Quantum states
Quantum states: general properties
- The purifications of two $\varepsilon$-close states need not be $\varepsilon$-close.
- The fidelity depends on more than just the difference of states.
- The closest diagonal state to a given state need not be the dephased original state.
- Not all states that are useful for teleportation violate Bell inequalities.
- Vanishing quantum discord is not necessary for completely positive maps. Originally, Shabani and Lidar claimed in this paper (arXiv) that the initial system-environment state having vanishing quantum discord is necessary (and sufficent) for the reduced dynamics of a system to be completely positive. A few years later Brodutch et al. presented a counterexample, cf. Section III in this paper (arXiv). Since then, Shabani and Lidar have published an erratum to their original paper.
- Not every approximate Markov chain is close to a Markov chain. The existence of a counterexample is proven, e.g., in Proposition 5.9 of this book (arXiv) as originally shown in this paper (arXiv)
- Even if two bipartite states are close the corresponding Schmidt bases need not have large overlap.
- Not every state which violates the separability criterion of the local uncertainty relations violates the realignment criterion for separability. (arXiv)
- Positive partial transpose does not imply separability beyond qubit-qutrit systems. (arXiv)
State transformations
- Even if two states are close together, there need not exist a channel close to the identity mapping one to the other state.; note that this is true for pure states but fails for mixed states
- The Alberti-Uhlmann theorem does not hold beyond qubits. A counterexample for two qutrit states is given in Proposition 6 of this paper (arXiv), and a qutrit-qubit counterexample is the subject of this paper
- Local unitary equivalent stabilizer states need not be local Clifford equivalent. This was known as the LU-LC conjecture to which a counterexample (arXiv)—followed by a family of counterexamples (arXiv)—of pairs of graph states (and, by extension, stabilizer states) have been found that are equivalent under local unitary operations, but not under local Clifford operations.
... from the perspective of operator theory
- The bounded operators on a Hilbert space $\mathcal H$ may be larger than $\mathcal H\otimes\mathcal H^*$. The reason for this is that $\mathcal H\otimes\mathcal H^*\simeq\mathcal B^2(\mathcal H)$ with the latter being the Hilbert-Schmidt operators, cf. this phys.SE answer; but in infinite dimensions there are bounded operators which are not Hilbert-Schmidt (a simple example here is the identity operator).
- Taking the positive part commutes with conjugation with a state.
