Pure state ‘really’ informationally complete with
rank-1 POVM
Yu Wang1,2 , Yun Shang1,3
1

arXiv:1711.07585v1 [quant-ph] 21 Nov 2017

2

Institute of Mathematics, AMSS, CAS, Beijing, 100190, China,
University of Chinese Academy of Sciences, Beijing, 100049, China
3
NCMIS, AMSS, CAS, Beijing, 100190, China.
shangyun602@163.com

Abstract. What is the minimal number of elements in a rank-1 positiveoperator-valued measure (POVM) which can uniquely determine any
pure state in d-dimensional Hilbert space Hd ? The known result is that
the number is no less than 3d − 2. We show that this lower bound is not
tight except for d = 2 or 4. Then we give an upper bound of 4d−3. For d =
2, many rank-1 POVMs with four elements can determine any pure states
in H2 . For d = 3, we show eight is the minimal number by construction.
For d = 4, the minimal number is in the set of {10, 11, 12, 13}. We show
that if this number is greater than 10, an unsettled open problem can be
solved that three orthonormal bases can not distinguish all pure states
in H4 . For any dimension d, we construct d + 2k − 2 adaptive rank-1
positive operators for the reconstruction of any unknown pure state in
Hd , where 1 ≤ k ≤ d.
Key words: Quantum state tomography, Pure state, Quantum measurement, Rank-1 operators.

1 Introduction
One of the central problems in quantum science and technology is the estimation
of an unknown quantum state, via the measurements on a large number of
copies of this state. Quantum state tomography is the process of determining an
arbitrary unknown quantum state with appropriate measurement strategies.
A quantum state ρ in d-dimensional Hilbert space Hd is described by a density matrix, namely by a positive semi-definite, unit-trace d × d matrix as Sd . A
generalized measurement can be described by a positive operator-valued measure (POVM) [1]. The POVM elements, Ek , satisfy the completeness condition:
P
k Ek = I. Performing this measurement on a system in state ρ, the probability
of the k-th outcome is given from the Born rule, pk = tr(ρEk ). If the statistics
of the outcome probabilities are sufficient to uniquely determine the state, the
POVM is regarded as informationally complete (IC) [2].
The IC-POVM can give a unique identification of an unknown state, which
should distinguish any pair of different states from the statistics of probabilities.
For example, we consider a POVM, { 41 (|0i±|1i)(h0|±h1|), 41 (|0i±i|1i)(h0|∓ih1|)}.

2

It is not an IC-POVM, as the statistics of the outcome probabilities for states
{|0i, |1i} under this measurement are the same. For any different quantum states
ρ1 , ρ2 ∈ Sd , an IC-POVM should distinguish them from the statistics of the
outcome probabilities. That is to say, we have tr(ρ1 Ek ) 6= tr(ρ2 Ek ) for some
elements Ek .
We know that a quantum state ρ in Hd is specified by d2 − 1 real parameters.
The number is reduced by one because tr(ρ) = 1. Caves et al. constructed an
IC-POVM which contains the minimal d2 rank-1 elements [3], i.e., multiples of
projectors onto pure states. If d + 1 mutually unbiased bases (MUBs) exist in
Hd , we can construct an IC-POVM with d(d + 1) elements [4]. MUBs have the
property that all inner products between projectors of different bases labeled by
i and j are equal to 1/d. Another related topic is the symmetric informationally
complete positive operator-valued measure (SIC-POVM) [5]. It is comprised of
d2 rank-1 operators. The inner products of all different operators are equal. This
SIC-POVM appears to exist in many dimensions.
For a state ρ in n-qubit system, d = 2n . Thus the cost of measurement resource with these measurement strategies grows exponentially with the increase
of number n. It is important to design schemes with lower outcomes to uniquely
determine the state. This is possible when we consider a priori information about
the states to be characterized.
Denote the rank of a density matrix for state ρ as k, 1 ≤ k ≤ d. And make
a decomposition that Sd = ⊕dk=1 Sd,k , where Sd,k is the set of all the density
matrices with rank k. When k = 1, the state in Sd,1 is pure. A pure state is
specified by d complex numbers, which correspond to 2d real numbers. For the
reason of normalization condition and freedom of a global phase, there are 2d − 2
independent real numbers totally.
Flammia, Silberfarb, and Caves [6] showed that any POVM with less than
2d elements can not distinguish all pair of different states ρ1 , ρ2 in Sd,1 , not
even in a subset S̃d,1 , where Sd,1 \ S̃d,1 is a set of measure zero. They gave a
definition of pure-state informationally complete (PSI-complete) POVM, whose
outcome probabilities are sufficient to determine any pure states (up to a global
phase), except for a set of pure states that is dense only on a set of measure zero.
That is to say, if a pure state was selected at random, then with probability 1 it
would be located in S̃d,1 and be uniquely identified. A PSI-complete POVM with
2d elements is constructed, but not all the elements in this POVM are rank-1.
They constructed another PSI-complete POVM with 3d − 2 rank-1 elements and
conjectured that there exists a rank-1 PSI-complete POVM with 2d elements.
Finkelstein proved this by a precise construction [7]. Moreover, he gave a
strengthened definition of PSIR-completeness, which indicates that all pure
states are uniquely determined. For any pair of different pure states ρ1 , ρ2 ∈ Sd,1 ,
a PSIR-complete POVM should distinguish them. He showed that a rank-1
PSIR-complete POVM must have at least 3d−2 elements and wondered whether
we could reach the lower bound of 3d − 2.
There are a series of studies on the relevant topic. For any pair of different
states ρ1 , ρ2 ∈ Sd,1 , Heinosaari, Mazzarella, and Wolf gave the minimal number

