From Quantum Tomography to Phase
Retrieval and Back
Michael Kech
Michael Wolf
Mathematical Physics
Department of Mathematics
Technische Universität München
Garching, Germany 80804
Email: kech@ma.tum.de
Mathematical Physics
Department of Mathematics
Technische Universität München
Garching, Germany 80804
Email: wolf@ma.tum.de
Abstract—This letter is devoted to the crossfertilization between the fields of phase retrieval and
quantum tomography. In the first part we discuss
topological aspects of quantum tomography which
turn out to imply lower bounds on the number
of frame vectors necessary to reconstruct signals
modulo phase. In the second part we generalize the
approach of Balan, Casazza and Edidin to construct
frames for phase retrieval to the context of quantum
tomography. In this process we obtain an extension
of their result to Parseval frames.
I. I NTRODUCTION
Quantum tomography, the task of reconstructing
a quantum state from the outcomes of an experiment, is one of the cornerstones of quantum
information science. Already for small systems it
is an extensive task [1] and with the degrees of
freedom growing exponentially in the system size,
this complexity becomes even more relevant when
considering bigger systems.
However, there is a way to at least partly circumvent this problem. In many cases not the whole state
space is of interest, but just some subset of it, e.g.
one might know that the state under investigation
is a pure state or reveals a particular symmetry (energy, particle number). This prior information can
significantly reduce the dimension of the relevant
set of states as compared to the set of all states.
The idea is to use this fact to reduce the number of
measurements that need to be performed [2], [3].
In this letter we present some of the main results
of [2],[4] and [5]. We especially focus on their
implications for the phase retrieval problem.
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
A. Quantum Tomography
Let us begin by introducing the mathematical
framework of quantum tomography.
Denote by h·, ·i the standard inner product on Cn
and by k · k2 the 2-norm. The real vector space of
hermitian operators on Cn is denoted by H(Cn )
and k · k∞ denotes the operator norm on H(Cn ).
Definition I.1. (Quantum State) A quantum state
on Cn is a hermitan operator ρ ∈ H(Cn ) with
ρ ≥ 0 and tr(ρ) = 1. S(Cn ) denotes the set of all
quantum states on Cn .
A physical system is completely characterized
by its quantum state and the goal of quantum
tomography is to reconstruct the quantum state of
a given system from the statistics of the outcomes
of a measurement.
In quantum mechanics measurements are generally modelled by Positive Operator Valued Measures (POVMs)[6]. For our purposes we use the
following notion of POVM.
Definition I.2. (POVM) An (m − 1)-dimensional
POVM on Cn is a multiset {P1 , ..., Pm } of m
positive
semidefinite operators on Cn such that
Pm
j=1 Pj = 1. The elements of a POVM are called
effect operators.
A POVM P := {P1 , ..., Pm } induces a linear
map
hP : H(Cn ) → Rm
ρ 7→ tr(P1 ρ), ..., tr(Pm ρ) .
Here, tr(Pi ρ) is the probability that the event i
occurs if the quantum system is in state ρ.
Note that S(Cn ) is a set of dimension n2 −1 and
hence, having no prior information, a POVM that
allows for a unique identification of an unknown
quantum state has to contain at least n2 effect
operators. However, the necessary number of effect
operators can reduce significantly if the quantum
state under investigation is not an arbitrary state but
is known to lie on a constrained subset of S(Cn ).
Definition I.3. Let R ⊆ S(H) be a subset. A
POVM P is called R-complete if hP |R is injective.
Essentially, this letter deals with the following
question: Given a subset R ⊆ S(Cn ), what is the
least dimension d(R) of an R-complete POVM?
We subdivide this question into two problems
which require technically very different methods.
First, in Section II, we focus on the problem of
finding lower bounds on d(R). The results we
obtain in this section give lower bounds on the
number of frame vectors necessary to identify a
signal modulo phase and they also apply to matrix
completion. Secondly, in Section III, we deal with
the problem of finding R-complete POVMs. Essentially, we take a more general but very similar
approach to [7] which in particular allows us to
extend their result to Parseval frames.
B. Phase Retrieval and Pure State Tomography
Rank one POVMs establish the relation of frames
to POVMs.
Definition I.4. (Rank One POVM) A POVM is rank
one if all effect operators have rank one.
Definition I.5. (Parseval Frames) A multiset
n
{v
P1m, ..., vm } 2⊆ C 2is called a Parseval frame if
i=1 hx, vi i = kxk2 .
For a Parseval frame {v1 , ..., vm } we find v1 v1† +
†
...+vm vm
= 1, i.e. we can associate in this way to
each Parseval frame F a POVM PF . Conversely for
each rank one POVM P there is a Parseval frame
F such that P = PF . In this sense a POVM is a
generalization of a Parseval frame.
