# An inequality on density matrix and its a generalization.

[section]1. Introduction

In a quantum system, the states are represented by density matrices, which are positive semi-definite matrices of trace 1 [1-4]. Recently, P. F. Renaud in  proved a matrix formulation of Gruss inequality which says that if A and B are n x n complex matrices and the numerical ranges W(A) and W(B) are contained in disks of radii R and S, respectively, then for every n x n density matrix T, the following inequality holds

[absolute value of Tr(TAB) - Tr(TA)Tr(TB)] [less than or equal to] 4RS. (1.1)

In this note, we introduce a generalized trace from a *-ideal I of a unital [C.sup.*]-algebra A into a unital Abelian [C.sup.*]-algebra B and prove some properties on it. Our main results extends the Gruss-type inequality (1.1).

Throughout this note, we assume that A is a unital [C.sup.*]-algebra with unit [1.sub.A] and I is a *-ideal of A, [I.sup.+] is the set of all positive elements in I and U (A) is the group of unitary elements of A. N denotes the set of all positive integers and C is the complex field. B is an unital Abelian [C.sup.*]-algebra with unit 1B and [OMEGA](B) is the character space of B that is the space of all characters of B. For every element b in B, put [absolute value of b] = [([b.sup.*]6).sup.1/2], called the absolute value of b. Let U : A [right arrow] B(H) be the universal representation. For a in A, define the numerical range W(a) of a to be that of operator U(a), that is,

W(a) = {{U(a)x,x) : x [member of] H, [parallel]x[parallel] = 1}.

From , we have

1 [parallel]a[parallel][less than or equal to] [omega](a) [less than or equal to] [parallel]a[parallel], [for all]a [member of] A, (1.2)

where [omega](a) := sup{[absolute value of ]{U(a)x,x)] : x [member of] H, [parallel]x[parallel] = 1}, called the numerical radius of a. It is known that if a is normal (i.e. [aa.sup.*] = [a.sup.*]a), then [omega](a) = [parallel]a[parallel].

Definition 1.1. A linear mapping [tau] : I [right arrow] B is called a generalized trace if it satisfies the following conditions:

(T1) [tau] is positive, i.e., a [member of] [I.sup.+] [??] [tau](a) [greater than or equal to] 0;

(T2) [tau] is unitary-stable, i.e., [for all]u [member of] U(A) and [for all]a [member of] I,

[tau]([u.sup.*]au) = [tau] (a);

(T3) [tau] is *-preserving, i.e., [for all]a [member of] I, [tau]([a.sup.*]) = ([tau][(a)).sup.*];

(T4) [tau] is sub-Jordan, i.e., [for all]a [member of] I, [tau]([a.sup.2]) [less than or equal to] ([tau][(a)).sup.2].

Clearly, if H is a Hilbert space over C, A = B(H) (the [C.sup.*]-algebra of all bounded linear operators on H), I = T(H) (the *-deal of all trace-class operators on H) or A = I = [M.sub.n] (C), then the usual trace Tr : I [right arrow] C is a generalized trace. To give a non-trivial example of generalized trace, let us consider such a [C.sup.*]-algebra A which has proper commutator ideal k(A) that is the closed ideal of Agenerated by all commutators ab - ba for all a and b in A. It is known that the Toeplitz algebra is such an algebra [6, pp. 102]. In this case, the quotient algebra A/k(A) is a nonzero unital Abelian [C.sup.*]-algebra. Let [pi] : A [right arrow] A/k(A) be the quotient homomorphism and [pi] be any character on A/k(A). Then for every x in A/k(A) with x [greater than or equal to] 1, the mapping

[tau] : A [right arrow] B, [tau](a) := [phi](x * [pi](a))[1.sub.B]

is a generalized trace. Moreover, if x is the unit of A/k(A) and t is a trace element of A (i.e., t [greater than or equal to] 0 and [tau](t) = [1.sub.B]), then

0 [less than or equal to] [tau](ta) [less than or equal to] [parallel]a[parallel][1.sub.B], [for all]a [greater than or equal to] 0. (1.3)

Another example is the matrix-trace tr from [C.sup.*]-algebra [M.sub.n](A)into A, see  for the details.

[section]2. Main results

In the sequel, let us assume that [tau]: I [right arrow] B is a generalized trace. With these notations, we have the following main results.

