Abstract
We introduce the notion of proper unitary connective-gate and we prove that entanglement cannot be characterized by such gates. We consider then a larger class of gates (called pseudo-unitary gates), which contains both the unitary and the anti-unitary quantum operations. By using a mixed language (a proper extension of the standard quantum computational language), we show how a logical characterization of entanglement is possible in the framework of a mixed semantics, which generalizes both the unitary and the pseudo-unitary quantum computational semantics.
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Notes
We recall that \(P_{0}^{(n)}\) and \(P_{1}^{(n)}\) represent, respectively, the falsity-property and the truth-property of the space \(\mathcal {H}^{(n)}\) (see Section 2 of the first part of this article).
We recall that \(\mathrm {p}(\rho ):=\text {tr}(P_{1}^{(n)}\,\rho )\), for any density operator ρ of \(\mathcal {H}^{(n)}\), where tr is the trace-functional (see Section 2 of the first part of this article).
See [5].
We recall that for any m, n ≥ 1 and for any density operator ρ of the Hilbert space \(\mathcal {H}^{(m+n)}\), AND (m, n) is defined as follows: \(\mathtt {AND}^{(m,n)}(\rho ):=\,\,{~}^{\mathfrak {D}} \mathtt {T}^{(m,n,1)} (\rho \otimes \,P_{0}^{(1)})\) (see Definition 7 of the first part of this article).
We recall that the concurrence of ρ is defined as follows: \( \mathcal {C}(\rho ):=\inf \left \{{\sum }_{i} w_{i} \mathcal {C}(P_{|{\psi _{i}}\rangle }): \rho ={\sum }_{i} w_{i} P_{|{\psi _{i}}\rangle }\right \}\), where \(\mathcal {C}(P_{|{\psi \rangle }})=\sqrt {2\Big (1-{\sum }_{i} {\lambda _{i}^{2}}\Big )}\) and the numbers λ i are eigenvalues of \(Red_{[m,n]}^{(1)}(P_{|{\psi \rangle }})\) (or, equivalently, of \(Red_{[m,n]}^{(2)}(P_{|{\psi \rangle }})\)) (see Section 3 of the first part of this article).
Notice that \(\text {tr}({\mathtt {SH}}^{(4k)}\underbrace {\rho \otimes \ldots \otimes \rho }_{2k}) \neq \text {tr}({~}^{\mathfrak {D}} {\mathtt {SH}}^{(4k)}\underbrace {\rho \otimes \ldots \otimes \rho }_{2k})= 1\).
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This article is dedicated to the memory of Peter Mittelstaedt. The authors will always remember the great scientist who has represented a noble example for all scholars of the IQSA-community.
Sergioli’s work has been supported by the Italian Ministry of Scientific Research within the FIRB project “Structures and dynamics of knowledge and cognition”, Cagliari unit F21J12000140001; Leporini’s work has been supported by the Italian Ministry of Scientific Research within the PRIN project “Automata and Formal Languages: Mathematical Aspects and Applications”.
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Dalla Chiara, M.L., Leporini, R. & Sergioli, G. Entanglement and Quantum Logical Gates. Part II.. Int J Theor Phys 54, 4530–4545 (2015). https://doi.org/10.1007/s10773-015-2667-2
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DOI: https://doi.org/10.1007/s10773-015-2667-2