A frame F := {v1 , ..., vm } induces a map MF :
Cn /∼ → Rm , [x] → (|hx, v1 i|2 , ..., |hx, vm i|2 )
where x ∼ y iff x = eiφ y for some φ ∈ R.
The connection between the phase retrieval problem and quantum tomography is established by the
set of pure quantum states, i.e. the set of quantum
states ρ ∈ S(Cn ) for which ρ2 = ρ. We denote the
set of pure quantum states on Cn by S1 (Cn ). Note
that S1 (Cn ) can be identified with P Cn−1 via the
map φ : P Cn−1 → S1 (Cn ), [v] 7→ vv † .
Lemma I.6. Let P be a POVM. P is S1 (Cn )
complete if and only if it is a S1 (Cn )-embedding.
This follows directly from Theorem 5 and
Lemma 1 of [2].
Lemma I.7. Let F be a Parseval frame. The
induced map MF is injective if and only if the associated rank one POVM PF is a S1 (Cn )-embedding.
Proof. First note that MF (x) = hPF ◦φ(x) for x ∈
P Cn−1 . Since P Cn−1 ⊆ Cn /∼, we conclude from
Lemma I.6 that hPF |S1 (Cn ) is a smooth embedding
if MF is injective. For the converse note that 1 ∈
spanR (PF ).
Remark . Note that in fact to every frame F :=
{v1 , ..., vm } for which MF is injective we can
associate a S1 (Cn )-embedding P , namely P :=
†
†
{1 − λ(v1 v1† + ... + vm vm
), λv1 v1† , ..., λvm vm
} for
+
λ ∈ R small enough.
II. L OWER B OUNDS
A. Immersions and Stability
Note that the induced map hP of a POVM P is
continuous. In fact, given some subset R ⊆ S(Cn ),
hP |R is continuous if we equip R with the subspace topology.
Thus, considering R as a topological space in its
own right, the smallest natural number k such that
there exists an injective and continuous map ψ :
R → Rk gives a lower bound on d(R). However, in
general there is very little know about the number
k if ψ is merely continuous. To our knowledge,
the only known methods to find lower bounds on k
exclusively rely on ψ to be an immersion, see e.g.
[8], [9], [10].
Definition II.1. (Immersion, Embedding) Let M, N
be smooth manifolds. A smooth mapping ψ : M →
N is an immersion if dψx is injective for all x ∈ M .
ψ is an embedding if ψ is both an immersion and
a homeomorphism onto its image.
Definition II.2. The immersion (embedding) dimension of a smooth manifold M is the smallest
number k such that there exists an immersion
(embedding) ψ : M → Rk .
The power of immersion theory comes from the
fact that the problem of finding the immersion
dimension of a smooth manifold can be related to
the theory of vector bundles in algebraic topology.
As a consequence the immersion dimension of a
smooth manifold can be bounded by its topological
invariants. Let us make two important remarks:
1) Note that an immersion need not be injective
and thus it is not clear how the immersion dimension is related to the number k.
Although there is very little known about
the relation between the immersion and the
embedding dimension of a smooth manifold
[11], in many cases the immersion dimension
is close to the embedding dimension and in
the scenarios we analyse this actually turns
out to be true.
2) Immersion theory requires a smooth structure, so we have to restrict to smooth submanifolds P ⊆ S(Cn ). Furthermore, given
a submanifold P ⊆ S(Cn ) and a POVM P ,
if the associated map hP |P is injective hP |P
need not be an immersion. However, we will
see in the following that requiring hP |P to
be an injective immersion is equivalent to
a natural stability property and thus, under
the premise of stability, the lower bounds
we obtain from immersion theory apply in
general.
Definition II.3. Let P ⊆ S(Cn ) be a smooth
submanifold. A POVM P is called P-embedding
if hP |P is a smooth embedding.
Lemma II.4. Let P ⊆ S(Cn ) be a smooth and
closed submanifold and let P be a POVM. hP |P
is an injective immersion if and only if P is Pembedding.
Proof. Note that P is compact as a closed subset
of the compact set S(Cn ). Using the standard fact
that a continuous and injective map ψ : M → N
between topological spaces N, M is a homeomorphism if M is compact, we conclude that hP |P is
a homomorphism onto its image.
Let us now state the main result of this section
which justifies using immersion theory as a tool to
find lower bounds on d(P) for smooth and closed
submanifolds P ⊆ S(Cn ).
Theorem II.5. (Stability) Let P ⊆ S(H) be a
closed submanifold and let P be a P-complete mdimensional POVM with associated linear map hP .
P is a P-embedding if and only if there is an
> 0 such that every m-dimensional POVM Q
with khP − hQ k∞ < is P-complete.
