What is a pseudopure state? In the paper titled "Experimental Implementation of the Quantum Baker’s Map" by Weinstein et al. (Phys. Rev. Let. 89 (2002)), the author says something like

[...] the pseudopure state corresponding to the state $ \left |000
 \right \rangle$.

But, what is a pseudopure state in general ? how it is different from a pure state ? and why do they call the state $ \left |000 \right \rangle$ pseudopure, isn't it a pure state ?
 A: A pseudo-pure state (PPS) is a state (in general, describing an $n$-qubit state) whose density matrix can be cast in the form
$$\rho_{PPS}=\frac{1-\epsilon}{2^n}\mathbb I_{2^n} + \epsilon |\psi\rangle\langle\psi|$$
where $|\psi\rangle$ is some pure state. It is still a mixed state (the trace of $\rho^2$ is not 1) but a small percentage $\epsilon$ of it is a pure state, and allows some manipulation analogous to the ones we do with pure states. For example if you consider plain time evolution of a density operator $\rho(0)$ at time $t=0$ through a unitary propagator $U(t)$:
$$\rho(0)\rightarrow \rho(t)=U(t)\rho(0)U^\dagger(t)$$
you see that only the pure part evolves whereas the completely mixed part remains unchanged:
$$\rho_{PPS}(t)=U^\dagger(t)\left(\frac{1-\epsilon}{2^n}\mathbb I_{2^n} + \epsilon |\psi\rangle\langle\psi|\right)U(t)=\frac{1-\epsilon}{2^n}\mathbb I_{2^n} + \epsilon U(t)|\psi\rangle\langle\psi|U ^\dagger(t)=$$
This is useful as the observables of interest in NMR are the magnetizations, which are obtained as the trace of the product $\rho(t)\sigma_i$ where the $\sigma_i^{(k)}$ are the Pauli matrices ($i=x,y,z$) related to the $k-$th spin of the NMR molecule, which are traceless (and by $\sigma_i^{(k)}$ I mean $\mathbb I_1\otimes\mathbb I_2\otimes\dots{}\otimes\mathbb I_{k-1}\otimes\sigma_i\otimes\mathbb I_{k+1}\otimes\dots{}\otimes\mathbb I_n$):
$$Tr\left[\rho_{PPS}(t)\sigma_i \right]=\frac{1-\epsilon}{2^n}\underbrace{Tr\left[\mathbb I_{2^n}\sigma_i^{(k)} \right]}_{=0}+\epsilon Tr\left[U(t)|\psi\rangle\langle\psi|U^\dagger (t)\sigma_i^{(k)} \right]$$
so only the pure term takes part in the calculation of the magnetization.
A: It is a state that is the starting point for a NMR quantum computation.
It is the effective pure state whose physical action is the same as the pure state.
To obtain a pseudo pure state:


*

*temporal averaging

*spatial averaging

*logical labeling

*cat benchmark


For
    example, temporal averaging[12] needs 2n
    1 (n is the quantity of qubits in system) permutations, and then adds up all
    the results to get the pseudopure state. Its experimental
    steps will increase exponentially with n, which makes the
    experiment become more complicated. On the other hand,
    the S/N ratio decreases exponentially with n. For spatial
    averaging[13], it uses a few PFG (pulse field of gradient) in
    the experimental scheme, which reduces the S/N ratio.
    Therefore, if a method not only reduces the complication,
    but also increases the S/N ratio, it is an ideal method no
    doubt.
     Recently, Vandersypen et al.[7,17] have proposed an
    idea for preparing pseudopure state in the multi-qubit system, which we called “CNOT gates combination”. That
    idea is analogous with temporal averaging[12]. Both use
    the summation of multi-experiment results to obtain the
    pseudopure state. However, CNOT gates combination
    uses less steps in experiment than temporal averaging.
    That idea takes CNOT gate, the most frequently used
    two-qubit gate in quantum computation circuit, as the basic element to prepare pseudopure state. Vandersypen et al.
    have realized 5-qubit order-finding[17] and 7-qubit Shor’s
    factoring algorithm[7] based on the pseudopure state obtained by that method. 


Please see here:
https://link.springer.com/article/10.1360/02tb9405
A pseudo-pure state is a highly mixed state.
Please see here:
https://arxiv.org/pdf/quant-ph/0012038.pdf
