In quantum information theory, the classical expectation value of an observable (self-adjoint element) in a quantum state (density element) is defined by , and the classical variance is expressed by . The Heisenberg uncertainty relation [5, 9] states that
where is a quantum state and and are two observables. The Heisenberg uncertainty relation gives a fundamental limit for the measurements of incompatible observables. A refinement of the Heisenberg uncertainty relation due to Schrödinger  is given by
where is the commutator of and and the classical covariance of and is defined by .
The third author, Furuichi, and Kuriyama  defined the one-parameter correlation and the one-parameter Wigner–Yanase skew information (is known as the Wigner–Yanase–Dyson skew information; see [4, 11]) for elements and , respectively, as follows:
where . They showed a trace inequality representing the relation between these two quantities as
In the case that , we get the classical notions of the correlation and the Wigner–Yanase skew information .
In this paper, we aim to replace the usual trace Tr by a tracial positive map between unital -algebras and to replace the functions and by functions and under certain conditions. These allow us to define the generalized covariance, the generalized variance, the generalized correlation and the generalized Wigner–Yanase–Dyson skew information related to the tracial positive maps and functions and .
The rest of the paper is organized as follows. In the next section, we provide some preliminaries and background material. In Section 3, we use some techniques in the noncommutative setting to give some Cauchy–Schwarz type inequalities for the generalized covariance and the generalized variance. Then we use them to extend inequalities (1.1) and (1.2) for tracial positive linear maps between -algebras. In Section 4, we present some inequalities and properties for the generalized correlation and the generalized Wigner–Yanase–Dyson skew information. In this section, we give a generalization of inequality (1.3) for tracial positive linear maps between -algebras. Finally, in Section 5, we establish some inequalities between variance and Wigner–Yanase–Dyson skew information. We indeed apply some arguments differing from the classical theory to investigate inequalities related to the generalized covariance, the generalized variance, the generalized correlation, and the generalized Wigner–Yanase–Dyson skew information.
Let us fix our notation and terminology used throughout the paper. Let stand for the -algebra of all bounded linear operators on a complex Hilbert space with the unit . An operator is called positive if for all , and we then write . An operator A is said to be strictly positive (denoted by ) if it is a positive invertible operator. Let be the Löwner order on the self-adjoint part of B(H). In the case that , is the same as the matrix algebra consist of all complex matrices. Due to the Gelfand–Naimark–Segal theorem, every -algebra can be regarded as a -subalgebra of for some Hilbert space . We use to denote -algebras. We denote by and the real and imaginary parts of , respectively. The self-adjoint part of is denoted by .
A linear map between -algebras is said to be -linear, if . It is positive, if whenever .
We say that is unital if and are unital and preserves the unit. A linear map is called -positive if the induced map given by is positive, where is the -algebra of
matrices with entries in . If is -positive for every , then is called completely positive. It is known that if the range of a positive linear map is commutative, then is completely positive; see [14, Theorem 1.2.4].
A map is called tracial if for all and in the domain of . The usual trace on the trace class operators acting on a Hilbert space is a tracial positive linear functional. It is known that every tracial positive map between -algebras is completely positive; see [3, page 57]. For a given closed two sided ideal of a -algebra , the commutativity of the quotient is equivalent to the existence of a tracial positive linear map satisfying and ; see  for more examples and implications of the definition. For a tracial positive linear map , a positive element is said to be a -density operator if . A unital -algebra is said to be injective whenever for every unital -algebra and for every self-adjoint subspace of , each unital completely positive linear map from into , can be extended to a completely positive linear map from into .
A pair of continuous real-valued functions defined on a set is called same monotonic if for every . Bourin  and Fujii  showed that for self-adjoint matrices and and for all continuous real-valued functions and on the spectrum of with the same monotonically.
If is a -subalgebra of , then a conditional expectation is a contractive positive linear map such that for every and all .
Our investigation is based on the following definition.
