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==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</math> (<math display="inline">\mathcal{H}</math>). The space of all linear operators in <math display="inline">\mathcal{H}</math> is denoted by the symbol <math display="inline">\mathcal{L}(\mathcal{H})</math> . In turn, this is a linear space. Moreover, <math display="inline">\mathcal{L}(\mathcal{H})</math> is the complex Hilbert space with the scalar product, <math display="inline"><A|B>=TrA^*B</math>. We consider linear operators acting in <math display="inline">\mathcal{L}(\mathcal{H})</math>. They are called ''superoperators.''

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

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>i\tfrac{d}{dt}\psi(t)=\widehat{H}\psi(t)(t), \psi(0)=\psi_0</math>
| width="33%" align="right" |<math>(3)</math>
|}



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.

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'': 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\frac{d\hat{\rho}}{dt}(t)=-i[\hat{\mathcal{H}},\hat{\rho}(t)], \hat{\rho}(0)=
\hat{\rho}_0</math>
| width="33%" align="right" |<math>(4)</math>
|}