1. Introduction
In this paper we introduce a new evolution equation in the matrix geometry such that the
norm is preserved. In [1], the author introduced the Ricci flow which exists globally when the initial matrix is a positive definite. The Ricci flow [2] [3] preserves the trace of the initial matrix and the flow converges the scalar matrix with the same trace as the initial matrix. In [4], we have introduced the heat equation, which also preserves the trace of the initial matrix. In [5]-[8], the authors introduce the norm preserving flows which are global flows and conver- ge to eigenfunctions. We know that the fidelity of quantum state is an important subject in quantum computation and quantum information [9] [10], the
norm flow we studied is very closed related to the fidelity. This is the motivation of the study of norm preserving flow in matrix geometry.
To introduce our new
norm flow in matrix geometry, we need to use some language from the book [11] and the papers [1] [4] [12]. Let
be two Hermitian matrices on
. Define
,
. We use
to denote the algebra of all
complex matrices which generated by
and
with the bracket
. Then
, which is the scalar multiples of the identity matrix
, is the commutant of the operation
. Sometimes we simply use 1 to denote the
identity matrix.
We define two derivations
and
on the algebra
by the commutators
![]()
and define the Laplacian operator on
by
![]()
where we have used the Einstein sum convention. We use the Hilbert-Schmidt norm
defined by the inner product
![]()
on the algebra
and let
. Here
is the Hermitian adjoint of the matrix
and
denotes the usual trace function on
. We now state basic properties of
,
and
(see also [1]) as follows.
Given a positive definite Hermitian matrix
. For any
, we define the Dirichlet energy
![]()
and the
mass
![]()
Let, for
,
![]()
Then the eigenvalues of the operator
correspond to the critical values of the Dirichlet energy
on the sphere
![]()
We consider the evolution flow
(1.1)
with its initial matrix
. Assume
is the solution to the flow above. Then
![]()
Since
, we know that
. Then
![]()
The aim of this paper is to show that there is a global flow to (1.1) with the initial data
and the flow preserves the positivity of the initial matrix.
2. Existence of the Global Flow
Firstly, we consider the local existence of the flow (1.1). We prefer to follow the standard notation and we let
, where
is a positive definite Hermitian matrix. Let
be such that
(2.1)
with the initial matrix
. Here
such that
. Then for
, we let
(2.2)
Formally, if the flow (2.1) exists, then we compute that
![]()
Then ![]()
In this section, our aim is to show that there is a global solution to Equation (2.1) for any initial matrix
with
.
Assume at first that
is any given continuous function and
is the corresponding solution of (2.1). Define
. Then
and we get
(2.3)
The Equation (2.3) can be solved by standard iteration method and we present it in below. Assume
and
are eigen-matrices and eigenvalues of
as we introduced in [4], such that
![]()
Note that ![]()
Assume that
is the solution to (2.3). Set
![]()
Then by (2.3), we obtain
![]()
Then
, and
.
Hence
![]()
and
(2.4)
solves (2.1) with the given
.
Next we define a iteration relation to solve (2.1) for the unknown
given by (2.2).
Define
such that it solves the equation
with
.
Let
be any integer. Define
such that
(2.5)
with
(2.6)
Then using the Formula (2.4), we get a sequence
.
We claim that
is a bounded sequence and
is also a bounded sequence.
It is clear that
. If this claim is true, we may assume
![]()
Then by (2.5) and (2.6), we obtain
![]()
and
![]()
which is the same as (2.1). That is to say,
obtained above is the desired solution to (2.1).
Firstly we prove the claim in a small interval
. Assume
and
on
,
. Then, by (2.5),
(2.7)
By (2.6), we obtain
. Then
![]()
By (2.7), we get
![]()
Then
.
Note that
such that
for any
. We have
![]()
Then
. Hence the claim is true in
.
Therefore, (2.1) has a solution in
. By iteration we can get a solution in
with
as the initial data. We can iterate this step on and on and we get a global solution to (2.1) with initial data
.
In conclusion we have the below.
Theorem 2.1 For any given initial matrix
with
, the Equation (2.1) has a global solution with
as its initial data and
for all
.
3. Positive Property Preserved by the Flow
In this section we show that positivity of the initial matrices is preserved along the flow. That is to say, we show that if the initial matrix is positive definite, then along the flow (2.1), the evolution matrix is also positive definite.
Theorem 3.1 Assume
, that is
is a Hermitian positive definite. Then
along the flow equation
![]()
with
, where
is given by (2.2).
Proof. By an argument as in [4], we know
is Hermitian matrix. Then we know that
for small
by continuity. Compute
![]()
where
.
Since
![]()
and
![]()
We know that
![]()
Then we have
![]()
Hence
and
,
. ![]()
Then the proof of Theorem 3.1 is complete.
Remark that by continuity, we can show that if
, then
along the flow (2.1).
Funds
The research is partially supported by the National Natural Science Foundation of China (No. 11301158, No.11271111).