Let be a tracial positive linear map from a -algebra into a unital -algebra , and let . Then for a continuous positive real-valued function and a continuous real-valued function which are defined on an interval containing the spectrum of with ,
are called the generalized covariance and the generalized variance of and , respectively. In addition, for continuous real-valued functions and , which are defined on an interval containing the spectrum of , the generalized correlation and the generalized Wigner–Yanase–Dyson skew information of two elements and are defined by
If we consider , and as a density operator, then and are the same classical covariance and variance, respectively. Moreover, for and , and are the one-parameter correlation and the one-parameter Wigner–Yanase–Dyson skew information, respectively.
In the case when and , we simply denote , and by , and , respectively. It is known that for every tracial positive linear map, the matrix
is positive, which is equivalent to
and is called the variance-covariance inequality; see .
To achieve our results we need the following known lemma. The reader may consult the survey paper .
[3, Lemma 2.1] Let be two operators in . Then the block matrix is positive if and only if .
3. Cauchy–Schwarz type inequalities related to uncertainty relation
In this section, we give some Cauchy–Schwarz type inequalities for the generalized covariance and the generalized variance. The results of this section are generalizations of Heisenberg’s uncertainty relation; see .
First, we state the variance-covariance inequality for the generalized covariance and variance. Its proof involves some standard matrix tricks but we prove it for the sake of convenience.
For every tracial positive linear map , the matrix
It follows from the three-positivity and the tracial property of that
The positivity of the above matrix implies that
Hence, by employing Lemma 2.2, we get
Now, we are ready to give a generalization for Schrödinger’s uncertainty relation for a tracial positive linear map.
Let be a tracial positive linear map from a -algebra into a unital -algebra , and let . If is a commutative subset of , then the matrices
are positive for all , and all continuous positive real-valued functions and on the spectrum with .
|(since is commutative)|
and an easy calculation shows that
Summing both sides of the above equalities, we get
Since and the range of is commutative, we get
By a continuity argument we can assume that and by using Lemma 3.1, we conclude that
|(since the range of is commutative)|
Now, Lemma 2.2 implies the positivity of the matrix
To prove the positivity of the second matrix, first note that the positivity of the matrix implies the positivity of
In addition, it follows from the commutativity of the range of that
whence we arrive at the second inequality. ∎
Next, we aim to give a generalization of Heisenberg’s uncertainty relation for a tracial positive linear map between -algebra. To get this result we need some lemmas.
Lemma 3.3 (Choi–Tsui).
[3, pages 59 – 60] Let be -algebras such that either one of them is -algebra or is an injective -algebra. Let be a tracial positive linear map. Then there exist a commutative -algebra for some compact Hausdorff space , tracial positive linear maps , and such that . Moreover, in case that is unital, then and can be chosen to be unital. In particular, is completely positive.
The next lemma is a consequence of the positivity of the matrix and two-positivity of .
Let be a tracial positive linear map from a -algebra into a unital -algebra . Then
for all and all .
The following theorem gives a generalization of Heisenberg’s uncertainty relation for tracial positive linear maps between -algebras.
Let be a -algebra and be a unital -algebra such that either one of them is -algebra or is an injective -algebra. If is a tracial positive linear map and , then the matrix
is positive for all , and all continuous positive real-valued functions and on the spectrum with .
4. Some properties of correlation
We intend to investigate the positivity of the generalized Wigner–Yanase–Dyson skew information. We extend the results of
 and  to the positivity of the generalized Wigner–Yanase–Dyson skew information.
It is easy to see that the same monotonicity of a pair of continuous real-valued functions defined on a set is equal to the validity of the inequality for every .
Let be tracial positive linear map between von Neumann algebras. Then
for each pair of same monotonic functions and defined on the spectrum of and each element . In particular,
Since is self-adjoint, there exists a sequence of self-adjoint operators converging to such that each term of the sequence has the spectral representation , whenever , are real numbers and ’s are mutually orthogonal projections with . Hence, we only need to prove inequality (4.1) for a such self-adjoint operator .
First note that . In addition,
Since , we can write Consequently,
Let be a tracial positive linear map between von Neumann algebras. Then
for each pair of same monotonic functions and defined on the spectrum of a self-adjoint element and each operator .
Define the map by
Clearly, is a tracial positive linear map. Let . Then the matrices and are self-adjoint elements of . Furthermore, the spectrums of and are equal. Hence,