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L2 norm preserving flow in matrix geometry
Linear Algebra and its Applications 487 (2015) 220–231
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
L2 norm preserving flow in matrix geometry ✩ Jiaojiao Li Department of Mathematics, Henan Normal University, Xinxiang, 453007, China
a r t i c l e
i n f o
Article history: Received 9 April 2015 Accepted 4 September 2015 Available online 25 September 2015 Submitted by R. Brualdi MSC: 58B34 15A57
a b s t r a c t In this paper, we study a new L2 norm preserving heat flow in matrix geometry. We show that if the initial data has trace zero and has unit L2 norm, this flow has a global solution and enjoys the entropy stability in any finite time. We show that as the time is approaching infinity, the flow has its limit as an eigen-matrix of the Laplacian operator. Interesting operator convex property of heat equation is also derived. © 2015 Elsevier Inc. All rights reserved.
Keywords: Global flow Norm conservation Entropy stability Operator convexity
1. Introduction In this paper we continue our study of evolution equation in matrix geometry [10]. We introduce the L2 norm preserving flow such that starting from any initial data with unit L2 norm and trace zero, we can produce a family of matrices of unit L2 norm such that their limit is an eigen-matrix of the Laplacian operator introduced in [10] and [6]. In [6], the author introduced the Ricci flow which exists globally when the initial matrix is positive definite. The Ricci flow preserves the trace of the initial matrix and the flow ✩ The research is partially supported by the National Natural Science Foundation of China (No. 11271111) and NSF of Henan Province No. 132300410052. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.laa.2015.09.011 0024-3795/© 2015 Elsevier Inc. All rights reserved.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
221
converges the scalar matrix with the same trace as the initial matrix. In [10], we have introduced the heat equation, which also preserves the trace of the initial matrix. The advantage of the heat equation is that the corresponding flow not only preserves the trace but also can be stated with any initial matrix. In [1,2,11], the authors introduced the norm preserving flows which are global flows and converge to eigenfunctions. In quantum information theory, we can see the interesting works [3–5,15], but we need evolution flows which preserve some norms like the L2 norm just like the normalized Ricci flow [8,7] preserves the volume and the curve flows [11–13] preserve length or area. This is the motivation of the study of norm preserving flow in matrix geometry. More precisely, we study again the matrix geometry model as in [10,6,14]. We consider two unitary n × n matrices u and v satisfying the commutation relation vu = quv, where q=e
2πim n
for an m ∈ {1, 2, · · · , n − 1}, such that m and n are relatively prime: note in particular that q n = 1, but q j = 1 for j = 1, · · · , n − 1. Concretely we can use ⎡ ⎢ ⎢ u=⎢ ⎢ ⎣
⎤
1
⎢ ⎥ ⎢ ⎥ ⎥ and v = ⎢ ⎢ ⎥ ⎣ ⎦
q ..
⎡
. q n−1
01
⎤
⎥ . 0 .. ⎥ ⎥ ⎥ .. . 1⎦ 1 0
where the blanks spaces are filled with zeroes. It is pointed out in [6] that v is obtained by Fourier transform of Zn = {0, 1, · · · , n − 1} from u, i.e. v = F ∗ uF , where F is the Fourier transform on Zn in a suitable basis, namely the n × n unitary matrix Fjk = q −jk /n1/2 for j, k = 0, 1, · · · , n − 1, and F ∗ denotes its Hermitian adjoint. It is also said that u and v generate the algebra Mn of all n × n complex matrices, namely the commutant of [u, v] consists of scalar multiples of the identity matrix, [u, v] = C1, where here and later we denote the n × n identity matrix simply as 1, since no confusion will arise. 2πi 2πi Let x, y be two Hermitian matrices on Cn . Define u = e n x , v = e n y . We use Mn to denote the algebra of all n × n complex matrices which generated by u and v with the bracket [u, v] = uv − vu. Then CI, which is the scalar multiples of the identity matrix I, is the commutant of the set {u, v}. We define two derivations δ1 and δ2 on the algebra Mn by the commutators δ1 := [y, ·], δ2 := −[x, ·].