3

of POVM elements to identify them [8]. The number is 4d − 3 − c(d)α(d), where
c(d) ∈ [1, 2] and α(d) is the number of ones appearing in the binary expansion of
d − 1; the results in papers [20, 21, 11] showed that four orthonormal bases, corresponding to four projective measurements, can distinguish all pure states. For
any pair of different states ρ1 ∈ Sd,1 , ρ2 ∈ Sd , Chen et al. showed that a POVM
must contain at least 5d − 7 elements to distinguish them [12]; Carmeli et al.
gave five orthonormal bases that are enough to distinguish them[13]. For a state
in Sd,k , it can be reconstructed with a high probability with rd log2 (d) outcomes
via compressed sensing techniques [14]. Goyeneche et al. [15] constructed five
orthonormal bases to determine all the coefficients of any unknown input pure
states. The first basis is fixed and used to determine a subset sd,1 ⊂ Sd,1 , where
the pure state belongs to. The other four bases are used to uniquely determine
all the states in sd,1 .
In this paper, we consider the pure-state version of informational completeness with rank-1 POVM. Firstly, we show that the lower bound of 3d − 2 is not
tight in most of the cases. It can be reached when d = 2 and possibly be reached
when d = 4. Then we show a result that there exist a large number of rank-1
PSIR-complete POVMs with 4d − 3 elements. Secondly, we make a discussion
about the rank-1 PSIR-complete POVMs when d = 2, 3, 4. For dimension d = 2
and d = 3, we construct the rank-1 PSIR-complete POVMs with the minimal
number of elements, which are 4 and 8 correspondingly. All the coefficients of
an unknown pure state in H2 and H3 can be calculated by these POVMs. For
dimension d = 4, the minimal number is in the range of {10, 11, 12, 13}. If it is
bigger than 10, an answer can be given to a related unsolved problem, i.e., three
orthonormal bases can not distinguish all pure states in H4 . Lastly, we construct
d + 2k − 2 rank-1 positive self-adjoint operators for the tomography of any input
pure states in Hd , here 1 ≤ k ≤ d. This is an adaptive strategy. For any input
pure state, we use d operators to determine a subset sd,1 ⊂ Sd,1 , where the pure
state belongs to. Together with the other 2k − 2 operators, we can uniquely determine all the pure states in sd,1 . Thus using this adaptive method, any input
pure states can be determined with at most 3d − 2 rank-1 operators.

2 The upper and lower bounds
In this section, we will give the upper and lower bounds of the minimal number
of elements in a rank-1 PSIR-complete POVM. Denote this minimal number as
m1 (d). It is in the range of [4d − 3 − c(d)α(d), 4d − 3].
2.1 Feasibility of 3d-2 for PSIR-complete
In this part, we show that a rank-1 PSIR-complete POVM with 3d − 2 elements
possibly exists when dimension d = 2 or 4. For the other dimensions, any rank-1
POVM with 3d − 2 elements cannot be PSIR-complete. Firstly, we introduce the
concept of PSIR-complete.

