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Gianfranco (talk | contribs) (Created page with "==3. Quantum instruments== ===3.1. A few words about the quantum formalism=== Denote by <math display="inline">\mathcal{H}</math> a complex Hilbert space. For simplicity, we assume that it is finite dimensional. Pure states of a system <math>S</math> are given by normalized vectors of <math display="inline">\mathcal{H}</math> and mixed states by density operators (positive semi-definite operators with unit trace). The space of density operators is denoted by <math>S...") |
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where <math display="inline">\hat{\mathcal{H}}</math> is system’s Hamiltonian. This equation implies that the pure state <math>\psi(t)</math> evolves unitarily <math>\psi(t)= \hat{U}(t)\psi_0</math>, where <math>\hat{U}(t)=e^{-it\hat{\mathcal H}}</math> is one parametric group of unitary operators,<math>\hat{U}(t):\mathcal{H}\rightarrow \mathcal{H}</math> . In quantum physics, Hamiltonian <math display="inline">\hat{\mathcal{H}}</math> is associated with the energy-observable. The same interpretation is used in quantum biophysics (Arndt et al., 2009). However, in our quantum-like modeling describing information processing in biosystems, the operator <math display="inline">\hat{\mathcal{H}}</math> has no direct coupling with physical energy. This is the evolution-generator describing information interactions. | where <math display="inline">\hat{\mathcal{H}}</math> is system’s Hamiltonian. This equation implies that the pure state <math>\psi(t)</math> evolves unitarily <math>\psi(t)= \hat{U}(t)\psi_0</math>, where <math>\hat{U}(t)=e^{-it\hat{\mathcal H}}</math> is one parametric group of unitary operators,<math>\hat{U}(t):\mathcal{H}\rightarrow \mathcal{H}</math> . In quantum physics, Hamiltonian <math display="inline">\hat{\mathcal{H}}</math> is associated with the energy-observable. The same interpretation is used in quantum biophysics (Arndt et al., 2009).<ref>Arndt M., Juffmann T., Vedral V. | ||
Quantum physics meets biology | |||
HFSP J., 3 (6) (2009), pp. 386-400, 10.2976/1.3244985</ref> However, in our quantum-like modeling describing information processing in biosystems, the operator <math display="inline">\hat{\mathcal{H}}</math> has no direct coupling with physical energy. This is the evolution-generator describing information interactions. | |||
Schrödinger’s dynamics for a pure state implies that the dynamics of a mixed state (represented by a density operator) is described by the ''von Neumann equation'': | Schrödinger’s dynamics for a pure state implies that the dynamics of a mixed state (represented by a density operator) is described by the ''von Neumann equation'': | ||