Proyecto de reconfiguración de la refinería de Minatitlán

m≥0A^mnVmq = − A^qn (n ≥ 0 , q < 0) . (6.3.19) A similar computation shows that the condition b_n ∣ 0 ⟩ = 0 reduces to

m∑<0A^∗mnV_pm = A^∗pn (n < 0 , p ≥ 0) . (6.3.20)

If the block of A where both indices are non-negative is invertible, Eq. (6.3.19) de-termines V uniquely. Similarly, if the block of A where both indices are negative is invertible, Eq. (6.3.20) determines V uniquely. If both blocks are invertible, it can be verified using unitarity of A that the ensuing two expressions for V are equivalent.

Working perturbatively in h, the invertibility assumptions hold, and using ((6.3.12)) we find

V = V⁽¹⁾ + O(h²), (6.3.21)

where

V_pq⁽¹⁾ = −A⁽¹⁾qp G_p^∗ = A^(1)∗pq G_q (p ≥ 0, q < 0). (6.3.22) We shall show in Section 6.4 that the normalisation constant N has the small h expansion

N = 1 − ¹₂∑

p,q∣V^pq∣² + O(h³) . (6.3.23)

6.4 Evolution of entangled states

In this section we study the evolution of Bell-type quantum states of modes within two cavities. We shall work perturbatively to quadratic order in h.

We specialize to the scenario where Rob is initially inertial, accelerates uniformly and then turns the engines off and travels with constant velocity. More complicated scenarios can be analyzed in a similar fashion adopting the techniques developed in the previous

Chapter 6: Kinematic entanglement degradation of fermionic cavity modes

chapter. Focusing first on Rob’s cavity only, we write out in Sec. 6.4.1 the pre-trip vacuum and states with a single (anti-)particle in terms of post-trip excitations on the appropriate vacuum. In Sec. 6.4.2 we address an entangled state where one field mode is controlled by Alice and one by Rob. In Sec. 6.4.3 we address a state of the type analysed in [60] where the entanglement between Alice and Rob is in the charge of the field modes.

6.4.1 Rob’s cavity: vacuum and single-particle states

Consider the initial vacuum ∣ 0 ⟩ in Rob’s cavity before the journey starts. We shall use (6.3.13) to express this state in terms of post-trip excitations over the post-trip vac-uum∣ ˜0 ⟩.

We expand the exponential in (6.3.13) as e^W = 1 + ∑

p,q

V_pq˜a^†_p˜b^†_q+¹2 ∑

p,q,i,j

V_pqV_ij˜a^†_p˜b^†_q˜a^†_i˜b^†_j+ O(h³). (6.4.1) We denote the final single-particle states by

∣˜1^k⟩⁺∶= ˜a^†k∣ ˜0 ⟩ (6.4.2) for k≥ 0 and by

∣˜1k⟩⁻∶= ˜b^†k∣ ˜0 ⟩ (6.4.3) for k < 0, so that the superscript ± indicates particles and antiparticles respectively.

From (6.4.1) we obtain where the ordering of the single-particle kets encodes the ordering of the fermion creation operators. It follows that the normalisation constant N is given by (6.3.23), and (6.3.13) gives Consider then in Rob’s cavity the state with exactly one pre-trip particle,

∣1^k⟩⁻∶= b^†k∣ 0 ⟩ for k < 0 or

∣1^k⟩⁺∶= a^†k∣ 0 ⟩ for k ≥ 0. (6.4.6)

Chapter 6: Kinematic entanglement degradation of fermionic cavity modes

Acting on the initial vacuum (6.4.5) by (6.3.15b) and the Hermitian conjugate of (6.3.15a) respectively, we find

We wish to consider a state where one cavity field mode is controlled by Alice and one by Rob. Concretely, we take

∣ φ^±init⟩AR+= ^√1₂( ∣ 0ˆk⟩_A∣ 0k⟩R ± ∣ 1ˆk⟩^κ_A∣ 1k⟩⁺R) , (6.4.8a)

∣ φ^±init⟩AR−= ^√1₂( ∣ 0^ˆk⟩_A∣ 0^k⟩R ± ∣ 1^ˆk⟩^κ_A∣ 1^k⟩⁻R) , (6.4.8b) where the superscripts ± indicate particles or antiparticles, so that κ = + for ˆk ≥ 0 and κ= − for ˆk < 0. Furthermore, we consider the two particle basis state of the two mode Hilbert space, corresponding to one excitation each in the modes ˆkin Alice’s cavity and k in Rob’s cavity, to be ordered as in (6.4.8). As pointed out in Ref. [67], making such a choice can lead to ambiguities in the entanglement. In fact, the fermionic Fock space is not naturally equipped with a tensor product structure. When defining vectors in the Fock space, the ordering of fermionic operators is uniquely defined unto an overall sign difference. In our case, the ambiguity amounts to a relative phase shift of π, i.e., a sign change, in (6.4.8), which does not affect the amount of entanglement. In other words, the states (6.4.8) are pure, bipartite, maximally entangled states of mode ˆk in Alice’s cavity and mode k in Rob’s cavity.

Chapter 6: Kinematic entanglement degradation of fermionic cavity modes

We form the density matrix for each of the states (6.4.8), express the density matrix in terms of Rob’s post-trip basis to order h²using (6.4.5) and (6.4.7), and take the partial trace over all of Rob’s modes except the reference mode k. All of Rob’s modes except k are thus regarded as environment, to which information is lost due to the acceleration.

