1. Introduction
Quantum coherence is not only a feature of quantum systems which arise due to superposition principle, but also is a kind of fundamental resources in quantum information and computation [
1,
2,
3,
4,
5,
6,
7,
8]. The resource theory of coherence is formulated with respect to a distinguished basis of a Hilbert space, which defines free states as the states that are diagonal in this basis [
3]. Several important quantifiers of quantum coherence have been introduced and assessed [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19]. Recently, it is shown that quantum coherence can be useful resource in quantum computation [
20,
21,
22,
23,
24], quantum metrology [
25], quantum thermodynamics [
26,
27,
28,
29,
30,
31] and quantum biology [
32,
33,
34].
Since the coherence of quantum states depends on the choice of the reference basis, it is natural to study the relationship among the coherence with respect to different bases. Cheng et al. [
35] first studied the situation of two specific coherence measures under mutual unbiased basis (MUB):
norm of coherence and relative entropy of coherence. They proposed the complementary relationship of the two coherence measures under any complete MUB set. Rastegin in [
36] discussed the uncertainty relation for the geometric measure of coherence under MUBs. Sheng et al. [
37] further studied the realization of quantum coherence through skewed information and the geometric measure under mutual unbiased bases. Recently, considered the standard coherence (SC), the partial coherence (PC) [
38,
39,
40] and the block coherence (BC) [
41,
42] as variance of quantum states under some quantum channel
, Zhang et al. [
43] proposed the concept of channel-based coherence of quantum states, called
-coherence, which contains the SC, PC and BC, but not contain the POVM-based coherence [
44,
45], and obtained some interesting results.
Usually, the coherence of an individual quantum state is discussed only when referring to a preferred basis. Considered sets of quantum states, Designolle et al. [
46] introduced the concept of set coherence for characterizing the coherence of a set of quantum states in a basis-independent way. Followed a resource-theoretic approach, the authors of [
46] defined the free sets of states as sets
of groups of states
such that there exists a choice of basis (equivalently, a unitary
U) for which all states
in the set
become diagonal. Clearly,
if and only if
is a commutative family of states, i.e.,
for all
.
Different from the discussions above, in this paper, we focus on the quantum incoherence based simultaneously on
k bases; equivalently, the coherence of a quantum state with respect to a basis contained in a given set
of
k orthonormal bases. In
Section 2, by defining the correlation function of two orthonormal bases
e and
f, we study the relationships between two sets of incoherent states with respect to
e and
f, and investigate the maximally coherent states with respect to
e and
f. In
Section 3, we discuss the strong incoherence and the weak coherence of a state with respect to a set of
k orthonormal bases and introduce a measure for the weak coherence. In
Section 4, we give a summary of this paper.
2. Correlation Function of Two Bases and Quantum Coherence
Let us consider a quantum system X, which is described by a d-dimensional Hilbert space H and let I denote the identity operator on H. We use and to denote the sets of all linear operators and all density operators (mixed states) on H, respectively. In quantum information theory, a positive operator valued measure (POVM) is a set of operators on H with for all and In particular, if for all i, then the POVM becomes a projective measurement (PM). For a rank-one PM P, there exists an orthonormal basis such that . In this case, we denote . We use the notation or to denote the conjugate of a complex number z.
For the fixed orthonormal basis
for
H,
denotes the set of incoherent states on
H w.r.t.
e, i.e., ones that have diagonal matrix representation under the basis
e. A quantum operation
on
is said to be an incoherent operation [
3] w.r.t
e if it admits an incoherent Kraus decomposition, i.e.,
with
We use to denote the set of incoherent operations w.r.t e on .
According to Ref. [
3], a coherence measure with respect to
e, called an
e-coherence measure, is a function
satisfying the following four conditions.
(1) Faithfulness: for all ; if and only if .
(2) Monotonicity: for any .
(3) Strong monotonicity: , for all operators in such that with , and if ; if .
(4) Convexity: for any states and any probability distribution .
A usual
-norm coherence measure [
3] of a state
with respect to a basis
e is defined by
Clearly,
Especially,
if and only if
for all
; in that case,
is called a
maximally coherent state with respect to
e.