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222
Define the Laplacian operator on Mn by ˆ = δ ∗ δ 1 + δ ∗ δ 2 = δ 2 + δ 2 = δμ δ μ , Δ 1 2 1 2 where we have used the Einstein sum convention. We use the Hilbert–Schmidt norm | · | defined by the inner product < a, b >:= τ (a∗ b) on the algebra Mn . Here a∗ is the Hermitian adjoint of the matrix a and τ denotes the ˆ (see also [6]) usual trace function on Mn . We now state basic properties of δ1 , δ2 and Δ as follows. For any c ∈ Mn , we define the Dirichlet energy D(c) =
< δμ c, δμ c >
and the L2 mass M (c) =< c, c > . Let, for c = 0, λ(c) =
D(c) . M (c)
ˆ correspond to the critical values of Then the eigenvalues of the Laplacian operator Δ the Dirichlet energy D(c) on the sphere Σ = {c ∈ Mn ; M (c) = 1}. I.e. assume that u is a critical point of D with |u| = 1. Then u + εη d D( )|ε=0 = 2 < δμ u, δμ η > −2D(u) < u, η >= 0 dε |u + εη| ˆ = D(u)u. That is to say, u is an eigenvector of for any η ∈ Mn , which implies that Δu 2 ˆ Δ. Hence, to preserve the L norm, it is natural to study the evolution flow ˆ + λ(c)c ct = −Δc
(1.1)
with its initial matrix c|t=0 = c0 ∈ Mn , c0 = 0. Assume that c = c(t) is the solution to the flow above. Then d ˆ > +2D(c). M (c) =< ct , c > + < c, ct >= −2 < c, Δc dt
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
ˆ >= D(c), we know that Since < c, Δc
d dt M (c)
223
= 0. Then
M (c) = M (c0 ). Denote by c¯ = τ (c) and Mn /CI := {c ∈ Mn ; c¯ = 0}. The aim of this paper is to show that there is a global flow to (1.1) and the flow has many very nice properties like entropy stability. We also study the operator convex preserving property for the heat equation. Our main results, Theorem 2.1, Theorem 3.2, Theorem 4.2, and Theorem 5.1 are contained in Sections 2, 3, 4 and 5 respectively. One open question is posed in the last section. 2. Existence of the global flow Firstly, we consider the local existence of the flow (1.1). We prefer to follow the ˆ Let a = a(t) ∈ Mn /CI be such that standard notation and we let Δ = −Δ. at = Δa + λ(t)a,
(2.1)
with the initial matrix a|t=0 = a0 . Here a0 ∈ Mn /CI such that |a|2 (0) = |a0 |2 = 1. Then for a = a(t), we let λ(t) =
|δμ a|2 < Δa, a > . =− |a|2 < a, a >
(2.2)
Formally, if the flow (2.1) exists, then we compute that d 2 |a| =< a, at > + < at , a >= 2 < a, Δa > +2λ(t) < a, a >= 0. dt Then |a|2 (t) = |a|2 (0) = 1, ∀t > 0. Note that, d a ¯(t) =< Δa(t), 1 > +λ(t)¯ a(t) = λ(t)¯ a(t), dt which implies that a ¯(t) = 0, ∀t > 0. Our main goal in this section is to show that there is a global solution to equation (2.1) for any initial matrix a0 ∈ Mn /CI with |a0 |2 = 1. function and a = a(t) is the Assume at first that λ(t) ≥ 0 is any given continuous − 0t λ(s)ds corresponding solution of (2.1). Define b(t) = e a(t). Then b(0) = a(0) and we get t t d d b(t) = e− 0 λ(s)ds (−λ(t))a(t) + e− 0 λ(s)ds a(t) dt dt
= −λ(t)a(t)e− = e−
t 0
λ(s)ds
t 0
λ(s)ds
Δa(t)
+ e−
t 0
λ(s)ds
(Δa(t) + λ(t)a(t))
224
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
= Δ(e−
t 0
λ(s)ds
a(t))
= Δb.
(2.3)
We now solve the equation (2.3) by the iteration method [11]. Assume (ϕi ) and (λi ) are eigen-matrices and eigenvalues of Δ such that −Δϕi = λi ϕi , < ϕi , ϕj >= δij . Note that λi ≥ 0. Assume that b = b(t) is the solution to (2.3). Set b=
< b, ϕi > ϕi :=
ui ϕi , ui ∈ C, ui =: ui (t).
By (2.3), we obtain (ui )t ϕi = Δ(ui ϕi ) = −ui λi ϕi . Then (ui )t = −ui λi , and ui = ui (0)e−λi t . Hence, b=
ui (0)e−λi t ϕi ,
and a(t) =
t
ui (0)e−λi t+
0
λ(s)ds
ϕi
(2.4)
solves (2.1) with the given λ(t) ≥ 0. Next we define an iteration relation to solve (2.1) for the unknown λ(t) given by (2.2). 0 ,a0 > Define a1 such that a1 solves the equation at = Δa + λ0 a with λ0 = − <Δa . Let k ≥ 1 be any integer. Define ak+1 such that (ak+1 )t = Δak+1 + λk (t)ak+1 , ak+1 (0) = a0 ,
(2.5)
with λk (t) = −
|δμ ak |2 < Δak , ak > = . < ak , ak > |ak |2
Then using the formula (2.4), we get a sequence (ak ). Fix T > 0. Note that (2.5) is equivalent to the integral equation
t ak+1 (t) =
[Δak+1 (s) + λk (s)ak+1 (s)]ds + a0 , 0
(2.6)
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for t ∈ [0, T ] and any continuous solution to this integral equation is also in C 1 [0, T ]. We claim that (ak (t)) ⊂ Mn /CI, 0 ≤ t ≤ T is a bounded t-Lipschitz (i.e. C 1− ) sequence and (λk (t)) is also a bounded Lipschitz sequence. It is clear that (ak ) ⊂ Mn /CI. If this claim is true, using the Arzela–Ascoli theorem that a bounded C 1− sequence has a subconvergent sequence, we may assume ˜ ak (t) → a(t), λk (t) → λ(t) in C α (0, T ), for any 0 < α < 1 and the limit is still Lipschitz and C 1 . By (2.5) and (2.6), we obtain ˜ at = Δa + λ(t)a and ˜ = − < Δa, a > , λ(t) < a, a > which is the same as (2.1). That is to say, a = a(t) obtained above is the desired solution to (2.1) with the initial data a0 . Firstly we prove the claim in the interval [0, T ]. By the norm equivalence we have |δμ a|2 ≤ B|a − a ¯|2 for some uniform constant B > 0. Assume |ak | ≤ A := e2BT and |λk | ≤ B on [0, T ]. Then by (2.5), 1 |ak+1 |2t = < ak+1 , (ak+1 )t > 2 = < ak+1 , Δak+1 > +λk |ak+1 |2 = −|δak+1 |2 + λk |ak+1 |2 . By (2.6), we obtain λk+1 =
|δμ ak+1 |2 |ak+1 |2 .
Then
|δμ ak+1 |2 = λk+1 |ak+1 |2 . By (2.7), we get 1 |ak+1 |2t = −λk+1 |ak+1 |2 + λk |ak+1 |2 2 = (λk − λk+1 )|ak+1 |2 .
