1. Introduction
Quantum fidelity [
1,
2] is a fundamental and indispensable tool in quantum information theory for quantifying the closeness between two quantum states that describe a quantum system. Among its various applications, quantum fidelity plays a crucial role in evaluating the success of key quantum communication tasks within quantum Shannon theory, including quantum teleportation [
3], quantum state merging [
4,
5], and quantum state redistribution [
6,
7]. To illustrate the importance of quantum fidelity, we focus on the task of quantum state merging. In this task, two users, Alice and Bob, initially possess separate parts
A and
B of a shared quantum state
. By employing local operations and classical communication assisted by shared entanglement, their objective is to merge Alice’s quantum state with Bob’s, resulting in the target state
, where
corresponds to Bob’s quantum system. Upon completion of the merging process, how can they ascertain the closeness of the resulting state to the desired target state? Without the aid of the quantum fidelity, it would be impossible to compare and assess the similarity between these states.
In this study, we consider the following inequality [
8]:
where
and
represent the quantum states of the bipartite system
, and
and
represent the reduced states of
and
corresponding to the quantum system
A. This inequality demonstrates that for any given pair of bipartite quantum states, the quantum fidelity on the bipartite quantum system
is always less than or equal to the quantum fidelity on the local quantum systems
A. To provide a simple illustration, let us examine the scenario of two EPR pairs [
9]:
where
and
are the computational basis of a two-dimensional quantum system. In this context, the quantum fidelity between
and
is found to be zero. However, when we evaluate their fidelity on the local quantum system
A, it becomes one. This intriguing observation implies that the quantum states
and
are indistinguishable on the local quantum system
A, indicating complete identity. However, on the bipartite quantum system
, they exhibit complete distinctness.
The inequality Equation (
1) is easy to understand, as discussed earlier. However, determining the conditions under which the fidelities in Equation (
1) become equal is difficult. This study focuses on overcoming this limitation by considering pure bipartite quantum states
and
. We aim to investigate the conditions for fidelity inequality as stated in Equation (
1) and provide explicit representations of pure bipartite quantum states that satisfy these conditions.
The remainder of this paper is organized as follows: In
Section 2, we introduce the definitions of global and local fidelities, along with the assumptions and lemmas that form the foundation of our main results.
Section 3 presents a comprehensive calculation of the global and local fidelities. In
Section 4, we present the conditions that establish the equivalence for fidelity equality.
Section 5 is devoted to presenting specific forms of pure bipartite quantum states that fulfill these equivalent conditions. Finally, in
Section 6, we discuss our findings, their implications, and outline potential avenues for future research.
2. Definitions, Assumptions, and Lemmas
In this section, we provide the definitions, assumptions, and lemmas that are employed throughout this work.
To begin, we consider finite-dimensional Hilbert spaces . The notation denotes a Hilbert space representing a quantum system X. The tensor product signifies a composite quantum system comprising two quantum systems A and B, which can be denoted as or simply . The dimension of the Hilbert space , denoted as , corresponds to the dimension of the quantum system X.
Let
denote the set of density operators on a Hilbert space
. In other words,
, where
denotes the set of all linear operators on
. The elements within
are referred to as quantum states. If a quantum state
can be expressed as a rank-1 projector, i.e., it can be represented as
where
is a normalized vector in the Hilbert space
, it is referred to as a pure state. Here, the unit vector
is also considered a pure quantum state. Quantum states that are not pure are referred to as mixed states, and they are denoted by
or
in this paper.
The trace,
, of a quantum state
operating on a Hilbert space
is defined as
where
represents any orthonormal basis of the Hilbert space
. For a bipartite quantum state
on a Hilbert space
, the partial trace over the Hilbert space
is defined as
where
denotes the identity matrix on the quantum system
A, and
represents any orthonormal basis of the Hilbert space
. In this scenario, the quantum state
obtained on the Hilbert space
is referred to as the reduced quantum state of
.
In this study, we focus on investigating the quantum fidelity [
8] between two quantum states
and
that represent the same quantum system. The quantum fidelity is defined as
In particular, when considering two pure quantum states
and
, the quantum fidelity can be straightforwardly calculated as
. We also investigate two pure quantum states
and
on the bipartite quantum system
, and with the assumption that
and
. For convenience, we use the notations
where
and
represent the reduced quantum states of pure bipartite quantum states
and
, respectively. When referring to the given quantum states
and
, we use the terms
and
to present the
global fidelity and the
local fidelity, respectively. Thus, the fidelity inequality in Equation (
1) can be expressed as
Finally, we introduce two lemmas that will be used in the subsequent sections.