4

Definition 1 : (PSI really-completeness [7]). A pure-state informationally really complete POVM on a d-dimensional quantum system Hd is a POVM whose
outcome probabilities are sufficient to uniquely determine any pure state (up to
a global phase).
As we introduced above, the PSIR-complete POVM can distinguish any pair
of different states ρ1 , ρ2 ∈ Sd,1 . Neglecting the restriction of rank-1, we denote
m0 (d) to be the minimal number of elements in a PSIR-complete POVM. Certainly, a rank-1 PSIR-complete POVM is PSIR-complete. Thus m1 (d) ≥ m0 (d).
From the result in [8], m0 (d) = 4d − 3 − c(d)α(d), where c(d) ∈ [1, 2] and α(d)
is the number of ones appearing in the binary expansion of d − 1. From the conclusion by Finkelstein, m1 (d) ≥ 3d − 2. But it is not clear when they are equal
or whether a greater number than 3d − 2 might be required. Now we compare
the size of m0 (d) and 3d − 2.
Let f (d) = 4d − 3 − c(d)α(d) − (3d − 2). By the definition of α(d), we have
log d ≥ α(d). So f (d) > d − 1 − 2 log d. Define g(d) ≡ d − 1 − 2 log d. Then
g 0 (d) = 1 − 2/d. If d > 2, it holds that g 0 (d) > 0. And when d = 8, g(8) = 1 > 0.
So when d ∈ [8, +∞), m0 (d) > 3d − 2. When d ∈ [2, 7], the true value of m0 (d)
is given in [8]. We have m0 (2, 3, 4, 5, 6, 7) = (4, 8, 10, 16, 18, 23). We compare this
with the value of 3d − 2: (4, 7, 10, 13, 16, 19). As a result, only when d = 2 or 4,
m0 (d) can be 3d − 2. For the other dimensions, m1 (d) ≥ m0 (d) > 3d − 2.
2.2 The upper bound of 4d − 3
In this section, we show that 4d − 3 is the upper bound of m1 (d). This upper bound is given by constructing rank-1 POVMs from the minimal sets of
orthonormal bases which can determine all pure states in Hd .
Definition 2 : Let B0 = {|φk0 i},· · · ,Bm−1 = {|φkm−1 i} be m orthonormal bases
of Hd , k = 0, · · · , d − 1. For different pure states ρ1 , ρ2 ∈ Hd , they are distinguishable if
tr(ρ1 |φkj ihφkj |) 6= tr(ρ2 |φkj ihφkj |)
(1)
for some |φkj i. If any pair of different pure states is distinguishable by B0 , · · · , Bm−1 ,
the bases {Bj } can distinguish all pure states [11].
Obviously m bases correspond to m · d rank-1 projections. Ejk = |φkj ihφkj |,
Pd−1
j = 0, · · · , m − 1, k = 0, · · · , d − 1. Since k=0 Ejk = I, we have tr(ρI) = 1 for
all pure state ρ. One projection for each basis can be left out as the probability
can be expressed by others. Thus m(d − 1) rank-1 self-adjoint operators can
distinguish all pure states. Can these operators be transformed to a rank-1 PSIRcomplete POVM?
From the proposition 3 in paper [8], we know that m(d − 1) self adjoint
operators can be used to construct a POVM with m(d − 1) + 1 elements.
Akj ≡ ( 21 I + 12 kEjk k−1 Ejk )/[m(d − 1)]. j = 0, · · · , m − 1, k = 0, · · · , d − 2. Then
P
O ≤ Akj ≤ I/m(d − 1) and by setting the new element A ≡ I − j,k Akj we get
a new POVM. This POVM have the same power with the self-adjoint operators
{Ek }, as there exists a bijection between the outcome probabilities of both sides.