Then |ak+1 |2 = e2 (λk −λk+1 )dt |a0 |2 = e2Bt ≤ A. ¯0 = 0. Then Recall that a d (¯ ak+1 ) =< Δak+1 , 1 > +λk < ak+1 , 1 >= λk a ¯k+1 , dt
(2.7)
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
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so
a ¯k+1 = e
λk dt
a ¯k+1 (0) = e
λk dt
a0 = 0.
Note that since a ¯k+1 = 0 and norm equivalence relation, λk+1 |ak+1 |2 = |δμ ak+1 |2 ≤ B|ak+1 − a ¯k+1 |2 = B|ak+1 |2 . Then λk+1 ≤ B. Hence the claim is true in [0, T ]. Because T > 0 is arbitrary, we get a global solution to (2.1) with initial data a0 . In conclusion we have the below. Theorem 2.1. For any given initial matrix a0 ∈ Mn /CI with |a0 |2 = 1, the equation (2.1) ¯(t) = 0 for all t > 0. has a global solution with a0 as its initial data and |a(t)|2 = 1 and a 3. Convergence of the flow a = a(t) at ∞ We start with the following interesting property of the L2 norm preserving flow. Proposition 3.1. Assume τ (a0 ) = a ¯0 = 0. Then a ¯(t) = 0, ∀t > 0. Proof. d a ¯ = τ (at ) dt = τ (Δa + λ(t)a) = τ (Δa) + λ(t)τ (a) = λ(t)¯ a, so a ¯(t) = a ¯(0)e
t 0
λ(s)ds
= 0. 2
We now prove that the global flow in Mn /CI converges to some eigen-matrix. Theorem 3.2. For any given initial matrix a0 ∈ Mn /CI with |a0 |2 = 1, the global solution to the equation (2.1) with a0 as its initial data sub-converges to an eigen-matrix of Δ with non-zero eigenvalue. Recall that we have the global flow a = a(t) such that at = Δa + λ(t)a,
a ¯=0
with λ(t) = |δμ a|2 ≥ C −1 |a|2 = C −1 , a(t)|t=0 = a0 , |a0 |2 = 1 and |a(t)|2 = 1.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
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Set (Δa)⊥ = Δa− < Δa, a > a, where < (Δa)⊥ , a >= 0. Note that λ(t) = − < Δa, a >, so by the Schwartz inequality, we obtain λ(t) ≤ |Δa|. Compute d λ(t) = (< δμ at , δμ a > + < δμ a, δμ at >) dt μ = −2 < Δa, at > = −2 < Δa, Δa + λ(t)a > = −2(Δa)2 − 2λ(t) < Δa, a > = −2(Δa)2 + 2 < Δa, a >2 = −2|(Δa)⊥ |2 ≤ 0. Then for any T > s > 0,
T λ(T ) + 2
|(Δa)⊥ |2 (t)dt = λ(s) ≤ λ(0).
s
Hence, by |a(t)| = 1 and the monotonicity property above, we may assume that for some sequence tj → ∞, |(Δa)⊥ |2 (tj ) → 0, a(tj ) → a∞ and |a∞ |2 = 1, a ¯∞ = 0, λ∞ = limtj →∞ λ(tj ) and (Δa∞ )⊥ = 0. The latter condition implies that < (Δa∞ )⊥ , a∞ >= 0, which is −Δa∞ = − < Δa∞ , a∞ > a∞ = λ∞ a∞ . Since λ∞ ≥ C −1 , we know that λ∞ is the non-zero eigenvalue of −Δ and then λ∞ > 0. This completes the proof of Theorem 3.2. 4. Positive and entropy stability properties preserved by the flow In this section, we show that positivity of the initial matrix 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.
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Theorem 4.1. Assume a0 > 0, that is a0 is a Hermitian positive definite. Assume that a(t) is a solution to the flow equation at = Δa + λ(t)a with a(0) = a0 , where λ(t) is given by (2.2). Then a(t) > 0, ∀t > 0 along the flow. Proof. Again we have
t a(t) = a0 +
[Δa(s) + λ(s)a(s)]ds. 0
By continuity, we know that a = a(t) > 0 for small t > 0. Compute d log det a =< a−1 , at >=< a−1 , Δa > +nλ(t) dt where n = τ (1). Note that < a−1 (δμ a)a−1 , δμ a > = τ (a−1 (δμ a)∗ a−1 , δμ a) = τ (a− 2 (δμ a)∗ a− 2 a− 2 δμ aa− 2 ) 1
1
1
1
= |a− 2 (δμ a)a− 2 |2 > 0, 1
1
and < a−1 , Δa >= − < δμ (a−1 ), δμ a >=< a−1 δμ a · a−1 , δμ a >= |a−1 δμ a|2 . We know that d log det a = |a−1 δμ a|2 + nλ(t) ≥ nλ(t) ≥ 0. dt Hence, we have a = a(t) > 0, ∀t > 0. 2 Remark that by continuity, we can show that if a0 ≥ 0, then a(t) ≥ 0 along the flow (2.1). Next, we study the norm stability and entropy stability of the flow. Our method is similar to that of [10] and we shall just be brief. Recall that there is a uniform constant C > 0 such that for any u, v ∈ Mn and with |u|2 = 1 = |v|2 , |λ(u) − λ(v)| ≤ C|u − v|.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
229
Given two initial matrices u0 , v0 . Let u, v be the corresponding solutions to (2.1) with initial datum u0 , v0 . Then as in [10], we have |u − v|2 (t) ≤ eC1 t |u − v|2 (0) := C1 (t)|u − v|2
(4.1)
for some uniform constant C1 > 0, which implies the HS norm stability of the flow in (2.1) in the finite time interval [0, T ]. For the trace norm stability of solutions, we recall that by the spectral theorem, we may let u − v = Q − P , where Q and P are positive definite operators with orthogonal support. Then the trace norm of u − v is T (u, v) = τ (P ) + τ (Q). For the positive definite solution u(t) > 0 along the flow (2.1). Then we can define the von Neumann entropy [15] by S(u) = −τ (u log u) for u = u(t) with u(0) = u0 . Then with minor modification to the argument as in [10], it follows that Theorem 4.2. Fix t > 0. If T (u(0), v(0)) ≤ solutions u(t), v(t) to (2.1) satisfy
1 C1 (t)e ,
u(0) > 0, v(0) > 0 in Mn , then the
|S(u(t)) − S(v(t))| ≤ C1 (t)T (u, v)(0) log d + η(C1 (t)T (u, v)(0)), where C1 (t) is the uniform constant in (4.1). 5. Operator convexity of the heat equation In this section we prove some nonlinear convexity properties of the initial matrix are preserved along the heat flow (5.1) below at = Δa.