From the review above, we find that quantum coherence relies on the choice of orthonormal bases. In what follows, we discuss the relationship between quantum coherence based on different reference bases. To do this, we let
and
be two orthonormal bases for
H and define
called the
correlation function between two bases
e and
f.
Recall that [
35] two orthonormal bases
e and
f for
H are said to be mutually unbiased if
for all
. Thus, when
e and
f for
H are mutually unbiased, it holds that
. More properties of the correlation function are given in the following theorem.
Theorem 1. Let e and f be two orthonormal bases for H. Then
(1)
(2) if and only if if and only if .
(3) if and only if e and f are mutually unbiased bases.
Proof. (1) Since
we get
for all
. So,
This shows that
Since
and
are two orthonormal bases for
H, there exists a
unitary matrix
such that
equivalently,
Hence,
, and using the Cauchy inequality yields that
Consequently,
(2) We see from Equation (
2) that
if and only if for any
i, there exists a unique
such that
and
for all
if and only if for any
i, there exists a unique
such that
, which is equivalent to
, i.e.,
.
(3) From the proof of (1), we see that if and only if , that is, e and f are mutually unbiased bases.
Suppose that
e and
f are mutually unbiased bases, then the coefficients
in (
3) satisfy
for all
Let
. Then it can be written as
with
for all
,
. Using Equation (
3) implies that
Since
and
, we see that
that is,
Since
is a
unitary matrix, we get
for all
, i.e.,
. Hence,
.
□
Remark 1. Suppose that , then there exists an i and such that andThen andSince for any , we get that This shows that there exists a state but . Similarly, there also exists a state but . From Theorem 1 and Remark 1, we get relationships between
and
as shown by the following
Figure 1.
It is clear that for any bases e and f. Especially, if they are mutually unbiased. However, even though e and f are not a pair of mutually unbiased bases, it is possible that see the following example.
Example 1. Let and be two orthonormal bases for withClearly, e and f are not a pair of mutually unbiased bases while . This example leads us to study the relationship between two bases
e and
f for
H such that
To do this, we let
and
be two bases for
H and
Since
are the eigenvalues of
, they can be rearranged as
in decreasing order, say,
. Thus, there exists a permutation matrix
such that
Suppose that
. Then
where
. Using Equation (
5) implies that
i.e.,
where
Since
are also the eigenvalues of
, they can be also rearranged as
in decreasing order. So, there exists a permutation matrix
such that
Thus,
Putting
yields that
Thus, when
, we see from Equation (
10) that for
,
and so
. Using Equation (
10) again yields that for
,
and so
, implying that
. Thus,
where
k means the number of different eigenvalues
of
and
is an
-doubly stochastic matrix, and
denotes the multiplicity of the
ith eigenvalue
.
Conversely, suppose that there exist
permutation matrices
and
such that
is of the form (
11) where
. Since the matrix
can be written as
where
we see from condition (
11) that
This implies that the subspaces generated by
and
are equal and so
Clearly,
.
As a conclusion, we arrive at the following.
Theorem 2. Let , and be two orthonormal bases for H and set . Then there exists a state in if and only if there exist two permutation matrices and such that the matrix is block-diagonal for some .
Example 2. Let and be two orthonormal bases for H such thatThenIt follows from Theorem 2 that there exists a state for example, Remark 2. From Theorem 2, we know that whether depends on the structure of the matrix C given by Equation (7). Since this, we call C the correlation matrix of the bases e and f and denote it by . Clearly, it can be written as the Hardamard product of the transition matrix from e to f and its conjugate matrix :where Theorem 2 also tells us that when for all , there do not exist permutation matrices and such that is block diagonal, so . Especially, for a pair of mutually unbiased bases e and f, when and , we have . Conversely, when is a maximally coherent state w.r.t. e, a question is: whether is also maximally coherent w.r.t. f. The follow example shows that the answer is negative.
