ast}\right)\right)$
$\displaystyle=\alpha\operatorname{tr}\left(\Xi_{k}\left(\rho-\rho_{\ast}\right)\right)$
$\displaystyle=\alpha(\delta_{\Xi}\left(\rho\right))_{k}$
$\displaystyle=\sum_{j=1}^{n}\operatorname{tr}\left(\pi_{0}\left(\Xi_{k}\right)\pi_{0}\left(\Pi_{j}\right)\right)\left(p^{\Pi}_{j}\left(\rho\right)-p^{\Pi}_{j}\left(\rho_{\ast}\right)\right)$
$\displaystyle=\sum_{j=1}^{n}\operatorname{tr}\left(\pi_{0}\left(\Xi_{k}\right)\pi_{0}\left(\Pi_{j}\right)\right)\operatorname{tr}\left(\pi_{0}\left(\Pi_{j}\right)\left(\rho-\rho_{\ast}\right)\right)\text{.}$

The set
$\left\\{\rho-\rho_{\ast}:\rho\in\mathcal{S}\left(\mathbb{C}^{d}\right)\right\\}$
generates $\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$, and, as $\Xi$ is
informationally complete, the same is true for
$\left\\{\pi_{0}\left(\Xi_{k}\right):k=1,\ldots,N\right\\}$, see Proposition
2. In consequence,

$\alpha\operatorname{tr}\left(AB\right)=\sum_{j=1}^{n}\operatorname{tr}\left(A\pi_{0}\left(\Pi_{j}\right)\right)\operatorname{tr}\left(\pi_{0}\left(\Pi_{j}\right)B\right)$
(25)

for every $A,B\in\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$. Now, from Theorem 3 we
deduce that $\pi_{0}\left(\Pi\right)$ is a tight frame in
$\mathcal{L}_{s}^{0}(\mathbb{C}^{d})$, and so, by Theorem 4, we get $(i)$.
Finally, in the same manner from $(i)$ we get (25) with $\alpha$ given by
(24). Let us consider an arbitrary (not necessarily IC) POVM
$\Xi=(\Xi_{k})_{k=1}^{N}$. Putting $A:=\pi_{0}\left(\Xi_{k}\right)$
($k=1,\ldots,N$) and $B:=\rho-\rho_{\ast}$
($\rho\in\mathcal{S}\left(\mathbb{C}^{d}\right)$), we can derive (23) in the
same way as above. ∎

It is an easy computation to show that for rank-$1$ equal-trace POVMs (22) and
(23) are equivalent. Hence, and from Corollary 9, we get the following result:

######  Corollary 19.

Assume that $\Pi=(\Pi_{j})_{j=1}^{n}$ is a rank-$1$ equal-trace POVM in
$\mathbb{C}^{d}$. Then the following conditions are equivalent:

  1. i.

$\Pi$ is morphophoric;

  2. ii.

$\Pi$ is a $2$-design POVM;

  3. iii.

for every POVM $\Xi=(\Xi_{k})_{k=1}^{N}$ (22) holds;

  4. iv.

for some IC-POVM $\Xi=(\Xi_{k})_{k=1}^{N}$ and $\alpha>0$ (22) holds.

Note that (23) follows from the morphophoricity of the measurement rather than
from its quantumness. Considering a classical fuzzy measurement of a classical
system that is morphophoric, we get exactly the same equation.

##  9 Conclusions

We believe that we have correctly identified and characterised morphophoricity
as the property of a quantum measurement that enables the extension of QBism
formalism to the broadest possible class of measurements. In case of rank-$1$
equal-trace morphophoric POVMs, i.e. $2$-design POVMs, both algebraic and
geometric properties of the resulting generalised qplexes are similar to the
SIC-POVM case. In the general case the situation is more complex. However,
even then we can recover some essential geometrical features of qplexes as
well as find a surprising generalisation of the quantum total law of
probability. Thus, we hope that our paper may become the first step towards an
abstract definition of a generalised qplex.

Finally, we would like to mention that figuring out how our qplex is located
inside the overall simplex may provide a way to solve the problems of
minimising and maximising the Shannon entropy of morphophoric POVMs. This in
turn leads to the understanding of the lower and upper bounds for the
uncertainty of the measurement results [10, 22, 23, 59, 74, 71, 76, 75].

## Acknowledgements

We thank Anna Szczepanek for numerous remarks that improve the readability of
the paper. AS is supported by Grant No. 2016/21/D/ST1/02414 of the Polish
National Science Centre. WS is supported by Grant No. 2015/18/A/ST2/00274 of
the Polish National Science Centre.

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