Lemma 1. For any two complex numbers α and β, we havewhere denotes the complex conjugate of β, k is a real number, and p is a real and non-negative value. Proof. (i) Assume that
holds for any two complex numbers
and
. Given that
and
are complex, they can be expressed as
and
using some real numbers
a,
b,
c, and
d. Notably,
Consequently, the assumption implies that
; thus,
. Therefore,
where
.
(ii) Assume that
holds for any two complex numbers
and
. AS
and
are complex, they can be represented as
and
based on some non-negative real numbers
,
,
, and
. Without loss of generality, we may assume that
. Observe that
,
, and
Therefore,
implies that
; thus,
. Consequently, we have
where
. For the inverse direction, assume that
holds for some non-negative
p. Since
,
is non-negative. Thus, we have
which completes the proof. □
Lemma 2. For any two vectors and represented aswe have the equalitywhere are complex coefficients, and indicates the computational basis of a d-dimensional Hilbert space. Proof. Consider the norm of the bipartite vector
, which is as follows:
In addition, the above quantity can be represented as
This completes the proof. □
3. Calculation of Global and Local Fidelities
In this section, we present the calculation of the global fidelity and the local fidelity for any two pure quantum states and . These calculations will be used in the next section.
Let us first consider the Schmidt decomposition [
8] of the quantum state
, which is given by
for some
. In this equation,
and
are orthonormal bases on the quantum systems
A and
B, respectively. Then, the quantum state
can be represented as
where
are complex numbers satisfying
Given that
and
are pure states,
can be calculated as
where the second equality arises from Equations (
28) and (
29). In addition, the reduced states
and
of the quantum states
and
can be represented as
Thus, the operator
is represented as
Consider an operator
L defined as
wherein the coefficients
are
In addition, let us consider an operator
M defined as
wherein the coefficients
are
Then,
M is positive, Hermitian, and has trace 1. Note that
L and
M satisfy the equality
.
Any operator
N, expressed as
that is positive, Hermitian, and has trace 1, has eigenvalues
and eigenvectors
given by
where
,
, and
and
are orthonormal vectors. Note that
and
.
Consequently, the eigenvalues
and
of
M are calculated as
and thus, the operator
L has the eigenvalues
and
. It follows that
Since the trace and determinant of operator
M, i.e.,
and
, respectively, are known, we have
among which the last equality arises from Lemma 2 and the rest can be obtained from the definitions of the coefficients
and
. Thus, the local fidelity
is represented as
4. Necessary and Sufficient Conditions
In this section, we present our main result, which establishes the necessary and sufficient conditions for the fidelity equality, i.e., .
Theorem 1 (necessary and sufficient conditions).
Let and be pure quantum states on a bipartite quantum system such that and . The quantum states and satisfy the fidelity equality, i.e., if and only if they satisfy the following four conditions: wherein the notations used are the same as those used in Equations (28) and (29), k is real, and p is real and non-negative. Proof. From Equation (
11) of Lemma 1, it suffices to demonstrate that the fidelity equality
holds if and only if the two quantum states
and
meet Equations (
63), (
64), and (
65) and the following condition:
(i) Assume that the equality
holds. Then, Equations (
33) and (
61) imply the following equation:
By applying the triangle inequality to the LHS, we obtain the following inequality:
If
holds, then the inequality in Equation (
69) becomes
By applying the inequality
to Equation (
70), we obtain
which is a contradiction. Consequently, we have the inequality
By applying this inequality to Equation (
69), we obtain the inequality
Thus, we have demonstrated that the equality
holds, which is the same as the first sufficient condition given as Equation (
63).
Second, we note that the LHS of Equation (
68) becomes
Therefore, the equality in Equation (
68) becomes
Here, the first inequality is obtained by eliminating a few of the non-negative terms, the second inequality arises from the reverse triangle inequality, and the last equality is obtained from the inequality in Equation (
72) and the first sufficient condition Equation (
63). This implies that
holds. Because any complex number
z satisfies the inequality
, we establish the second sufficient condition presented in Theorem 1.
To obtain the third sufficient condition, presented as Equation (
65), we use Equation (
78) as follows:
where the inequality is obtained by eliminating a few of the non-negative terms and applying the reverse triangle inequality and the last equality arises from the first sufficient condition given as Equation (
63). From Equations (
72) and (
64), we have
which yields the third sufficient condition given as Equation (
65).