(5.1)
Here we have assumed that the solution is Hermitian symmetric, i.e. a∗ = a. Recall that we say f : Mn → Mn is operator convex if for any Hermitian symmetric matrices A and B, we have μf (A) + (1 − μ)f (B) − f (μA + (1 − μ)B) ≥ 0,
for μ ∈ (0, 1).
230
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
Let f (x) = x2 . Then we compute, Δa2 = δμ (δμ a2 ) = δμ (δμ a · a + aδμ a) = δμ2 a · a + 2(δμ a)2 + aδμ2 a = Δa · a + 2(δμ a)2 + aΔa and ∂t (a2 ) = aat + at a. By (∂t − Δ)a = 0, we obtain (∂t − Δ)a2 = −2(δμ a)2 . Note that τ (a−2 Δa2 ) ≥ 0. Hence, we have d log det(a2 ) = τ (a−2 (a2 )t ) dt = τ (a−2 Δa2 ) − 2τ (a−2 (δμ a)2 ) = τ (a−2 Δa2 ) + 2τ (a−1 δμ a · (δμ a)∗ a−1 ) = τ (a−2 Δa2 ) + 2τ (a−1 δμ a · (a−1 δμ a)∗ ) ≥ 0. Then we deduce that a2 > 0 provided a20 > 0. Assume that λ > 0 is any positive constant. Recall that (λ + a)−1 > 0 if and only if λ + a > 0. Note that, by λ + a0 > 0 and applying Theorem 3.1 in [10] to a + λ which satisfies (5.1), we have λ + a > 0. Then (λ + a)−1 > 0, ∀t > 0. Thus we have proved the below. Theorem 5.1. If f is either given by f (a) = a2 or f (a) = (λ + a)−1 with λ > 0, and if f (a0 ) > 0, then f (a) > 0, ∀t > 0. We conjecture that for f being operator convex, if f (a0 ) > 0, then f (a) > 0, ∀t > 0 along the flow. The reason is given as follows. According to [9], we have that any continuous operator convex real function on [0, ∞] can be expressed as
f (x) = f (0) + ax + bx2 +
∞ λ x −1+ )dμ (λ), ( 1+λ x+λ 0
where a, b ≥ 0, dμ is a nonnegative measure.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
231
1 Since for f (x) = x, x2 , x+λ , we already know that if f (a0 ) > 0, then f (a) > 0, ∀t > 0. Hence, for any Hermitian positive definite matrix a0 > 0 with f (a0 ) > 0, then
f (a) > 0, ∀t > 0. However, we cannot give a rigorous proof and we leave it as an open problem. Acknowledgements The author would like to thank Prof. Li Ma for helpful suggestions and discussion. The author would like to thank the referees very much for suggestions. References [1] L. Caffarelli, F.H. Lin, An optimal partition problem for eigenvalues, J. Sci. Comput. 31 (2007) 5–18. [2] L. Caffarelli, F.H. Lin, Non-local heat flows preserving the L2 energy, Discrete Contin. Dyn. Syst. 23 (2009) 49–64. [3] B. Li, Z.H. Yu, S.M. Fei, X.Q. Li-Jost, Time optimal quantum control of two-qubit systems, Sci. China Ser. G 56 (2013) 2116–2121. [4] H. Zhao, X.H. Zhang, S.M. Fei, Z.X. Wang, Characterization of four-qubit states via Bell inequalities, Chin. Sci. Bull. 58 (2013) 2334–2339. [5] L. Zhang, S.M. Fei, J. Zhu, Unifying treatment of discord via relative entropy, Internat. J. Theoret. Phys. 52 (2013) 1946–1955. [6] R. Duvenhage, Noncommutative Ricci flow in a matrix geometry, J. Phys. A 47 (2014) 045203. [7] X. Dai, L. Ma, Mass under Ricci flow, Comm. Math. Phys. 274 (2007) 65–80. [8] R. Hamilton, The Ricci flow on surfaces, in: Mathematics and General Relativity: Proceedings of the AMSCIMSCSIAM Joint Summer Research Conference, in: Contemp. Math., vol. 71, American Mathematical Society, Providence, RI, 1988, pp. 237–262. [9] I.H. Kim, Modulus of convexity for operator convex functions, J. Math. Phys. 55 (2014) 082201. [10] J.J. Li, Heat equation in a model matrix geometry, C. R. Math. Acad. Sci. Paris 353 (4) (2015) 351–355. 58J35 (58B34). [11] L. Ma, L. Cheng, Non-local heat flows and gradient estimates on closed manifolds, J. Evol. Equ. 9 (2009) 787–807. [12] L. Ma, L. Cheng, A non-local area preserving curve flow, Geom. Dedicata 171 (1) (2013) 231–247. [13] L. Ma, A.Q. Zhu, On a length preserving curve flow, Monatsh. Math. 165 (2012) 57–78. [14] J. Madore, An Introduction to Noncommutative Differential Geometry and Its Physical Applications, second edition, Cambridge University Press, Cambridge, 1999. [15] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, 2004.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
L2 norm preserving flow in matrix geometry ✩ Jiaojiao Li Department of Mathematics, Henan Normal University, Xinxiang, 453007, China
a r t i c l e
i n f o
Article history: Received 9 April 2015 Accepted 4 September 2015 Available online 25 September 2015 Submitted by R. Brualdi MSC: 58B34 15A57
a b s t r a c t In this paper, we study a new L2 norm preserving heat flow in matrix geometry. We show that if the initial data has trace zero and has unit L2 norm, this flow has a global solution and enjoys the entropy stability in any finite time. We show that as the time is approaching infinity, the flow has its limit as an eigen-matrix of the Laplacian operator. Interesting operator convex property of heat equation is also derived. © 2015 Elsevier Inc. All rights reserved.
