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==3. Quantum instruments== | |||
===3.1. A few words about the quantum formalism=== | ===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.'' | 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.'' | ||
Latest revision as of 09:25, 27 September 2022
3. Quantum instruments
3.1. A few words about the quantum formalism
Denote by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{H}} a complex Hilbert space. For simplicity, we assume that it is finite dimensional. Pure states of a system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} are given by normalized vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{H}} and mixed states by density operators (positive semi-definite operators with unit trace). The space of density operators is denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{H}} ). The space of all linear operators in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{H}} is denoted by the symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}(\mathcal{H})} . In turn, this is a linear space. Moreover, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}(\mathcal{H})} is the complex Hilbert space with the scalar product, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle <A|B>=TrA^*B} . We consider linear operators acting in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\tfrac{d}{dt}\psi(t)=\widehat{H}\psi(t)(t), \psi(0)=\psi_0} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3)} |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \hat{\mathcal{H}}} is system’s Hamiltonian. This equation implies that the pure state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)} evolves unitarily Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)= \hat{U}(t)\psi_0} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{U}(t)=e^{-it\hat{\mathcal H}}} is one parametric group of unitary operators,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{U}(t):\mathcal{H}\rightarrow \mathcal{H}} . In quantum physics, Hamiltonian Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \hat{\mathcal{H}}} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\hat{\rho}}{dt}(t)=-i[\hat{\mathcal{H}},\hat{\rho}(t)], \hat{\rho}(0)= \hat{\rho}_0} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (4)} |