An elementary introduction to the geometry of quantum states with a picture book
6.2 Directions associated with states with substantial purity are rare Our aim in this section is to show that the probability f ...
By H ̈older inequality ∫ DN dΩrN (^2) − 1 ≤ (∫ DN dΩr(N (^2) −1)p ) 1 /p(∫ DN dΩ 1q ) 1 /q (6.12) Picking p= N^2 + 1 N^2 − 1 , q ...
the intersection ofM half-spaces, each of which contains the origin. This can be formulated as^13 cα·x≤ 1 , α= 1,...M To decide ...
7.3 Entangled pure states Pure bi-partite states can be put into equivalence classes labeled by the Schmidt numbers, [1], leadin ...
The projection on|β〉can be written in terms ofσμas: |β〉〈β|= 1 M M∑− 1 jk=0 |jj〉〈kk| = 1 M^3 M∑− 1 jk=0 M∑^2 − 1 μ,ν=0 〈kk|σμ⊗σνt ...
ρ≥0 whileρptis not a positive matrix. Bothρ,ρpt≥0 butρis not separable. In the case thatρis a pure state or^14 N 1 N 2 ≤6, onl ...
7.6 Entanglement witnesses An entanglement witness for a given partition, (N 1 ,...,Nn), is a Hermitian matrix W so that Tr(Wρ)≥ ...
Figure 14: The separable states (green) and the corresponding witnesses, (Schematic). Sis indeed an entanglement witness since 〈 ...
are orthogonal projections. PS projects on the states that are symmetric under swap, andPAon the anti-symmetric ones. Hence, Tr ...
|β〉〈β| ρpt ρ WS Figure 15: The figure shows the maximally entangled state |β〉〈β|on the unit circle, its witnessρpt, the entangle ...
whereZ,Xare the standard 2×2 Pauli matrices. The numerator in Eq. (7.33) takes the form (^1) M/ 2 ⊗ ( 12 ⊗ 12 +aZ⊗Z+ 12 ⊗bX ) ⊗ ...
A The average purity of quantum states In section 6.2 we quoted a result of [24], Eq. (6.8), which allows to explicitly compute ...
The reason one can explicitly compute such integral is that the measure factors dμN,K=dμ|ρ|⊗dμΩ (A.5) Trρ=‖ψ‖^2 depends only on ...
inNdimensions C= { x ∣ ∣ ∣x= (x 1 ,...,xN), |xj|≤ 1 2 , j= 1,...,N } (B.1) The radius of inertia ofCis re^2 =E ( x^2 ) = ∑ E ( x ...
References [1] Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information. AAPT, 2002. [2] Ingemar Bengtsso ...
[16] Ryszard Horodecki, Pawe l Horodecki, Micha l Horodecki, and Karol Horodecki. Quantum entanglement. Reviews of modern physic ...
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