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But not all of the elements are rank-1. The following conversion can keep the
elements of transformed POVM to be rank-1.
Rank-1 conversion: Given n rank-1 positive self-adjoint operators {Ek :
Pd
k = 1, · · · , n}, G =
by {Fk : k =
k=1 Ek > 0, a rank-1 POVM denoted
Pn
1, · · · , n} can be constructed. Fk = G−1/2 Ek G−1/2 and k=1 Fk = I.
From the discussion in [6, 7] , if positive operators {Ek } are informationally complete with respect to generic pure states (a set of measure zero can
be neglected), and they can determine all (normalized and unnormalized) pure
states in this set, {Fk } is a PSI-complete POVM. Furthermore, if positive operators {Ek } are informationally complete with respect to all pure states, can the
converted POVM {Fk } be PSIR-complete? Here we give a sufficient condition.
Theorem 1. Let {Ek } be a set of rank-1 positive self-adjoint operators, whose
outcome probabilities are sufficient to uniquely determine all pure states (up to
a global phase). Some of the elements satisfy the following condition:
X
Ek = I.
(2)
k∈B

After the rank-1 conversion, we will get a rank-1 PSIR-complete POVM {Fk }.
Proof. Here we prove that any pair of different pure states is distinguishable by
this POVM.
Let ρ1 and ρ2 be an arbitraryPpair of different pure states. Define qi =
−1/2
tr(G−1 ρi ) for i = 1, 2. As G = I + k∈B
/ Ek , we have det(G) 6= 0. So G, G
−1
−1
and G are of full rank. And qi = tr(G ρi ) 6= 0.
Then define another pair of pure states σi = G−1/2 ρi G−1/2 /qi , i = 1, 2. For
any k,
tr(Fk ρi ) = qi tr(Ek σi ).
(3)
When pure states σ1 and σ2 are the same, as ρ1 6= ρ2 , the number q1 should
not be equal to q2 . Thus tr(Fk ρ1 ) 6= tr(Fk ρ2 ) for any k.
When pure states σ1 and σ2 are different, by the assumption of {Ek }, there
exists some Ek satisfyingPtr(Ek σ1 ) 6= tr(EP
k σ2 ). If q1 = q2 , then tr(Fk ρ1 ) 6=
tr(Fk ρ2 ). If not,
we
have
tr(E
σ
)
=
σ2 ) = 1. As we have the
k
1
k∈B
k∈B tr(EkP
P
P
assumption k∈B tr(Ek ) = I. Then k∈B tr(Fk σ1 ) 6= k∈B tr(Fk σ2 ). Thus it
can also be deduced that tr(Fk ρ1 ) 6= tr(Fk ρ2 ) for some k ∈ B.
So the POVM {Fk } can distinguish the different pure states ρ1 , ρ2 ∈ Sd,1 .
This indicates that given a set of outcome probabilities {pk }, there is a unique
pure state ρ such that pk = tr(ρFk ) for all k. For any other different pure state
σ, we can always get tr(σFk ) 6= pk for some k. Thus {Fk } is enough to uniquely
determine any pure states from the other different pure states. With the prior
knowledge that the state is pure, it can be uniquely determined.
Remark: In this proof, we consider the case where {Ek } can distinguish all
pure states. We can make a extension to this theorem. If {Ek } can distinguish
all different states ρ1 , ρ2 ∈ Hd , and the equation 2 still holds, then the conversed
POVM {Fk } can is informationally complete with respect to all quantum states,
pure or mixed.

6

Theorem 2. Assume that m orthonormal bases can distinguish all pure states in
Hd , a large number of PSIR-complete POVMs with m(d − 1) + 1 rank-1 elements
can be constructed.
Proof. Denote these orthonormal bases as {Bj }, j = 0, · · · , m − 1. The elements
in basis Bj are {|φkj i}, k = 0, · · · , d − 1. Now we pick up m(d − 1) + 1 elements
from these bases. We can randomly choose one basis Bj and keep all the elements
Pd−1
in it. The corresponding projectors satisfy k=0 |φkj ihφkj | = I. Then we select
d − 1 elements at random from each one of the other bases. Thus we will get a
set of m(d − 1) + 1 elements. There are m · dm−1 collections totally.
Each collection will correspond to m(d − 1) + 1 rank-1 projectors, which can
distinguish all pure states. They satisfy the condition in Theorem 1. After the
rank-1 conversion, we will get a rank-1 PSIR-complete POVM with m(d − 1) + 1
elements. Moreover, we can construct a large number of PSIR-complete POVMs
for each collection. Denote the projectors to be {E1 , · · · , Ed , Ed+1 , · · · , Em(d−1)+1 },
Pd
where k=1 Ek = I. We can multiply Ej by an arbitrary non-negative number
ej , where j = d + 1, · · · , m(d − 1) + 1. So a new set of operators is constructed,
{E1 , · · · , Ed , ed+1 · Ed+1 , · · · , em(d−1)+1 · Em(d−1)+1 }. They also satisfy the condition in Theorem 1. The proof is complete.
Various researches focus on the minimal number of orthonormal bases that
can distinguish all pure states [17, 18, 19, 20, 21]. This problem is almost solved.
The minimal number of orthonormal bases is summarized in [11]. Moreover, four
bases are constructed from a sequence of orthogonal polynomials. For dimension
d = 2, at least three orthonormal bases are needed to distinguish all pure quantum states. For d = 3 and d ≥ 5, the number is four. For d = 4, four bases are
enough but it is not clear whether three bases can also distinguish.
So we can give the upper bound of m1 (d). When d = 2, m1 (2) = 4. When
d ≥ 3, m1 (d) = 4d − 3.