Proposition 2.1.

(a) [for all]a [member of] I, [for all]b [member of] A and n [member of] N, we have [tau]([(ab).sup.n]) = [tau]([(ba).sup.n]),

(b) [tau]([(ab.sup.)n]) [greater than or equal to] 0, [for all]a, b [member of] [I.sup.+], n [member of] N,

(c) [[absolute value of [tau]([b.sup.*]a)].sup.2] [less than or equal to] [tau]([a.sup.*]a)[tau]([b.sup.*]b), [for all]a,b [member of] I,

(d) 0 [less than or equal to] [tau](ab) [less than or equal to] [tau](a)[tau](b), [for all]a, b [member of] [I.sup.+],

(e) 0 [less than or equal to] [tau]([a.sup.n]) [less than or equal to] ([tau][(a)).sup.n], [for all]a [member of] [I.sup.+], n [member of] N.

Proof. (a) [for all]u [member of] U(A) and [for all]a [member of] I, from the property (T2) we have

[tau](ua) = [tau]([u.sup.*](ua)u) = [tau](au).

It follows from this and the property A =span(U(A)) that the statement (a) is true for n = 1. Thus, [for all]a [member of] I, [for all]b [member of] A and n [member of] N, we obtain

[tau]([(ab).sup.n]) = [tau]((a * b[(ab).sup.n-1]) = [tau](b[(ab).sup.n-1] * a) = [tau]([(ba).sup.n]).

(b) It is easy to see that [for all]a, b [member of] [I.sup.+] and [for all]n [member of] N, [(ab).sup.n-1] a is in [I.sup.+]. Thus, by (a) and (T1) we have

[tau]([(ab).sup.n]) = [tau]([b.sup.1/2] * [(ab).sup.n-1] a * [b.sup.1/2]) [greater than or equal to] 0.

(c) Let [phi] be an arbitrary element of [OMEGA](B) and define <a, b> = [phi]([tau]([b.sup.*]a)). Then we obtain asemi-inner product on I. Thus, [for all]a, b [member of] I, we have [[absolute value of <a,b>].sup.2] [less than or equal to] <a, a><b, b>, i.e.,

[[absolute value of [phi]([tau]([b.sup.*]a))].sup.2] [less than or equal to] [phi]([tau]([a.sup.*]a))[phi]([tau]([b.sup.*]b)),

thus,

[phi][[absolute value of [tau]([b.sup.*]a)].sup.2]) [less than or equal to] [phi]([tau]([a.sup.*]a)[tau]([b.sup.*]b)).

This shows the desired inequality.

(d) [for all]a, b [member of] [I.sup.+], we see from (b), (c) and (T4) that

0 [less than or equal to] [tau](ab) [less than or equal to] [[[tau]([a.sup.*]a)[tau]([b.sup.*)b) ].sup.1/2] [less than or equal to] [tau](a)[tau](b).

(e) It is from (d). The proof is then completed.

Theorem 2.2. Let t be a trace element of I (i.e., t [greater than or equal to] 0 and [tau](t) = [1.sub.B]) satisfying (1.3), then

(a) For all a and b in A,

[[absolute value of [tau]([tab.sup.*])].sup.2] [less than or equal to] [tau]([taa.sup.*])[tau]([tbb.sup.*]). (2.1)

(b) For all a, b [member of] A, we have

[[absolute value of [tau](tab) - [tau](ta)[tau](tb)] [less than or equal to] dist(a, [C1.sub.A]) * dist(b, [C1.sub.A]) * [1.sub.B] [less than or equal to] [parallel]a[parallel] [parallel]b[parallel][1.sub.B]. (2.2)

(c) If a, b [member of] A such that W(a), W(b) are contained in the disks of radii r, s, respectively, then we have

[[absolute value of [tau](tab) - [tau](ta)[tau](tb)] [less than or equal to] 4rs * [1.sub.B]. (2.3)

In addition, if a, b are normal, then

[[absolute value of [tau](tab) - [tau](ta)[tau](tb)] [less than or equal to] rs * [1.sub.B]. (2.4)

Proof. Note that for any x and y in B, x [less than or equal to] y if and only if [phi](x) [less than or equal to] [phi](y)([for all][phi] [member of] [OMEGA](B). Let [phi] [member of] 0(B), define {*, *) : A x A [right arrow] C as