The proof of this result can be found in [4]. It is
a more general version of a similar result obtained
in [12] 1 .
B. Lower Bounds for Phase Retrieval
By Lemma I.6 a S1 (Cn )-complete POVM P is
a S1 (Cn )-embedding and hence hP ◦ φ gives an
embedding of P Cn−1 in Euclidean space. Thus,
lower bounds on the immersion dimension of complex projective space immediately transfer to lower
bounds on the dimension of S1 (Cn )-embeddings
and by the remark after Lemma I.7 also to lower
bounds on the number of frame vectors necessary
to identify signals modulo phase.
Bounds on the immersion dimension of projective space in Euclidean space were thoroughly
studied in the mathematical literature. Here we state
the lower bounds obtained in [9] directly applied to
the phase retrieval problem.
Theorem II.6. [9] Let F := {v1 , ..., vm } be a
frame. If MF is injective, then


4n − 4 − 2α(n) + 1, ∀n
m > 4n − 4 − 2α(n) + 2, n odd, α(n) = 2mod4


4n − 4 − 2α(n) + 3, n odd, α(n) = 3mod4,
where α(n) denotes the number of 1‘s in the binary
expansion of n − 1.
C. Lower Bounds for Matrix Recovery
Denote by Hr (Cn ) := {X ∈ H(n) : rank(X) ≤
r} the hermitian matrices of rank at most r. In
this section we give bounds on the dimension of a
POVM P for which hP |Hr (Cn ) is injective 2 .
Note that since Hr (Cn ) is not a smooth manifold
our approach does not apply directly. To circumvent
this problem we first identify a smooth submanifold
P ⊆ S(Cn ) which is contained in Hr (Cn ). Let P
be a POVM such that hP |Hr (Cn ) is injective. If P is
stable with respect to Hr (Cn ) in a sense similar to
Theorem II.5, it is certainly stably P-complete and
thus a P-embedding by Theorem II.5. This implies
that lower bounds on the immersion dimension of
P also give lower bounds on the dimension of P .
1 The
connection is made by Lemma I.6.
matrix recovery, one typically considers sets of operators
instead of POVMs. Thus, it might be worth noting that Proposition II.9 gives the same bounds if we deal with sets of hermitian
operators instead of POVMs.
2 In
Let us now construct the submanifold P. For
r ∈ {1, ..., n}, define the rank r matrix Dr ∈
2
diag(1, 2, ..., r, 0, ...).
Hr (Cn ) by Dr := r(r+1)
Furthermore let S(Dr ) := {U Dr U † : U ∈ U (n)},
where U (n) denotes the unitary group. Note that
S(Dr ) ⊆ Hr (Cn ).
Lemma II.7. S(Dr ) is diffeomorphic to the complex flag manifold U (n)/(U (1)r × U (n − r))
Proof. Note that S(Dr ) is the orbit of Dr with
respect to the action by conjugation of U (n) on
H(Cn ). The isotropy group of D(s) under this
action is U (1)r ×U (n−r). Factoring the orbit map
over this isotropy group yields a diffeomorphism
(Theorem 3.62 of [13])
U (n)/(U (1)r × U (n − r)) ' S(Cn )s .
Ssn
can be identified with a complex flag
Thus,
manifold and the lower bounds on the immersion
dimension of complex flag manifolds obtained in
[14] directly convert to lower bounds on d(Ssn ).
To state the result of [14] we need the following
functions.
DefinitionP
II.8. Let n ∈ N, k ∈ {0, 1, ..., n}.
n−1
α1 (n) := i=0 α(i),
β(n, k) := α1 (n) − α1 (k) − α1 (n − k),
Let {n1 , ..., nk } be a partition of n.PLet K be
some subset of {1, ..., k} and set d := i∈K ni .
Proposition II.9. [14] The complex flag manifold
U (n)/U (n1 ) × ... × U (nk ) cannot be immersed
in Euclidean Space of dimension 4d(n − d) −
2β(n, d) − 1 and it cannot be embedded in Euclidean space of dimension 4d(n − d) − 2β(n, d).
To be able to estimate the quality of these lower
bounds let us also give the following result about
the upper bounds.
Proposition II.10. For 1 ≤ r ≤ n/2, there exists
a POVM P of dimension 4r(n − r) − 1 such that
hP |Hr (Cn ) is injective.
This proof of this result can be found in [2].
