By James M. McCracken
This ebook is a short advent to destructive quantum channels, i.e., linear, trace-preserving (and constant) quantum maps that aren't thoroughly confident. The flat and sharp operators are brought and defined. entire positivity is gifted as a mathematical estate, however it is argued that entire positivity isn't really a actual requirement of all quantum operations. Negativity, a degree of the inability of whole positivity, is proposed as a device for empirically checking out whole positivity assumptions.
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Extra resources for Negative Quantum Channels: An Introduction to Quantum Maps that are Not Completely Positive
Speciﬁcally, we will use constructors to take vectors of states to superoperators or speciﬁc kinds of matrices. 13 For example, a superoperator S constructed using the tomography vector E would be written using some superoperator constructor S as S DSˇE : e columns of S will be deﬁned by “columnized”¹¹ versions of linear combinations of the members of E . e superoperator S is a matrix of complex values constructed from rearranged elements ¹¹e concept of columnizing matrices will be discussed in depth in the next section.
4j j D 1 ˙ c0 C c1 C c2 2c0 c3 C c3 0 : 2 2 e main idea of tomography is to ﬁnd a complete description of the qubit channel using only valid states that can be created in the lab. Consider the following example. 6 e experimentalist might be looking for the state Â Ã a b D c d 32 2. TOMOGRAPHY that can be written down as Â Ã Â Ã Â Ã Â Ã 1 0 0 1 0 0 0 0 Da Cb Cc Cd : 0 0 0 0 1 0 0 1 e matrices associated with the coeﬃcients a and d are valid density matrices ( 0 and 1 , respectively) and can, therefore, be created in the lab.
ED. ˝ 1; ˝ 2; ˝ 3; ˝ 4/ and E˝ D. x1 ˙ y1 ; x2 ˙ y2 ; x3 ˙ y3 ; x4 ˙ y4 / : e dot product of a vector of states with a vector of complex numbers is deﬁned in analogy to the normal dot product between vectors as cE E D 4 X i ci i D c1 1 C c2 2 C c3 3 C c4 4 ; where E is a qubit vector of states and cE is a vector of complex numbers. A superoperator is a linear operator that takes an operator to another operator. , to take density operators to other density operators). 12 It will be shown in the next section that a superoperator representing a quantum channel can be completely characterized in the lab using the states in a tomography vector.