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Channel-State duality with centers (2404.16004v2)

Published 24 Apr 2024 in quant-ph, hep-th, math-ph, and math.MP

Abstract: We study extensions of the mappings arising in usual channel-state duality to the case of Hilbert spaces with a direct sum structure. This setting arises in representations of algebras with centers, which are commonly associated with constraints, and it has many physical applications from quantum many-body theory to holography and quantum gravity. We establish that there is a general relationship between non-separability of the state and the isometric properties of the induced channel. We also provide a generalisation of our approach to algebras of trace-class operators on infinite dimensional Hilbert spaces.

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  19. Note1. For the |Ψ±⟩ketsuperscriptΨplus-or-minus\mathinner{|{\Psi^{\pm}}\rangle}| roman_Ψ start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ⟩ states, the corresponding operator is X⁢σ1𝑋superscript𝜎1X\sigma^{1}italic_X italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT, so multiplied by a Pauli matrix.
  20. Note2. We will for simplicity work in this section with bounded operators on finite dimensional Hilbert spaces.
  21. Note3. Notice that, in particular, this is a pendent of properties in local QFT, as formalised by Haag duality, where subsystems are identified and distinguished by their localization on the spacetime manifold.
  22. Note4. We assume here that the algebras contain a unit, so an identity operator.
  23. Note5. This 2-out-of-3 property appears to be due to the relatively rigid way entanglement shows itself in pure states, as manifested through there being a (mostly unique) measure of entanglement for pure states, which is not the case for mixed states.
  24. Note6. These simply take two factors in a tensor product and swap them, 𝒮⁢|a⟩⊗|b⟩=|b⟩⊗|a⟩tensor-product𝒮ket𝑎ket𝑏tensor-productket𝑏ket𝑎\mathcal{S}\mathinner{|{a}\rangle}\otimes\mathinner{|{b}\rangle}=\mathinner{|{% b}\rangle}\otimes\mathinner{|{a}\rangle}caligraphic_S start_ATOM | italic_a ⟩ end_ATOM ⊗ start_ATOM | italic_b ⟩ end_ATOM = start_ATOM | italic_b ⟩ end_ATOM ⊗ start_ATOM | italic_a ⟩ end_ATOM.
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  26. Note8. This assumes an extension map and associated partial trace operation have been chosen.
  27. Note9. The usual issues of domains apply, but as long as the extensions are bounded, we may neglect them.
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