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3. Quantum instruments

3.1. A few words about the quantum formalism

Denote by ${\textstyle {\mathcal {H}}}$ a complex Hilbert space. For simplicity, we assume that it is finite dimensional. Pure states of a system $S$ are given by normalized vectors of ${\textstyle {\mathcal {H}}}$ and mixed states by density operators (positive semi-definite operators with unit trace). The space of density operators is denoted by $S$ ( ${\textstyle {\mathcal {H}}}$ ). The space of all linear operators in ${\textstyle {\mathcal {H}}}$ is denoted by the symbol ${\textstyle {\mathcal {L}}({\mathcal {H}})}$ . In turn, this is a linear space. Moreover, ${\textstyle {\mathcal {L}}({\mathcal {H}})}$ is the complex Hilbert space with the scalar product, ${\textstyle <A|B>=TrA^{*}B}$ . We consider linear operators acting in ${\textstyle {\mathcal {L}}({\mathcal {H}})}$ . They are called superoperators.

The dynamics of the pure state of an isolated quantum system is described by the Schrödinger equation:

i{\tfrac {d}{dt}}\psi (t)={\widehat {H}}\psi (t)(t),\psi (0)=\psi _{0}

(3)

where ${\textstyle {\hat {\mathcal {H}}}}$ is system’s Hamiltonian. This equation implies that the pure state $\psi (t)$ evolves unitarily $\psi (t)={\hat {U}}(t)\psi _{0}$ , where ${\hat {U}}(t)=e^{-it{\hat {\mathcal {H}}}}$ is one parametric group of unitary operators, ${\hat {U}}(t):{\mathcal {H}}\rightarrow {\mathcal {H}}$ . In quantum physics, Hamiltonian ${\textstyle {\hat {\mathcal {H}}}}$ is associated with the energy-observable. The same interpretation is used in quantum biophysics (Arndt et al., 2009).^[1] However, in our quantum-like modeling describing information processing in biosystems, the operator ${\textstyle {\hat {\mathcal {H}}}}$ 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:

{\frac {d{\hat {\rho }}}{dt}}(t)=-i[{\hat {\mathcal {H}}},{\hat {\rho }}(t)],{\hat {\rho }}(0)={\hat {\rho }}_{0}

(4)

↑ Arndt M., Juffmann T., Vedral V. Quantum physics meets biology HFSP J., 3 (6) (2009), pp. 386-400, 10.2976/1.3244985

[1] Arndt M., Juffmann T., Vedral V. Quantum physics meets biology HFSP J., 3 (6) (2009), pp. 386-400, 10.2976/1.3244985

[1]

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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'':