ewNotes.bib

@FOOTNOTE{Note1,key="Note1",note=" A quantum (mixed) state $\rho $ can be represented by a density matrix which is a Hermitian, positive semidefinite operator (matrix) of trace one. If the rank of $\rho $ is 1, then the state is a pure state. "}
@FOOTNOTE{Note2,key="Note2",note=" The partial transpose (PT) operation acting on subsystem $A$ is defined as $\op {k_A,k_B}{l_A,l_B}^{\intercal _A} := \op {l_A,k_B}{k_A,l_B}$ where $\qty {\ket {k_A,k_B}}$ is a product basis of the joint system $\protect \mathcal  {H}_{AB}$. "}
@FOOTNOTE{Note3,key="Note3",note="A matrix (operator) is positive, semidefinite (PSD) if all its eigenvalues are non-negative."}
@FOOTNOTE{Note4,key="Note4",note=" The Euclidean norm of a matrix $A$ is defined as $\norm {A}_2:=\protect \sqrt  {\Tr (A^\dagger A)}$. The trace norm of $A$ is defined as $\norm {A}_{\Tr }\equiv \norm {A}_{1}:=\Tr (\abs {A})\equiv \Tr (\protect \sqrt  {A^\dagger A})$. Correspondingly, trace distance between two density matrices is $d_{\tr }(\rho ,\rho ') : = \protect \frac  {1}{2} \norm {\rho -\rho '}_1$. "}
@FOOTNOTE{Note5,key="Note5",note=" The Bell (CHSH) inequality (witness): $\vb {O}_{\protect \textup  {CHSH}}=\qty (\protect \mathds  {1}, a b, a b', a' b, a' b' )$ with $a = Z, a' = X, b = (X-Z)/\protect \sqrt  {2}, b = (X+Z)/\protect \sqrt  {2}$ and $\vb {w}_{\protect \textup  {CHSH}} = \qty (\pm 2, 1, -1, 1, 1)$ "}
@FOOTNOTE{Note6,key="Note6",note="In other words, the trace distance $\norm {\rho _{\protect \textup  {pre}}-\op {\psi _\protect \textup  {tar}}}_1 < \protect \sqrt  {1-\alpha }$ because the fidelity and trace distance are related by the inequalities $1-F\le d_{\tr }(\rho ,\rho ') \le \protect \sqrt  {1-F^2}$ (c.f. \protect \nameref  {prm:weak_membership problem_for_separability})"}
@FOOTNOTE{Note7,key="Note7",note=" $W_{\protect \text  {GHZ}_3} = \protect \frac  {1}{8} \qty ( 3*III - XXX- \protect \textup  {Perm}(IZZ) + \protect \textup  {Perm}(XYY))$ where $ZZI\equiv Z\otimes Z\otimes I$ and $\protect \textup  {Perm}(IZZ)\equiv ZZI + ZIZ+ IZZ$ for readability. "}
@FOOTNOTE{Note8,key="Note8",note="For example, the observables $ZZI$, $ZIZ$, and $IZZ$ can be measured by one local measurement setting $ZZZ$."}
@FOOTNOTE{Note9,key="Note9",note="The corresponding white noise regime for W state is $p_\protect \text  {noise}\in [8/21,0.791)$"}
@FOOTNOTE{Note10,key="Note10",note=" Denote $O_{\sigma }\in \qty {I,X,Y,Z}^{\otimes n}$ for a Pauli observable. Denote $\vb {x}_{\rho ,\protect \bm  {\sigma }}:=(\Tr (\rho O_{\sigma _1}),\protect \dots  ,\Tr (\rho O_{\sigma _M}))$ for a vector of expectations of $M$ Pauli observables $\protect \bm  {\sigma }\subseteq \qty {I,X,Y,Z}^n$ measured on $\rho $. "}
@FOOTNOTE{Note11,key="Note11",note="Quantum state tomography refers to the task of recovering the density matrix of an unknown $D$-dimensional state $\rho $ within error tolerance $\epsilon $, given the ability to prepare and measure copies of $\rho $."}
@FOOTNOTE{Note12,key="Note12",note="Adaptive measurements are the intermediate between independent measurements and collective (entangled) measurements, in which the copies of $\rho $ are measured individually, but the choice of measurement basis can change in response to earlier measurements."}
@FOOTNOTE{Note13,key="Note13",note="The notation $\protect \tilde  {\protect \mathcal  {O}}$ hides a polylog factor. A full tomography requires estimate $D^2$ measurements (observables) with additive error $\epsilon \ll 1/D$ for all $E_i$, so the sample complexity of shadow tomography is compatible with lower bounds of full quantum state tomography."}
@FOOTNOTE{Note14,key="Note14",note="The open-source code for classical shadow with the code from \protect \url  {https://github.com/hsinyuan-huang/predicting-quantum-properties}"}