The relevant partial traces of Rob’s matrix elements depend on the sign of the mode label k. Throughout this work, we use the notation Tr_¬k to emphasize that we are performing a trace over all degrees of freedom (mode numbers) except k and analogously for Tr_¬k,k^′. For k≥ 0, corresponding to (6.4.8a), we find

Tr_¬k∣ 0k⟩ ⟨ 0k∣ = (1 − fk⁻) ∣ ˜0k⟩ ⟨˜0k∣ + fk⁻∣˜1k⟩⁺⁺⟨˜1k∣ , (6.4.9a) Tr_¬k∣ 0^k⟩⁺⟨1^k∣ = (G^k+ A⁽²⁾kk) ∣ ˜0^k⟩⁺⟨˜1^k∣ , (6.4.9b) Tr_¬k∣1k⟩⁺⁺⟨1k∣ = (1 − fk⁺) ∣˜1k⟩⁺⁺⟨˜1k∣ + fk⁺∣ ˜0k⟩ ⟨˜0k∣ , (6.4.9c) where we have used (6.3.22) and introduced the abbreviations

f_k⁺∶= ∑

p≥0∣A⁽¹⁾_pk ∣², f_k⁻∶= ∑

q<0∣A⁽¹⁾_qk∣². (6.4.10) For k< 0, corresponding to (6.4.8b), we find similarly

Tr_¬k∣ 0^k⟩ ⟨ 0^k∣ = (1 − fk⁺) ∣ ˜0^k⟩ ⟨˜0^k∣ + fk⁺∣˜1^k⟩⁻⁻⟨˜1^k∣ , (6.4.11a) Tr_¬k∣ 0k⟩⁻⟨1k∣ = (Gk^∗+ A^(2)∗_kk ) ∣ ˜0k⟩⁻⟨˜1k∣ , (6.4.11b) Tr_¬k∣1^k⟩⁻⁻⟨1^k∣ = (1 − fk⁻) ∣˜1^k⟩⁻⁻⟨˜1k∣ + fk⁻∣ ˜0k⟩ ⟨˜0k∣ . (6.4.11c)

6.4.3 States with entanglement between opposite charges

We finally consider the state in the initial region of the form

∣ χ^±init⟩AR = ^√1₂( ∣ 1k⟩⁺A∣ 1^k′⟩⁻R ± ∣ 1^k′⟩⁻A∣ 1^k⟩⁺R) , (6.4.12) where the meaning of the subscripts and superscripts is as described for Eq. (6.4.8), indicating that k≥ 0 and k^′< 0. In this state Alice and Rob each have access to both of the modes k and k^′, and the entanglement is in the charge of the field modes, similarly to the states considered in [60]. While superselection rules do not allow for states with linear combinations of different charges, our cavity scenario does not lead to inconsistencies.

In fact, is is easy to see that the state (6.4.12) is not a superposition of different charges in any of the cavities.

We form the reduced density matrix to order h² as in Sec. 6.4.2, but now the partial tracing over Rob’s modes excludes both mode k and mode k^′. The relevant matrix

Chapter 6: Kinematic entanglement degradation of fermionic cavity modes

elements take the form

Tr_¬k,k^′∣1^k′⟩⁻⁻⟨1^k′∣ = fk^−′∣ ˜0k⟩⁺∣ ˜0k^′⟩⁻⁻⟨˜0k^′∣⁺⟨˜0k∣ + (1−fk^−′−fk⁻+∣A⁽¹⁾kk^′∣²) ∣ ˜0k⟩⁺∣ ˜1k^′⟩⁻⁻⟨˜1k^′∣⁺⟨˜0k∣ + (fk⁻−∣A⁽¹⁾_kk^′∣²) ∣ ˜1k⟩⁺∣ ˜1k^′⟩⁻⁻⟨˜1k^′∣⁺⟨˜1k∣ + (∑

q<0

G_kG_k^∗′A^(1)∗_qk A⁽¹⁾_qk^′∣ ˜0k⟩⁺∣ ˜0k^′⟩⁻⁻⟨˜1k^′∣⁺⟨˜1k∣ + h.c.) , (6.4.13a) Tr_¬k,k^′∣1^k⟩⁺⁺⟨1^k∣ = fk⁺∣ ˜0^k⟩⁺∣ ˜0^k′⟩⁻⁻⟨˜0^k′∣⁺⟨˜0^k∣ + (1−fk^+′−fk⁺+∣A⁽¹⁾kk^′∣²) ∣ ˜1^k⟩⁺∣ ˜0^k′⟩⁻⁻⟨˜0^k′∣⁺⟨˜1^k∣

+ (fk^+′−∣A⁽¹⁾kk^′∣²) ∣ ˜1^k⟩⁺∣ ˜1^k′⟩⁻⁻⟨˜1^k′∣⁺⟨˜1^k∣ − (∑

p≥0

G_kG_k^∗′A^(1)∗pk A⁽¹⁾pk^′∣ ˜0^k⟩⁺∣ ˜0^k′⟩⁻⁻⟨˜1^k′∣⁺⟨˜1^k∣ + h.c.) , (6.4.13b) Tr_¬k,k^′∣1^k⟩⁺⁻⟨1^k′∣ = (Gk^∗G_k^∗′∣A⁽¹⁾_kk^′∣²+ A^∗kkA^∗k^′k^′) ∣ ˜1^k⟩⁺∣ ˜0k^′⟩⁻⁻⟨˜1k^′∣⁺⟨˜0k∣ , (6.4.13c) where in (6.4.13c) A^∗kkA^∗k^′k^′ is kept only to order h² in the small h expansion

A^∗kkA^∗k^′k^′ = Gk^∗G_k^∗′+ Gk^∗′A^(2)∗kk + Gk^∗A^(2)∗k^′k^′ + O(h³) . (6.4.14)