Bipartite entanglement and the arrow of time

Frembs, Markus

doi:10.1103/physreva.107.022218

Cited by 2 publications

(15 citation statements)

References 53 publications

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“…In the next section, we will show that theorem 1 and theorem 2 obtain under the restriction to Jordan algebras. In contrast, we will see that theorem 3 does depend on the choice of complex structure [20] (equivalently, dynamical correspondence [2] or time orientation [27,29]) and thus on the choice of C * -algebra L(H) or L(H) op .…”

Section: Since Transposition Is An Order-reversing Involution On L(h)...mentioning

confidence: 87%

“…3 Following notation in [44] and related work [27][28][29][30], we use * instead of † for the (Hermitian) adjoint.…”

Section: Jamiołkowski's Isomorphism and Theoremmentioning

confidence: 99%

“…The two (associative) products in equation ( 5) are readily distinguished from one another by the sign in front of the commutator. This sign obtains a physical interpretation as the direction of time as follows (see [2,23,[27][28][29]): in L(H), the evolution of the system (in the Heisenberg picture) is given by the map…”

Section: Since Transposition Is An Order-reversing Involution On L(h)...mentioning

confidence: 99%

“…Ψ describes the unitary evolution in L(H) (as opposed to anti-unitary evolution [8,67]). With [27][28][29], we therefore call Ψ the canonical time orientation on L(H), and Ψ * (t, a) := Ψ(−t, a) for all a ∈ J (H) sa and t ∈ R the reverse time orientation on L(H). The relation with the choice of sign in equation ( 5) follows by differentiation:…”

Section: Since Transposition Is An Order-reversing Involution On L(h)...mentioning

confidence: 99%

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Variations on the Choi–Jamiołkowski isomorphism

Frembs,

Cavalcanti

2024

J. Phys. A: Math. Theor.

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The Choi-Jamiolkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C∗-algebras they arise from, which are distinguished by their order of composition.A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism [M.-D. Choi, Lin. Alg. Appl., 10, 285 (1975)] further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, cruciallydepends on the dynamical (compositional) structure in C∗-algebras. This in turn lies at the heart of the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiolkowski [A. Jamiolkowski, Rep. Math. Phys., 3, 275 (1972)]. Here, we overcome this subtle but pervasive issue in a number of ways: first, we prove a versionof Choi’s theorem for Jamiolkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C∗-algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.

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Section: Since Transposition Is An Order-reversing Involution On L(h)...mentioning

confidence: 87%

“…3 Following notation in [44] and related work [27][28][29][30], we use * instead of † for the (Hermitian) adjoint.…”

Section: Jamiołkowski's Isomorphism and Theoremmentioning

confidence: 99%

Section: Since Transposition Is An Order-reversing Involution On L(h)...mentioning

confidence: 99%

Section: Since Transposition Is An Order-reversing Involution On L(h)...mentioning

confidence: 99%

See 2 more Smart Citations

Variations on the Choi–Jamiołkowski isomorphism

Frembs,

Cavalcanti

2024

J. Phys. A: Math. Theor.

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“…Theorem 7 was foreshadowed for finite type I factors, and with a focus on classifying quantum from non-signalling correlations in [31]. Its close connection with entanglement classification is discussed in [53]. For a broader perspective on the significance of contextuality in the foundations of quantum mechanics we refer to [12].…”

Section: Discussionmentioning

confidence: 99%

Gleason’s theorem for composite systems

Frembs,

Döring

2023

J. Phys. A: Math. Theor.

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Gleason’s theorem [A. Gleason, J. Math. Mech., 6, 885 (1957)] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that ﬁnitely additive measures on the projection lattice P(H) extend to positive linear functionals on the algebra of bounded operators B(H). Over many years, and by the eﬀort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type I2 factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: ﬁrst, we extend its scope to dilations in the sense of Naimark [M. A. Naimark, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., 41, 359 (1943)] and Stinespring [W. F. Stinespring, Proc. Am. Math. Soc., 6, 211 (1955)] and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition [E. M. Alfsen and F. W. Shultz, Commun. Math. Phys., 194, 87 (1998)]. We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.

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Bipartite entanglement and the arrow of time

Cited by 2 publications

References 53 publications

Variations on the Choi–Jamiołkowski isomorphism

Variations on the Choi–Jamiołkowski isomorphism

Gleason’s theorem for composite systems

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