In table I, the lower bounds of Proposition II.9
are compared to the upper bounds of Proposition
II.10.
l\r
2
3
5
24/34;39
6
28/40;47
7
32/50;55
51/76;83
8
36/60;63
57/90;95
9
40/66;71
63/98;107
10
44/72;79
4
88/134;143
69/110;119
96/148;159
TABLE I
D IMENSION OF Hr (Cl+r )/ L OWER BOUNDS ON EMBEDDING
DIMENSION FOR U (l + r)/U (l) × U (1)r (P ROPOSITION
II.9); U PPER BOUND ON DIMENSION OF POVM P FOR
WHICH hP |H (Cl+r ) IS INJECTIVE (P ROPOSITION II.10).
r
III. U PPER B OUNDS AND R ESTRICTED
M EASUREMENT S CHEMES
In this section we give a method to deal with the
problem of finding POVMs that are complete with
respect to a given subset R ⊆ S(Cn ). Let us make
a short remark at this point: Since an arbitrary measurement might not be feasible for experimental
implementation, one should restrict to the subset of
implementable measurement schemes. For example
this could be the restriction to von Neumann measurements or to local measurements when dealing
with a multipartite system. The method given here
can deal with constrained measurement schemes.
The method is inspired by the approach taken in
[7] to find frames for the phase retrieval problem.
Similar to [7], it relies on the following observation:
A POVM P := {Q1 , ..., Qm } is R-complete with
respect to a subset R ⊆ S(Cn ) if the equations
tr(Qi x) = 0,
i ∈ {1, ..., m},
(1)
have no solution for x ∈ ∆(R) − {0}, where
∆(R) := {x − y : x, y ∈ R}.
For a given subset R ⊆ S(Cn ), we want to
characterize non-injective POVMs via the equations
(1) and use the dimension theory of semi-algebraic
geometry to show that these have zero measure.
Therefore, the sets of POVMs are assumed to be
real semi-algebraic sets M containing POVMs of
a fixed dimension m. An example is given by
the restriction to the set of m-dimensional rank
one POVMs. Furthermore, in order to ensure that
the equations (1) in fact become real algebraic
equations, we have to replace ∆(R) − {0} by a
suitable semi-algebraic set. We do this by constructing a real semi-algebraic set 0 ∈
/ D ⊆ H(Cn ) 3
with the following property: If there is a POVM
P := {Q1 , ..., Qm } ∈ M and an X ∈ ∆(R) − {0}
with
tr(Qi X) = 0
, i ∈ {1, ..., m},
then there is X 0 ∈ D with
tr(Qi X 0 ) = 0
, i ∈ {1, ..., m}.
(2)
If a semi-algebraic set 0 ∈
/ D ⊆ H(Cn ) has this
property, we say that D represents ∆(R) − {0}.
The solutions of the equations (2) characterize
the non-injective POVMs: Let M̃ be the semialgebraic set obtained from M × D by imposing the equations (2). By construction, the noninjective POVMs are contained in the projection
of M̃ ⊆ M × D on the first factor with the
canonical projection π1 : M × D → M. But if
dim M̃ < dim M, we find dim π1 (M̃) < dim M
4
and thus the non-injective measurements have
measure zero in M.
In order for this approach to be efficient, we need
to guarantee that the equations (2) are independent.
In this case dim M̃ < dim M is equivalent to
m > dim D and thus the quality of our result is
determined by how low-dimensional we can choose
the semi-algebraic set D.
Deploying this procedure, we can prove the
following result.
Theorem III.1. Let R ⊆ S(Cn ) be a subset and let
D be a semi-algebraic set that represents ∆(R) −
{0}. If m > dim D, almost all m-dimensional rank
one POVMs are R-embeddings.
This is a special case of a more general theorem,
which will appear in [5]. Theorem III.1 implies the
following statement.
Corollary III.2. Let m ≥ 4n − 4. For almost
all Parseval frames F := {v1 , ..., vm } on Cn the
induced map MF is injective.
This corollary generalizes the main result of [16]
to Parseval frames. Let us sketch its proof here,
more details will appear in [5].
3 Here we identify H(Cn ) with n2 -dimensional real affine
space.
4 π maps semi-algebraic sets semi-algebraic sets and does
1
not increase the dimension. See Theorem 2.2.1 and Proposition
2.8.6 of [15].
Proof. Note that ∆(S1 (Cn )) ⊆ H2 (Cn ). To construct a set D that represents ∆(S1 (Cn )) − {0}
one can additionally impose the equations tr(X) =
0, tr(X 2 ) = 1, X ∈ H2 (Cn ). The first one
considers that states have unit trace and the second
one considers that the equations (2) are invariant
under X → λX for λ ∈ R. Both of these equations
do reduce the dimension of H2 (Cn ) and we find
dim D = dim H2 (Cn )−2 = 2(2n−2)−2 = 2n−6.
Using Theorem III.1 we get d(S1 (Cn )) > 2n − 6.
Finally, Lemma I.7 concludes the proof.
ACKNOWLEDGMENT
We thank Teiko Heinosaari and Péter Vrana for
discussions and their contribution to results this
letter is based on.
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