Example 3. Let and be a pair of mutually unbiased bases for wherechooseThen is maximally coherent with respect to f but is incoherent w.r.t. e, while for the statewe haveTherefore, is both maximally coherent w.r.t. e and f. The following theorem shows that there must exist a maximally coherent state w.r.t. any two bases for .
Theorem 3. Let and be two orthonormal bases for . Then there exists a state such that Proof. First, we observe that
if and only if
and
if and only if
Suppose that
then
is a unitary matrix, which is given.
For a state
of the form given by (
14), then
. We compute that
Thus,
if and only if
if and only if
since
.
Since
U is a unitary matrix, it can be represented as
where
, and
s.t.
. The last condition implies that
for some integer
n. Taking
implies that
and so there exists a real number
such that second equation in (
17) holds. Since
, the first equation in (
17) holds too. Hence,
This shows that the state
defined by Equation (
14) with
satisfies
that is,
□
3. Weak Coherence
In this section, we turn to discuss the weak coherence of quantum states. To this, we use to denote a set of k orthonormal bases for H, i.e.,
Definition 1. We say that is strongly incoherent (S-incoherent) w.r.t. if ρ is incoherent w.r.t. any basis in . Otherwise, we say that ρ is weakly coherent (W-coherent) w.r.t. .
Denoted by
the set of all S-incoherent states of
H w.r.t.
. Clearly,
Definition 2. Let Φ be a quantum operation on . Then Φ is said to be an S-incoherent operation (SIO) w.r.t. (or -incoherent operation (IO)) if for all , that is, for each , Φ has a family of Kraus operators such that Denoted by
the set of all SIOs w.r.t.
, then
Similar to the definition of the standard coherence measure, let us introduce the concept of a -coherence measure.
Definition 3. A function is said to be a -coherence measure if the following four conditions are satisfied:
(1) Faithfulness: ; if and only if
(2) Monotonicity: for every and for every
(3) Strong monotonicity: for each for every and every with a family Kraus operators where and for , and for .
(4) Convexity: where and is a probability distribution.
The following theorem gives a method for constructing a -coherence measure from k-coherence measures
Theorem 4. Let be -coherence measures. Then the function defined byis a -coherence measure. Proof. (1) Let
. Since
for all
we have
Furthermore,
(2) Let
For each
, since
is an
-coherence measure and
, we get
for all
, and so
(3) Let
,
with families of Kraus operators
. Put
and
for
, and
for
. For each
, since
is an
-coherence measure and
, we get
This implies that for each
,
(4) Let
and let
be a probability distribution. Since
is an
-coherence measure, we have
for all
, and therefore,
Using Definition 3 yields that the function
defined by Equation (
18) becomes a
-coherence measure. □
Using Theorem 4 yields that the function
defined by
is a
-coherence measure. We see from property (
1) that
for all states
of the system. A state
is said to be
maximally coherent w.r.t.
if
. Clearly, a state
is maximally coherent
if and only if it is maximally coherent w.r.t. each
.
Remark 3.(1) ; Especially, if there exist two mutually unbiased bases in , then , that is, if and only if .
(2) Theorem 3 implies when and , there exists a maximally coherent state ρ w.r.t. , that is, .
(3) The following theorem means that when and is a complete set of mutually unbiased bases, there does not exist necessarily a maximally coherent state w.r.t. .
It was proved in [
47] that the maximal number
of mutually unbiased bases for
H is
if the dimension
d of
H is a prime-power. Thus,
, i.e., there exists a complete set of three mutually unbiased bases for
.
Theorem 5. Let where be any orthonormal basis for , and withThen and g are mutually unbiased bases pairwise for and for all states ρ of , that is, there does not exist a state ρ such that Proof. Obviously,
and
g are mutually unbiased bases pairwise for
. Suppose that there exists a state
such that Equation (
20) holds, i.e.,
Then under the three bases, we have
where
Since
, we conclude from Equation (
21) that
. Substituting
in Equation (
23) with
and comparing the coefficient of
in Equations (
22) and (
23), we find that
Similarly, substituting
in Equation (
24) with
and comparing the coefficient of
in Equations (
22) and (
24), we find that
Combining Equations (
25) and (
26) yields that
, a contradiction. □