3 Rank-1 PSIR-complete POVMs for H2 , H3 and H4
In this section, we will present some results about the rank-1 PSIR-complete
POVMs for lower dimensions d. In Figure 1, we show the relations between
different kinds of informationally complete POVM. An IC-POVM is a PSIRcomplete POVM.
3.1 d=2
For dimension d = 2, four is the minimal number of elements in a rank-1 PSIRcomplete POVM. One example showed in [6] is the following:
E c = ac I + bc nc · σ, c = 1, 2, 3, 4.

(4)
√

The
ac = bc = 1/4.√n1 = √
(0, 0, 1), n2 = (2 2/3, 0, −1/3), n3 =
√ parameter:
√
4
(− 2/3, 2/3, −1/3), n = (− 2/3, − 2/3 − 1/3). σ = (σx , σy , σz ). This is

7

1

2

3
4

Fig. 1. The relations of different kinds of informationally complete POVM. The labels {1, 2, 3, 4} stand for SIC-POVM, IC-POVM, PSIR-complete POVM, PSI-complete
POVM respectively. For example, an IC-POVM is a PSIR-complete POVM.

also a SIC-POVM. It can distinguish all quantum states in H2 , pure or mixed.
There are two SIC-POVMs for d = 2 introduced in paper [5]. The other SICPOVM is used [22], which shows the efficiency of qubit tomography.
Now we can construct 12 rank-1 IC-complete POVMs with four elements.
There are three mutually unbiased bases for d = 2.
√
√
B0 = {|0i, |1i}, B1 = {(|0i ± |1i)/ 2}, B2 = {(|0i ± i|1i)/ 2}.
(5)
These three mutually unbiased bases can distinguish all quantum states in H2 .
We can select four elements as introduced in Theorem 2. There are 12 collections
the elements for one collection are |0i, |1i, (|0i +
√ totally. For example,
√
|1i)/ 2, (|0i + i|1i)/ 2. The corresponding rank-1 projectors are |0ih0|, |1ih1|,
(|0i + |1i)(h0| + h1|)/2 and (|0i + i|1i)(h0| − ih1|)/2. After the rank-1 conversion,
we will get a rank-1 POVM with 4 elements. Interestingly, this POVM is the
special case when d = 2 constructed by Caves et al. [3].
3.2 d=3
For dimension d = 3, there are four mutually unbiased bases. By Theorem 2, we
have 4 × 33 collections with 9 elements. We can construct rank-1 IC-complete
POVMs with 9 elements from each selection. By a reference to Heinosaari et al.
[8], m0 (3) = 8. So the minimal number of elements is either 8 or 9 for a rank-1
PSIR-complete POVM. Now we show that this number is 8 by constructing 8
rank-1 operators satisfying Theorem 1. After the rank-1 conversion, we will get
a PSIR-complete POVM with 8 elements. The operators are as follows:
E0 = |0ih0|, E1 = |1ih1|, E2 = |2ih2|, E3 = (|0i + |1i)(h0| + h1|), E4 =
(|0i+i|1i)(h0|−ih1|), E5 = (|0i+|2i)(h0|+h2|), E6 = (|0i+|1i+|2i)(h0|+h1|+h2|),
E7 = (|0i + |1i + i|2i)(h0| + h1| − ih2|).
P2
Let an arbitrary unknown pure state in H3 be |φi = k=0 ak eiθk |ki. Let ak
be non-negative real numbers for k = 0, 1, 2. As eiπ = −1, we can modify the
value of θk to guarantee ak ≥ 0. Let θk be in the range of [0, 2π), as eiθk =

