quantum entanglement and gravity
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Lawrence Crowell
Jun 24, 2021, 8:12:19 AM6/24/21
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https://www.youtube.com/watch?v=vgYQglmYU-8
Quantum entropy is S = -k Tr[ρ log(ρ)] and this is larger for the density matrix ρ = |ψ〉〈ψ| pertaining to an entangled system. One can see how this formula works for a microcanonical case so that ρ_n = 1/n, a probability p = 1/n and the trace a sum from 0 to N. We then write the entropy as S = -k sum_n[1/n log(1/n)] which give the summation to N is S = k log(N).
To see how entanglement enters into this consider the Taylor expansion of the logarithm,
log(ρ) = ρ – 1 – ½(ρ – 1)^2 + ⅓(ρ – 1)^3 - … .
where if the density matrix has off diagonal terms the quadratic and higher powers of the density matrix will contribute. The quantum phase of a system is in the off diagonal elements, where the quantum phase defines entanglements etc, then the square of the density matrix will give contributions from these terms.
This plays a role with quantum gravitation. The Bekenstein rule S = k A/4ℓ_p^2 defines the entropy of a black hole with event horizon area A = 4πR_s^2 for R_s = 2GM/c^2 the Schwarzschild radius. The Planck length ℓ_p = √(Għ/c^3) is the minimal length a quantum bit can be isolated. The Bekenstein rule is a classical principle. For quantum gravitation, or in a semi-classical or O(ħ) expansion of quantum gravitation, Bekenstein’s rule will become
S = k A/4ℓ_p^2 + 〈quantum corrections〉.
This last part with quantum corrections has to do with how spacetime is built up from quantum entanglements. Sorry, I cannot go into this in much greater detail in a short post. It is a bit deep.
Quantum entropy is S = -k Tr[ρ log(ρ)] and this is larger for the density matrix ρ = |ψ〉〈ψ| pertaining to an entangled system. One can see how this formula works for a microcanonical case so that ρ_n = 1/n, a probability p = 1/n and the trace a sum from 0 to N. We then write the entropy as S = -k sum_n[1/n log(1/n)] which give the summation to N is S = k log(N).
To see how entanglement enters into this consider the Taylor expansion of the logarithm,
log(ρ) = ρ – 1 – ½(ρ – 1)^2 + ⅓(ρ – 1)^3 - … .
where if the density matrix has off diagonal terms the quadratic and higher powers of the density matrix will contribute. The quantum phase of a system is in the off diagonal elements, where the quantum phase defines entanglements etc, then the square of the density matrix will give contributions from these terms.
This plays a role with quantum gravitation. The Bekenstein rule S = k A/4ℓ_p^2 defines the entropy of a black hole with event horizon area A = 4πR_s^2 for R_s = 2GM/c^2 the Schwarzschild radius. The Planck length ℓ_p = √(Għ/c^3) is the minimal length a quantum bit can be isolated. The Bekenstein rule is a classical principle. For quantum gravitation, or in a semi-classical or O(ħ) expansion of quantum gravitation, Bekenstein’s rule will become
S = k A/4ℓ_p^2 + 〈quantum corrections〉.
This last part with quantum corrections has to do with how spacetime is built up from quantum entanglements. Sorry, I cannot go into this in much greater detail in a short post. It is a bit deep.
LC
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