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How does this relate to the MIC-POVM representation of states?
In the QuantumFramework, we have a transformation that turns a quantum state into a probability distribution in a minimal d^2 dimension, not of order d^3, like in your paper. Then, the Schrodinger equation (technically Liouville–von Neumann equation) can also be written in probability vector form.
We’ve made this a special case for evolution in phase space. For example, given random Hamiltonian and initial state, you can inspect what equations it generates for NDSolve:
equations = Normal @ QuantumEvolve[QuantumOperator["RandomHermitian"],
QuantumWignerMICTransform[QuantumState["RandomMixed"]], {t, 0, 1},
"ReturnEquations" -> True]
For a qubit, the initial vector would be a 4-dimension probability distribution.
Of course, not every 4-dimension probability distribution would correspond to a valid qubit.
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Dear Nikolay,
thank you for your comment. I’m not familiar with this particular representation, but if you’re referring to works such as [2006.13727] Minimal informationally complete measurements for probability representation of quantum dynamics, then it seems to me that this is fundamentally different. Our aim is to discuss Leggett-Garg inequalities, and thus we need to be able to distinguish physical observables (observed and disturbed). Our probability vector representation allows for such a distinction. On the other hand, the MIC-POVM measurement cannot distinguish the observables since the POVM measurement provides all of the probabilities at once by measuring quantum devices with larger dimensions. Consequently, the MIC-POVM representation cannot be used to investigate assumptions underlying the formulation of Leggett-Garg inequalities such as noninvasive measurability.
Lastly, please note that the dimension of probability vectors associated with generic N-level qudit states in our construction is N(N^2-1). For more details, please refer to Sec. II of [2312.16281] Probability vector representation of the Schrödinger equation and Leggett-Garg-type experiments.
All the best,
Sebastian
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You may find it helpful. I’ve formulated the temporal version of the CHSH game in analogy to the spatial one, which I’ve studied here: On quantum amplitudes, correlations and negativity - Staff Picks - WOLFRAM COMMUNITY . It is mainly based on this paper: https://arxiv.org/pdf/1005.3421.pdf.
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