Keywords: Global flow Norm conservation Entropy stability Operator convexity
1. Introduction In this paper we continue our study of evolution equation in matrix geometry [10]. We introduce the L2 norm preserving flow such that starting from any initial data with unit L2 norm and trace zero, we can produce a family of matrices of unit L2 norm such that their limit is an eigen-matrix of the Laplacian operator introduced in [10] and [6]. In [6], the author introduced the Ricci flow which exists globally when the initial matrix is positive definite. The Ricci flow preserves the trace of the initial matrix and the flow ✩ The research is partially supported by the National Natural Science Foundation of China (No. 11271111) and NSF of Henan Province No. 132300410052. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.laa.2015.09.011 0024-3795/© 2015 Elsevier Inc. All rights reserved.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
221
converges the scalar matrix with the same trace as the initial matrix. In [10], we have introduced the heat equation, which also preserves the trace of the initial matrix. The advantage of the heat equation is that the corresponding flow not only preserves the trace but also can be stated with any initial matrix. In [1,2,11], the authors introduced the norm preserving flows which are global flows and converge to eigenfunctions. In quantum information theory, we can see the interesting works [3–5,15], but we need evolution flows which preserve some norms like the L2 norm just like the normalized Ricci flow [8,7] preserves the volume and the curve flows [11–13] preserve length or area. This is the motivation of the study of norm preserving flow in matrix geometry. More precisely, we study again the matrix geometry model as in [10,6,14]. We consider two unitary n × n matrices u and v satisfying the commutation relation vu = quv, where q=e
2πim n
for an m ∈ {1, 2, · · · , n − 1}, such that m and n are relatively prime: note in particular that q n = 1, but q j = 1 for j = 1, · · · , n − 1. Concretely we can use ⎡ ⎢ ⎢ u=⎢ ⎢ ⎣
⎤
1
⎢ ⎥ ⎢ ⎥ ⎥ and v = ⎢ ⎢ ⎥ ⎣ ⎦
q ..
⎡
. q n−1
01
⎤
⎥ . 0 .. ⎥ ⎥ ⎥ .. . 1⎦ 1 0
where the blanks spaces are filled with zeroes. It is pointed out in [6] that v is obtained by Fourier transform of Zn = {0, 1, · · · , n − 1} from u, i.e. v = F ∗ uF , where F is the Fourier transform on Zn in a suitable basis, namely the n × n unitary matrix Fjk = q −jk /n1/2 for j, k = 0, 1, · · · , n − 1, and F ∗ denotes its Hermitian adjoint. It is also said that u and v generate the algebra Mn of all n × n complex matrices, namely the commutant of [u, v] consists of scalar multiples of the identity matrix, [u, v] = C1, where here and later we denote the n × n identity matrix simply as 1, since no confusion will arise. 2πi 2πi Let x, y be two Hermitian matrices on Cn . Define u = e n x , v = e n y . We use Mn to denote the algebra of all n × n complex matrices which generated by u and v with the bracket [u, v] = uv − vu. Then CI, which is the scalar multiples of the identity matrix I, is the commutant of the set {u, v}. We define two derivations δ1 and δ2 on the algebra Mn by the commutators δ1 := [y, ·], δ2 := −[x, ·].
