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As the transposition map preserves eigenvalues, the spectrum of () is the same as the spectrum of , and in particular () must still be positive semidefinite. Thus must also be positive semidefinite. This proves the necessity of the PPT criterion.
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,
The fidelity between two quantum states and , expressed as density matrices, is commonly defined as: [1] [2] (,) = ().The square roots in this expression are well-defined because both and are positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem.
The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation:
Print/export Download as PDF; Printable version; In other projects ... In mathematics, positive semidefinite may refer to: Positive semidefinite function ...
A result of Albrecht Pfister [8] shows that a positive semidefinite form in n variables can be expressed as a sum of 2 n squares. [9] Dubois showed in 1967 that the answer is negative in general for ordered fields. [10] In this case one can say that a positive polynomial is a sum of weighted squares of rational functions with positive ...
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form =, where is a unitary matrix and is a positive semi-definite Hermitian matrix (is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size.
The trace distance is defined as half of the trace norm of the difference of the matrices: (,):= ‖ ‖ = [() † ()], where ‖ ‖ [†] is the trace norm of , and is the unique positive semidefinite such that = (which is always defined for positive semidefinite ).