I Partial trace and the reduced density matrix

yucheng
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TL;DR Summary
Trace paradox?
From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix:
$$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b \rangle - \langle \phi_b| \rho_{AB} V | \phi_b \rangle = Tr_B(V \rho_AB) - Tr_B(\rho_AB V) = 0???$$
 
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yucheng said:
TL;DR Summary: Trace paradox?
I have corrected your LaTeX formula to make it readable and meaningful:
$$i \hbar \dot{\rho}_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle $$
$$= \Sigma_b \langle\phi_b | V \rho_{AB} | \phi_b \rangle - \langle \phi_b| \rho_{AB} V | \phi_b \rangle = Tr_B(V \rho_{AB}) - Tr_B(\rho_{AB} V) = 0???$$
 
By the way, there is also another instructive trace paradox. Since ##[x,p]=i\hbar 1##, we have
$${\rm Tr} [x,p] ={\rm Tr}(i\hbar 1)=i\hbar {\rm Tr}1=i\hbar\infty$$
but also
$${\rm Tr} [x,p] ={\rm Tr} (xp) - {\rm Tr} (px) =0$$
so
$$0=i\hbar\infty$$
Can you resolve this one? :wink:

Hint: The solution of this paradox is entirely unrelated to the solution of the previous one. The key is to understand the meaning of ##{\rm Tr}1=\infty##, can we pretend that it is actually a big but finite number?
 
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@Demystifier
OMG I did not even realize it was published! I thought I just left it as a draft, but thanks for replying!

After doing some other problems, I realized that a partial trace is defined for a composite Hilbert space, which means that taking the trace with respect to the ##| b \rangle## basis breaks the common argument for commutation under the trace i,e, using the resolution of the identity because we have ##\Sigma \langle b' |\rho_{AB}| a,b \rangle \langle a,b| V |b' \rangle## instead.
 
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Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
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