Abstract
Fix a point in a finite-dimensional complex vector space and consider the sequence of iterates of this point under the composition of a unitary map with the orthogonal projection on the hyperplane orthogonal to the starting point. We prove that, generically, the series of the squared norms of these iterates sums to the dimension of the underlying space. This leads us to construct a (device-dependent) dimension witness for quantum systems which involves the probabilities of obtaining certain strings of outcomes in a sequential yes–no measurement. The exact formula for this series in non-generic cases is provided as well as its analogue in the real case.
Highlights
Theorem 1 offers the possibility of estimating the dimension of the system from the statistics of a projective measurement performed on this system
We fix a unit vector z ∈ R2 and let P stand for the orthogonal projection on Θ := span{z}⊥
The preceding three lemmas pave the way for the following result, which is a crucial step in the proof of theorem 1, and plays a key role in studying the symbolic dynamics generated by the quantum system under consideration [15, section 1.3]
Summary
We swap the antipodal cities A and B for the Bloch vectors rz0, rz1 , where rz1 = −rz0 , related to the elements of the orthonormal projective basis {|z0 , |z1 } of CP1, and travels along parallels for the rotation Oλ through the angle λ about the N–S axis of S2. U we denote the (projective) unitary operator corresponding to this rotation via rUz = Oλ(rz), where |z ∈ CP1 [6, p 88]. Theorem 1 offers the possibility of estimating (from below) the dimension of the system from the statistics of a projective measurement performed on this system.
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