We present some generalizations of quantum information inequalities involving tracial positive
linear maps bet
ween
w the MathML source">C⁎-algebras. Among several results,
we establish a noncommutative Heisenberg uncertainty relation. More precisely,
we sho
w that if
w the MathML source">Φ:A→B is a tracial positive
linear map bet
ween
w the MathML source">C⁎-algebras,
w the MathML source">ρ∈A is a Φ-density element and
w the MathML source">A,B are self-adjoint operators of
w the MathML source">A such that
w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si6.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=25e7c2cb63b6bac84c54935d71534d76">lineImage" height="20" width="185" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306230-si6.gif"> for some scalers
w the MathML source">0<m<M, then under some conditions
where
w the MathML source">Km,M(ρ[A,B]) is the Kantorovich constant of the operator
w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si10.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=61bcfc5c79430e8681c00e84548eedd4">lineImage" height="20" width="88" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306230-si10.gif"> and
w the MathML source">Vρ,Φ(X) is the generalized variance of
X. In addition,
we use some arguments differing from the scalar theory to present some inequalities related to the generalized correlation and the generalized Wigner–Yanase–Dyson ske
w information.