<x,y> := [phi]([tau]([txy.sup.*])). (2.5)

Thus, we have the following Cauchy-Schwarz inequality: [for all]x, y [member of] A,

[[absolute value of <x,y>].sup.2] [less than or equal to] <x, x> <y,y>. (2.6)

Especially, we have [for all]x [member of] A,

[[absolute value of <x, [1.sub.A]>].sup.2] [less than or equal to] <x,x><[l.sub.A], [1.sub.A]> = <x,x>. (2.7)

(a) [for all][phi][member of][OMEGA](B), from (2.6) we obtain

[phi]([[absolute value of [tau]([tab.sup.*])].sup.2]) = [[absolute value of <a,b>].sup.2] [less than or equal to] <a,a><b,b>

= [phi]([tau]([taa.sup.*]))[phi]([tau]([tbb.sup.*]))

= [phi]([tau]([taa.sup.*])[tau]([tbb.sup.*])).

Hence, (2.1) has been proved.

(b) Fixed a [phi] [member of] [OMEGA](B). It suffices to prove that [for all][lambda], [mu] [member of] C,

[phi]([absolute value of ][tau](tab)) - [tau](ta)[tau](tb)]) [less than or equal to] [parallel]a [lambda][parallel][parallel]b - [mu][parallel]. (2.8)

Let [lambda], [mu] [member of] C and define [a.sub.[lambda]] = a - [lambda],[b.sub.[mu]] = b - [mu]. Then from (2.6) and (2.7) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This shows that (2.8) holds.

(c) Let a, b [member of] A such that W(a), W(b) are contained in closed disks D([[lambda].sub.0], r), D([[mu].sub.0], s) with radii r, s and centered [[lambda].sub.0], [[mu].sub.0], respectively. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, from (1.2) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.9)

From (2.2) and (2.9), we obtain (2.3). In the case where a,b are normal, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are normal. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore (2.2) yields (2.4). This completes the proof.

Remark 2.1. For a subset E of A and a character[phi] [member of] [OMEGA](B), define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then by Theorem 2.2 (c), we get [delta](A, t, [phi]) [less than or equal to] 4. If Nor(A) is a set of all normal elements of A, then from Theorem 2.2 (c), we see that [delta](Nor(A),t, [phi]) [less than or equal to] 1.

Remark 2.2. For each [phi] [member of] [OMEGA](B), define [F.sub.t](a,b) = [tau](tab) - [tau](ta)[tau](tb), then we obtain a bilinear mapping [F.sub.t] : A x A [right arrow] B. Using (2.2), we see that [F.sub.t] is continuous and satisfies [parallel][F.sub.t][parallel] [less than or equal to] 1.

References

 Guilu Long, Yang Liu and Chuan Wang, Allowable generalizedquantum gates, Commun. Theor. Phys., 51(2009), 65-67.

 Guilu Long, Yang Liu, Duality computing in quantum computers, Commun. Theor. Phys., 50(2008), 1303-1306.

 G. L. Long, Mathematical theory of duality computer in the density matrix formalism, Quantum Information Processing, Quantum Information Processing, 6(2007), 49- 54.

 Gui Lu Long, Yang Liu, Duality quantum computing, Front. Comput. Sci. China, 2(2008), 167-178.

 P. F. Renaud, A matrix formulation of Gruss inequality, Linear Algebra and its Applications, 335(2001), 95-100.

 G. J. Murphy, [C.sup.*]-Algebras and Operator Theory, Academic press, Inc., San Diego, 1990.

 J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980.

 H. X. Cao, Z. B. Xu, J. H. Zhang, et al, Matrix-trace on [C.sup.*]- algebra M"(A), Linear Algebra and Its Applications, 345(2002), No. 1-3, 255-260.

 This work is partly supported by the National Natural Science Foundation of China (No. 10571113, 10871224).

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, P.R.China E-mail: hbmeng@snnu.edu.cn
Author: Printer friendly Cite/link Email Feedback Meng, Hongbing Scientia Magna Report 9CHIN Jan 1, 2010 2126 M class Q composition operators. A proof of Smarandache-Patrascu's theorem using barycentric coordinates. Abelian groups Inequalities (Mathematics) Matrices Matrices (Mathematics)