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
222
Define the Laplacian operator on Mn by ˆ = δ ∗ δ 1 + δ ∗ δ 2 = δ 2 + δ 2 = δμ δ μ , Δ 1 2 1 2 where we have used the Einstein sum convention. We use the Hilbert–Schmidt norm | · | defined by the inner product < a, b >:= τ (a∗ b) on the algebra Mn . Here a∗ is the Hermitian adjoint of the matrix a and τ denotes the ˆ (see also [6]) usual trace function on Mn . We now state basic properties of δ1 , δ2 and Δ as follows. For any c ∈ Mn , we define the Dirichlet energy D(c) =
< δμ c, δμ c >
and the L2 mass M (c) =< c, c > . Let, for c = 0, λ(c) =
D(c) . M (c)
ˆ correspond to the critical values of Then the eigenvalues of the Laplacian operator Δ the Dirichlet energy D(c) on the sphere Σ = {c ∈ Mn ; M (c) = 1}. I.e. assume that u is a critical point of D with |u| = 1. Then u + εη d D( )|ε=0 = 2 < δμ u, δμ η > −2D(u) < u, η >= 0 dε |u + εη| ˆ = D(u)u. That is to say, u is an eigenvector of for any η ∈ Mn , which implies that Δu 2 ˆ Δ. Hence, to preserve the L norm, it is natural to study the evolution flow ˆ + λ(c)c ct = −Δc
(1.1)
with its initial matrix c|t=0 = c0 ∈ Mn , c0 = 0. Assume that c = c(t) is the solution to the flow above. Then d ˆ > +2D(c). M (c) =< ct , c > + < c, ct >= −2 < c, Δc dt
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
ˆ >= D(c), we know that Since < c, Δc
d dt M (c)
223
= 0. Then
M (c) = M (c0 ). Denote by c¯ = τ (c) and Mn /CI := {c ∈ Mn ; c¯ = 0}. The aim of this paper is to show that there is a global flow to (1.1) and the flow has many very nice properties like entropy stability. We also study the operator convex preserving property for the heat equation. Our main results, Theorem 2.1, Theorem 3.2, Theorem 4.2, and Theorem 5.1 are contained in Sections 2, 3, 4 and 5 respectively. One open question is posed in the last section. 2. Existence of the global flow Firstly, we consider the local existence of the flow (1.1). We prefer to follow the ˆ Let a = a(t) ∈ Mn /CI be such that standard notation and we let Δ = −Δ. at = Δa + λ(t)a,
(2.1)
with the initial matrix a|t=0 = a0 . Here a0 ∈ Mn /CI such that |a|2 (0) = |a0 |2 = 1. Then for a = a(t), we let λ(t) =
|δμ a|2 < Δa, a > . =− |a|2 < a, a >
(2.2)
Formally, if the flow (2.1) exists, then we compute that d 2 |a| =< a, at > + < at , a >= 2 < a, Δa > +2λ(t) < a, a >= 0. dt Then |a|2 (t) = |a|2 (0) = 1, ∀t > 0. Note that, d a ¯(t) =< Δa(t), 1 > +λ(t)¯ a(t) = λ(t)¯ a(t), dt which implies that a ¯(t) = 0, ∀t > 0. Our main goal in this section is to show that there is a global solution to equation (2.1) for any initial matrix a0 ∈ Mn /CI with |a0 |2 = 1. function and a = a(t) is the Assume at first that λ(t) ≥ 0 is any given continuous − 0t λ(s)ds corresponding solution of (2.1). Define b(t) = e a(t). Then b(0) = a(0) and we get t t d d b(t) = e− 0 λ(s)ds (−λ(t))a(t) + e− 0 λ(s)ds a(t) dt dt
= −λ(t)a(t)e− = e−
t 0
λ(s)ds
t 0
λ(s)ds
Δa(t)
+ e−
t 0
λ(s)ds
(Δa(t) + λ(t)a(t))
224
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
= Δ(e−
t 0
λ(s)ds
a(t))
= Δb.
(2.3)
We now solve the equation (2.3) by the iteration method [11]. Assume (ϕi ) and (λi ) are eigen-matrices and eigenvalues of Δ such that −Δϕi = λi ϕi , < ϕi , ϕj >= δij . Note that λi ≥ 0. Assume that b = b(t) is the solution to (2.3). Set b=
< b, ϕi > ϕi :=
ui ϕi , ui ∈ C, ui =: ui (t).
By (2.3), we obtain (ui )t ϕi = Δ(ui ϕi ) = −ui λi ϕi . Then (ui )t = −ui λi , and ui = ui (0)e−λi t . Hence, b=
ui (0)e−λi t ϕi ,
and a(t) =
t
ui (0)e−λi t+
0
λ(s)ds
ϕi
(2.4)
solves (2.1) with the given λ(t) ≥ 0. Next we define an iteration relation to solve (2.1) for the unknown λ(t) given by (2.2). 0 ,a0 > Define a1 such that a1 solves the equation at = Δa + λ0 a with λ0 = − <Δa
(2.5)
with λk (t) = −
|δμ ak |2 < Δak , ak > = . < ak , ak > |ak |2
Then using the formula (2.4), we get a sequence (ak ). Fix T > 0. Note that (2.5) is equivalent to the integral equation
t ak+1 (t) =
[Δak+1 (s) + λk (s)ak+1 (s)]ds + a0 , 0
(2.6)
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
225
for t ∈ [0, T ] and any continuous solution to this integral equation is also in C 1 [0, T ]. We claim that (ak (t)) ⊂ Mn /CI, 0 ≤ t ≤ T is a bounded t-Lipschitz (i.e. C 1− ) sequence and (λk (t)) is also a bounded Lipschitz sequence. It is clear that (ak ) ⊂ Mn /CI. If this claim is true, using the Arzela–Ascoli theorem that a bounded C 1− sequence has a subconvergent sequence, we may assume ˜ ak (t) → a(t), λk (t) → λ(t) in C α (0, T ), for any 0 < α < 1 and the limit is still Lipschitz and C 1 . By (2.5) and (2.6), we obtain ˜ at = Δa + λ(t)a and ˜ = − < Δa, a > , λ(t) < a, a > which is the same as (2.1). That is to say, a = a(t) obtained above is the desired solution to (2.1) with the initial data a0 . Firstly we prove the claim in the interval [0, T ]. By the norm equivalence we have |δμ a|2 ≤ B|a − a ¯|2 for some uniform constant B > 0. Assume |ak | ≤ A := e2BT and |λk | ≤ B on [0, T ]. Then by (2.5), 1 |ak+1 |2t = < ak+1 , (ak+1 )t > 2 = < ak+1 , Δak+1 > +λk |ak+1 |2 = −|δak+1 |2 + λk |ak+1 |2 . By (2.6), we obtain λk+1 =
|δμ ak+1 |2 |ak+1 |2 .
Then
|δμ ak+1 |2 = λk+1 |ak+1 |2 . By (2.7), we get 1 |ak+1 |2t = −λk+1 |ak+1 |2 + λk |ak+1 |2 2 = (λk − λk+1 )|ak+1 |2 .
Then |ak+1 |2 = e2 (λk −λk+1 )dt |a0 |2 = e2Bt ≤ A. ¯0 = 0. Then Recall that a d (¯ ak+1 ) =< Δak+1 , 1 > +λk < ak+1 , 1 >= λk a ¯k+1 , dt
(2.7)
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
226
so
a ¯k+1 = e
λk dt
a ¯k+1 (0) = e
λk dt
a0 = 0.
