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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 [5] 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 [7], 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 [8] 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

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

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

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

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

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

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

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

[8] 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.

[1] 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
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Date:Jan 1, 2010
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