8

ei(θk +2tπ) for integer t. For the freedom choice of global phase, we let θ0 = 0.
The outcome probabilities can be calculated as follows:
tr(Ek |φihφ|)=a2k , for k = 0, 1, 2.
tr(E3 |φihφ|)=a20 + a21 + 2a0 a1 cos θ1 ,
tr(E4 |φihφ|)=a20 + a21 + 2a0 a1 sin θ1 ,
tr(E5 |φihφ|)=a20 + a22 + 2a0 a2 cos θ2 ,
tr(E6 |φihφ|)=a20 + a21 + a22 + 2a0 a1 cos θ1 + 2a0 a2 cos θ2 + 2a1 a2 cos θ1 cos θ2 +
2a1 a2 sin θ1 sin θ2 ,
tr(E7 |φihφ|)=a20 + a21 + a22 + 2a0 a1 cos θ1 + 2a0 a2 sin θ2 + 2a1 a2 cos θ1 sin θ2 −
2a1 a2 sin θ1 cos θ2 .
The coefficients of ak can be calculatedpby Ek , where k = 0, 1, 2. As the
coefficient ak is non-negative, we have ak = tr(Ek |φihφ|). The remaining task
is to determine θk .
When only one element in {a0 , a1 , a2 } is nonzero, it is the trivial case. The
state can be |0i, |1i or |2i.
When two elements in {a0 , a1 , a2 } are nonzero, the state can also be determined. For example, a0 = 0 and a1 , a2 6= 0. We can write the state as
|φi = a1 |1i + a2 eiθ2 |2i. The global phase of θ1 is extracted. So θ1 = 0. The
remaining unknown coefficient θ2 can be calculated by the effect of E6 and E7 .
If a1 = 0 and a0 , a2 6= 0, the state is |φi = a0 |0i + a2 eiθ2 |2i. The coefficient θ2
can be calculated by the effect of E5 and E7 . If a2 = 0 and a0 , a1 6= 0, coefficient
θ1 can be calculated by the effect of E3 and E4 .
When all elements in {a0 , a1 , a2 } are nonzero, we let θ0 = 0. Now we determine the remaining coefficients θ1 and θ2 . From the effect of E3 and E4 ,
cos θ1 and sin θ1 can be calculated correspondingly, thus the coefficient θ1 can
be uniquely determined. After we know the values of ak and θ1 , we can calculate
cos θ2 by the effect of E5 . At the same time, sin θ2 can be calculated by the
effect of E6 or E7 , as cos θ1 and sin θ1 can not be both zero. Then θ2 is uniquely
determined.
Thus any pure state in H3 can be uniquely determined by the eight rank-1
positive self-adjoint operators. These operators satisfies the condition in Theorem 1. After the rank-1 conversion, we will get a PSIR-complete POVM with
eight elements. By a reference to Heinosaari et al. [8], such POVM is one of the
minimal possible resource.
3.3 d=4
For dimension d = 4, the known result is that m0 (4) = 10 [8]. There are five
mutually unbiased bases. Thus we can construct many rank-1 IC-POVMs with
16 elements. Four orthonormal bases can distinguish all pure states in H4 [11].
By theorem 1 and theorem 2, we can construct many PSIR-complete POVMs
with 13 elements. So the true value of m1 (4) is in the range of {10, 11, 12, 13}.
It is still not clear whether three bases can distinguish all pure states in
H4 . No results show three bases would fail and no results give the potential
support. There are some partial answers to this question. Three orthonormal

9

bases consisting solely of product vectors are not enough. In fact, even four
product bases are not enough [11]. Eleven is the minimum number of Pauli
operators needed to uniquely determine any two-qubit pure state [23].
We can conclude that there is no gap between m0 (d) and m1 (d) when d =
2, 3. If a gap exists when d = 4, three orthonormal bases are not enough to
distinguish all pure states. Consider the contrapositive form. If three orthonormal
bases can distinguish all pure states in H4 , we can construct a PSIR-complete
POVM with 10 elements by Theorem 2.