Note that since a ¯k+1 = 0 and norm equivalence relation, λk+1 |ak+1 |2 = |δμ ak+1 |2 ≤ B|ak+1 − a ¯k+1 |2 = B|ak+1 |2 . Then λk+1 ≤ B. Hence the claim is true in [0, T ]. Because T > 0 is arbitrary, we get a global solution to (2.1) with initial data a0 . In conclusion we have the below. Theorem 2.1. For any given initial matrix a0 ∈ Mn /CI with |a0 |2 = 1, the equation (2.1) ¯(t) = 0 for all t > 0. has a global solution with a0 as its initial data and |a(t)|2 = 1 and a 3. Convergence of the flow a = a(t) at ∞ We start with the following interesting property of the L2 norm preserving flow. Proposition 3.1. Assume τ (a0 ) = a ¯0 = 0. Then a ¯(t) = 0, ∀t > 0. Proof. d a ¯ = τ (at ) dt = τ (Δa + λ(t)a) = τ (Δa) + λ(t)τ (a) = λ(t)¯ a, so a ¯(t) = a ¯(0)e
t 0
λ(s)ds
= 0. 2
We now prove that the global flow in Mn /CI converges to some eigen-matrix. Theorem 3.2. For any given initial matrix a0 ∈ Mn /CI with |a0 |2 = 1, the global solution to the equation (2.1) with a0 as its initial data sub-converges to an eigen-matrix of Δ with non-zero eigenvalue. Recall that we have the global flow a = a(t) such that at = Δa + λ(t)a,
a ¯=0
with λ(t) = |δμ a|2 ≥ C −1 |a|2 = C −1 , a(t)|t=0 = a0 , |a0 |2 = 1 and |a(t)|2 = 1.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
227
Set (Δa)⊥ = Δa− < Δa, a > a, where < (Δa)⊥ , a >= 0. Note that λ(t) = − < Δa, a >, so by the Schwartz inequality, we obtain λ(t) ≤ |Δa|. Compute d λ(t) = (< δμ at , δμ a > + < δμ a, δμ at >) dt μ = −2 < Δa, at > = −2 < Δa, Δa + λ(t)a > = −2(Δa)2 − 2λ(t) < Δa, a > = −2(Δa)2 + 2 < Δa, a >2 = −2|(Δa)⊥ |2 ≤ 0. Then for any T > s > 0,
T λ(T ) + 2
|(Δa)⊥ |2 (t)dt = λ(s) ≤ λ(0).
s
Hence, by |a(t)| = 1 and the monotonicity property above, we may assume that for some sequence tj → ∞, |(Δa)⊥ |2 (tj ) → 0, a(tj ) → a∞ and |a∞ |2 = 1, a ¯∞ = 0, λ∞ = limtj →∞ λ(tj ) and (Δa∞ )⊥ = 0. The latter condition implies that < (Δa∞ )⊥ , a∞ >= 0, which is −Δa∞ = − < Δa∞ , a∞ > a∞ = λ∞ a∞ . Since λ∞ ≥ C −1 , we know that λ∞ is the non-zero eigenvalue of −Δ and then λ∞ > 0. This completes the proof of Theorem 3.2. 4. Positive and entropy stability properties preserved by the flow In this section, we show that positivity of the initial matrix 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.
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228
Theorem 4.1. Assume a0 > 0, that is a0 is a Hermitian positive definite. Assume that a(t) is a solution to the flow equation at = Δa + λ(t)a with a(0) = a0 , where λ(t) is given by (2.2). Then a(t) > 0, ∀t > 0 along the flow. Proof. Again we have
t a(t) = a0 +
[Δa(s) + λ(s)a(s)]ds. 0
By continuity, we know that a = a(t) > 0 for small t > 0. Compute d log det a =< a−1 , at >=< a−1 , Δa > +nλ(t) dt where n = τ (1). Note that < a−1 (δμ a)a−1 , δμ a > = τ (a−1 (δμ a)∗ a−1 , δμ a) = τ (a− 2 (δμ a)∗ a− 2 a− 2 δμ aa− 2 ) 1
1
1
1
= |a− 2 (δμ a)a− 2 |2 > 0, 1
1
and < a−1 , Δa >= − < δμ (a−1 ), δμ a >=< a−1 δμ a · a−1 , δμ a >= |a−1 δμ a|2 . We know that d log det a = |a−1 δμ a|2 + nλ(t) ≥ nλ(t) ≥ 0. dt Hence, we have a = a(t) > 0, ∀t > 0. 2 Remark that by continuity, we can show that if a0 ≥ 0, then a(t) ≥ 0 along the flow (2.1). Next, we study the norm stability and entropy stability of the flow. Our method is similar to that of [10] and we shall just be brief. Recall that there is a uniform constant C > 0 such that for any u, v ∈ Mn and with |u|2 = 1 = |v|2 , |λ(u) − λ(v)| ≤ C|u − v|.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
229
Given two initial matrices u0 , v0 . Let u, v be the corresponding solutions to (2.1) with initial datum u0 , v0 . Then as in [10], we have |u − v|2 (t) ≤ eC1 t |u − v|2 (0) := C1 (t)|u − v|2
(4.1)
for some uniform constant C1 > 0, which implies the HS norm stability of the flow in (2.1) in the finite time interval [0, T ]. For the trace norm stability of solutions, we recall that by the spectral theorem, we may let u − v = Q − P , where Q and P are positive definite operators with orthogonal support. Then the trace norm of u − v is T (u, v) = τ (P ) + τ (Q). For the positive definite solution u(t) > 0 along the flow (2.1). Then we can define the von Neumann entropy [15] by S(u) = −τ (u log u) for u = u(t) with u(0) = u0 . Then with minor modification to the argument as in [10], it follows that Theorem 4.2. Fix t > 0. If T (u(0), v(0)) ≤ solutions u(t), v(t) to (2.1) satisfy
1 C1 (t)e ,
u(0) > 0, v(0) > 0 in Mn , then the
|S(u(t)) − S(v(t))| ≤ C1 (t)T (u, v)(0) log d + η(C1 (t)T (u, v)(0)), where C1 (t) is the uniform constant in (4.1). 5. Operator convexity of the heat equation In this section we prove some nonlinear convexity properties of the initial matrix are preserved along the heat flow (5.1) below at = Δa.