By applying the first three conditions to the equality in Equation (
68), we deduce the last condition given as Equation (
67). This condition is equivalent to the fourth sufficient condition stated in Theorem 1, based on Equation (
11) of Lemma 1.
(ii) We assume the aforementioned four conditions to prove the converse of Theorem 1. Note that
where the first equality is obtained by applying the third necessary condition given as Equation (
65) to the local fidelity
given by Equation (
61), the first and fourth conditions stated in Equations (
63) and (
67) lead to the second equality, and the third and fourth equalities arise from the second condition given as Equation (64) and from Equation (
76), respectively. □
Theorem 1 implies the following corollary, which is nothing but the contrapositive of Theorem 1.
Corollary 1. Let and be pure quantum states on a bipartite quantum system such that and . The quantum states and satisfy the fidelity inequalityif and only if they fail to satisfy at least one of four necessary and sufficient conditions outlined in Theorem 1, where and are defined in Equations (28) and (29), respectively. By employing Theorem 1 or Corollary 1, one can readily verify whether a pair of pure quantum states and satisfies the fidelity equality . As a special case of Theorem 1, if the quantum state is separable, then the four equivalence conditions are reduced to a single condition, as follows.
Corollary 2. If is separable, then the fidelity equality holds if and only if the following condition holds:where is defined in Equation (29). Proof. In Equation (
28), if
is separable, then
, and thus, we have
. Assuming that
holds, the first necessary and sufficient condition in Theorem 1 implies that
. Furthermore, from the third necessary and sufficient condition in Theorem 1, we have that
for any
.
For the inverse, let us assume that
holds for any
. Note that for
, the global fidelity
and the local fidelity
are given by
which implies that
because
for any
. □
5. Representations for Fidelity Equality
Based on the primary results presented in
Section 4, we provide specific forms of the quantum state
when the quantum states
and
satisfy
.
If
is a separable state, denoted as
, Corollary 2 implies that the other quantum state
is represented as follows:
where
for any
. This representation shows that
is the linear combination of the orthogonal states
and
. Furthermore, these states are also orthogonal to each other in subsystem
A. Specifically, when we consider subsystem
A,
and
become
and
, respectively. Therefore, in this case, the quantum states
have no effects on the global and local fidelities, while
and its coefficient
determine them, i.e.,
.
On the contrary, let us consider the case that
is entangled, i.e.,
in Equation (
28). Then, the third necessary and sufficient condition of Theorem 1 implies that
where
. From the first, second, and fourth conditions in Theorem 1, along with Lemma 1, the coefficients
have the following relations:
where
k,
, and
are real numbers, and
p,
, and
are non-negative real numbers. Thus, the quantum state
in Equation (
98) becomes
where the coefficient
is a complex number defined as
.
Remark 1. The coefficient p in the representation of the quantum state in Equation (103) determines its entanglement properties. Specifically, given by Equation (98) is separable if and only if holds. Therefore, of Equation (103) is separable if and only if . Consequently, for the case of , the representation in Equation (103) simplifies to 6. Conclusions
In this study, we have explored quantum fidelity and its fundamental properties. Specifically, we have focused on bipartite pure quantum states
and
, where the dimension of quantum system
A is two and the dimension of system
B is arbitrary. We have introduced the global fidelity
and the local fidelity
for these quantum states in
Section 2. We have established the inequality
but the conditions under which these fidelities are equal remained unknown. In
Section 4, we have provided the necessary and sufficient conditions for the fidelity equality
. Additionally, in
Section 5, we have presented specific representations of the quantum state
when
is satisfied by
and
.
In this study, our analysis was based on the assumption that the bipartite quantum states for calculating quantum fidelities are pure, and we have considered a fixed dimension of two for subsystem
A. However, for future research, we propose investigating the necessary and sufficient conditions for fidelity equality in general bipartite states. Moreover, it would be valuable to explore the relationships between the amount of entanglement and fidelity equality, as quantum entanglement plays a crucial role in quantum communication tasks, although our current work does not focus on it. To the best of our knowledge, there is a lack of research addressing the connection between entanglement and fidelity equality. Therefore, elucidating these relationships would contribute significantly to the field. Additionally, we suggest examining a specific scenario in which one of our target states corresponds to the the isotropic state [
10] or the Werner state [
11].