I wrote this as a response to Bob's post on firewalls etc. I felt I should
send this in a separate thread,
A central aspect in the formalism of entanglement is the Schmidt
decomposition. A general tensor product of states is
|ψ⟩ = sum_{ij}C_{ij}|i⟩⊗|j⟩.
Now consider the intermediary state |φ_i⟩ = sum_jC_{ij}|j⟩, which forms |ψ⟩
= sum_i|i⟩⊗|φ_i⟩. The density matrix ρ = |ψ⟩⟨ψ| has the trace with respect
to the φ states
Tr_φ(ρ) = sum_{ij} ⟨φ_j |φ_i⟩ |i⟩⟨j|,
and we have ⟨φ_j |φ_i⟩ = p_iδ_{ij}. Clearly then we can write
|ψ⟩ = sum_{ij}√(p_i)|i⟩⊗|j⟩.
This is also a polar form of the wave function.
This Schmidt form is with
√(p_i) = sum_jC_{ij}
where this amplitude transforms as
C_{ij} → C’_{ij} = U_{ik}√(p_k)V_{kj},
The two unitary matrices U_{ik} and V_{kj} are transformations of the two
states |i⟩ and |j⟩ and the transformation of the amplitude C_{ij} is given
by the product. A simple case is with a simple spin model of SU(2) so the
product of the two unitary groups is SU(2)×SU(2) = SO(4).
We can use this with density matrices as a set of diagonal plus
off-diagonal states. Consider, the N×N density matrix
ρ_{ij} = p_iδ_{ij} + σ_iτ_j, i and j =1 … N^2 - 1
where the first term corresponds to maximally mixed states for p_i = 1/N
and the second terms are the off-diagonal quantum phase. Here σ_i is a
generator of an SU(N) group and τ_j a B;och vector. The space of all unit
trace Tr(ρ) = 1 of all N×N density matrices is a manifold designated by ℳ.
This manifold is the intersection, in the set of all Hermitian matrices, of
a positive cone P with the hyperplane parallel to all linear traceless
operators. This is a convex set defined by the set of all projectors onto a
one-dimensional subspace. This the defines the projective geometry ℂP^{N-1}
in the Hilbert space of ℳ.
There is more geometry I could discuss but will defer to later. This
involves some subtle issues with the relationship between the diagonal
trace terms and the off diagonal term corresponding to quantum phases.
The group theoretic implications of this are then interesting. Consider the
rotation of the Bloch vector τ_i → τ’_i according the unitary
transformation of the density matrix
τ’_i = ½Tr(ρ’)σ_i = [ σ_kU_{ki}σ_lU^†_{lj} ] τ_j = sum_jO_{ij} τ_j.
We then have SO(N^2 - 1) matrices that are associated with SU(N), or more
properly that SU(N)/ℤ_N is a subset of SO(N^2 - 1). To consider this let N
= 4, then we have that SU(4)/ ℤ_4 is a subset of SO(15). SO(15) is fixed in
a frame of SO(16) and this corresponds to a U(1) fibration over SU(4) as
U(4) = SU(4)×U(1).
The Hopf fibration defines the sphere S^4 = O(6)/O(5) or that O(6) ≈ U(4)
is the 4-sphere with an O(5) fibration. If we shift to a hyperbolic setting
then we have O(4,2)/SO(5,1) = AdS_5. with the quotient on the O(4,2) =
U(2,2). We then clearly have a correspondence with the orthogonal group
SO(16). The correspondence to AdS_5×S^5 is then with the Cartan
decomposition SO(32) → SO(16)×120 and the corresponding unitary group is
U(2,2,ℂ) in complex conformal relativity. There is a conservation of
information between the U(2,2) and SO(16), where the first pertains to
conformal gravitation and the latter a gauge field theory.
There is the issue of a “sleight of hand” where the unitary group is in
split form, corresponding to spacetime with Lorentzian metric and the
orthogonal group is Euclidean and corresponds to a gauge group. The claim
here is that for the Lorentzian group a difficulty is this leads to
negative probabilities. However, this really is not as bad as one might
think. Coherent states, such as with laser photons or condensates, have
this feature. These forms of quantum states have both a Riemannian and
symplectic geometric structure. These over-complete quantum states give a
way that classical-like structure can emerge from quantum physics. The
central feature of pure state quantum mechanics is linearity of Hilbert
space of states and operators. The transition to nonlinearity with this
conservation of information, say qubits ↔ spatial or spacetime information,
is a feature of how state collapse and the stability of classical states
does not violate conservation of information.
I am going to try to respond to a post by Bruce on the Born rule as time
permits.
Cheers LC
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