(5.1)
Here we have assumed that the solution is Hermitian symmetric, i.e. a∗ = a. Recall that we say f : Mn → Mn is operator convex if for any Hermitian symmetric matrices A and B, we have μf (A) + (1 − μ)f (B) − f (μA + (1 − μ)B) ≥ 0,
for μ ∈ (0, 1).
230
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
Let f (x) = x2 . Then we compute, Δa2 = δμ (δμ a2 ) = δμ (δμ a · a + aδμ a) = δμ2 a · a + 2(δμ a)2 + aδμ2 a = Δa · a + 2(δμ a)2 + aΔa and ∂t (a2 ) = aat + at a. By (∂t − Δ)a = 0, we obtain (∂t − Δ)a2 = −2(δμ a)2 . Note that τ (a−2 Δa2 ) ≥ 0. Hence, we have d log det(a2 ) = τ (a−2 (a2 )t ) dt = τ (a−2 Δa2 ) − 2τ (a−2 (δμ a)2 ) = τ (a−2 Δa2 ) + 2τ (a−1 δμ a · (δμ a)∗ a−1 ) = τ (a−2 Δa2 ) + 2τ (a−1 δμ a · (a−1 δμ a)∗ ) ≥ 0. Then we deduce that a2 > 0 provided a20 > 0. Assume that λ > 0 is any positive constant. Recall that (λ + a)−1 > 0 if and only if λ + a > 0. Note that, by λ + a0 > 0 and applying Theorem 3.1 in [10] to a + λ which satisfies (5.1), we have λ + a > 0. Then (λ + a)−1 > 0, ∀t > 0. Thus we have proved the below. Theorem 5.1. If f is either given by f (a) = a2 or f (a) = (λ + a)−1 with λ > 0, and if f (a0 ) > 0, then f (a) > 0, ∀t > 0. We conjecture that for f being operator convex, if f (a0 ) > 0, then f (a) > 0, ∀t > 0 along the flow. The reason is given as follows. According to [9], we have that any continuous operator convex real function on [0, ∞] can be expressed as
f (x) = f (0) + ax + bx2 +
∞ λ x −1+ )dμ (λ), ( 1+λ x+λ 0
where a, b ≥ 0, dμ is a nonnegative measure.
J. Li / Linear Algebra and its Applications 487 (2015) 220–231
231
1 Since for f (x) = x, x2 , x+λ , we already know that if f (a0 ) > 0, then f (a) > 0, ∀t > 0. Hence, for any Hermitian positive definite matrix a0 > 0 with f (a0 ) > 0, then
f (a) > 0, ∀t > 0. However, we cannot give a rigorous proof and we leave it as an open problem. Acknowledgements The author would like to thank Prof. Li Ma for helpful suggestions and discussion. The author would like to thank the referees very much for suggestions. References [1] L. Caffarelli, F.H. Lin, An optimal partition problem for eigenvalues, J. Sci. Comput. 31 (2007) 5–18. [2] L. Caffarelli, F.H. Lin, Non-local heat flows preserving the L2 energy, Discrete Contin. Dyn. Syst. 23 (2009) 49–64. [3] B. Li, Z.H. Yu, S.M. Fei, X.Q. Li-Jost, Time optimal quantum control of two-qubit systems, Sci. China Ser. G 56 (2013) 2116–2121. [4] H. Zhao, X.H. Zhang, S.M. Fei, Z.X. Wang, Characterization of four-qubit states via Bell inequalities, Chin. Sci. Bull. 58 (2013) 2334–2339. [5] L. Zhang, S.M. Fei, J. Zhu, Unifying treatment of discord via relative entropy, Internat. J. Theoret. Phys. 52 (2013) 1946–1955. [6] R. Duvenhage, Noncommutative Ricci flow in a matrix geometry, J. Phys. A 47 (2014) 045203. [7] X. Dai, L. Ma, Mass under Ricci flow, Comm. Math. Phys. 274 (2007) 65–80. [8] R. Hamilton, The Ricci flow on surfaces, in: Mathematics and General Relativity: Proceedings of the AMSCIMSCSIAM Joint Summer Research Conference, in: Contemp. Math., vol. 71, American Mathematical Society, Providence, RI, 1988, pp. 237–262. [9] I.H. Kim, Modulus of convexity for operator convex functions, J. Math. Phys. 55 (2014) 082201. [10] J.J. Li, Heat equation in a model matrix geometry, C. R. Math. Acad. Sci. Paris 353 (4) (2015) 351–355. 58J35 (58B34). [11] L. Ma, L. Cheng, Non-local heat flows and gradient estimates on closed manifolds, J. Evol. Equ. 9 (2009) 787–807. [12] L. Ma, L. Cheng, A non-local area preserving curve flow, Geom. Dedicata 171 (1) (2013) 231–247. [13] L. Ma, A.Q. Zhu, On a length preserving curve flow, Monatsh. Math. 165 (2012) 57–78. [14] J. Madore, An Introduction to Noncommutative Differential Geometry and Its Physical Applications, second edition, Cambridge University Press, Cambridge, 1999. [15] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, 2004.