4 Adaptive d + 2k − 2 rank-1 operators for any dimensions
Goyeneche et al. took an adaptive method to demonstrate that any input pure
state in Hd is unambiguously reconstructed by measuring five observables, i.e.,
projective measurements onto the states of five orthonormal bases [15]. Thus ∼
5d rank-1 operators are needed. The adaptive method is that the choice of some
measurements is dependent on the result of former ones. The fixed measurement
basis is the standard, B0 = {|0i, · · · , |d − 1i}. We measure the pure state with
this basis first. The results of this basis will determine a subset sd,1 ⊂ Sd,1 , where
the input pure state belongs to. They construct four bases {B1 , B2 , B3 , B4 } to
determine all pure states in sd,1 .
Pd−1
Let an arbitrary unknown input pure state in Hd be |φi = s=0 as eiθs |si,
where as is a non-negative real number and θs ∈ [0, 2π) for s = 0, · · · , d − 1. We
can extract the global phase to let one phase θs be 0.
Now we construct d + 2k − 2 adaptive rank-1 positive self-adjoint operators
to determine this pure state, where 1 ≤ k ≤ d. Thus at most 3d − 2 rank-1
elements are enough by adaptive strategy.
The first d operators to be measured with are
Es = |sihs|, s = 0, · · · , d − 1.

(6)
p
We can calculate the amplitudes as by the effect of Es , as = tr(Es |φihφ|).
Then we keep track of the sites {s} of nonzero amplitudes {as } to determine a
subset sd,1 . Let k be the number of nonzero amplitudes, 1 ≤ k ≤ d.
For example, the sites of nonzero amplitudes are {0, · · · , d − 1}. Then k = d.
Pd−1
The subset sd,1 is { k=0 ak eiθk |ki : ak 6= 0}. The remaining 2d − 2 rank-1
operators are follows:
Fs = (|0i + |si)(h0| + hs|), Gs = (|0i + i|si)(h0| − ihs|);
where s = 1, · · · , d − 1.
We extract the global phase to make θ0 = 0. We have the equations:
(
tr(Fs |φihφ|) = a20 + a2s + 2a0 as cos θs ,
tr(Gs |φihφ|) =

a20 + a2s + 2a0 as sin θs .

(7)

(8)

10

From the assumption and measurement results of Es , all the amplitudes
as are nonzero and known. Then cos θs and sin θs can be calculated by the
effect of Fs and Gs . All the coefficients θs can be uniquely determined. Thus all
coefficients of the unknown pure state in Hd are calculated.
The operators Es and Gs appear in the construction of PSI-complete POVM
given by Finkelstein [7]. And operators Es , Fs /2 and Gs /2 are some part of d2
rank-1 elements in the IC-POVM constructed by Caves et al. [3]. In fact, Fs and
Gs can be the other types to calculate θs . For example, Fs = (|1i+|si)(h1|+hs|),
Gs = (|1i + i|si)(h1| − ihs|), s = 0, 2, · · · , d − 1.
Now consider the general case, the sites of nonzero amplitudes are {n0 , · · · , nk−1 }.
Pk−1
The subset sd,1 is { j=0 anj eiθnj |nj i : anj 6= 0}. The remaining 2k − 2 projections are as follows:
Fs = (|n0 i + |ns i)(hn0 | + hns |), Gs = (|n0 i + i|ns i)(hn0 | − ihns |);

(9)

where s = 1, · · · , d − k − 1. Let the phase θn0 = 0. With similar analysis, we can
uniquely calculate cos θj and sin θj by the effect of Fj and Gj . All the phases θs
and amplitudes as of |φi can be uniquely determined.

5 Conclusions
We analyse the minimal number of elements in rank-1 PSIR-complete POVM.
The bound is in [4d − 3 − c(d)α(d), 4d − 3]. The lower bound of 3d − 2 is not
tight except for d = 2, 4. For d = 2, we construct many rank-1 POVMs with four
elements which can distinguish all quantum states. For d = 3, we show that eight
is the minimal number in a PSIR-complete POVM by construction. For d = 4,
if m1 (4) > 10, we can give a answer to an unsolved problem. Three orthonormal
bases can not distinguish all pure states in H4 . Finally, we construct d + 2k − 2
adaptive rank-1 positive self-adjoint operators to determine any input states in
Hd , where 1 ≤ k ≤ d. Thus we can determine an arbitrary unknown pure state
in Hd with at most 3d − 2 rank-1 operators by adaptive strategy.

Acknowledgements
This work was partially supported by National Key Research and Development
Program of China under grant 2016YFB1000902, National Research Foundation
of China (Grant No.61472412), and Program for Creative Research Group of
National Natural Science Foundation of China (Grant No. 61621003).

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