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Update: This is a clarification following Peter Shor's remark: Among the biparticle pure states, the maximally entangled states have the property that their partial trace with respect to one system is … As a generalization, I wish to know the local orbits of the states within a fixed biparticle density matrix orbit whose partial trace has the maximal von Neumann entropy relative to all other states in …

(There is another procedure to obtain mixed states using a partial trace on a composite system, but I am not interested on this here). …

But how is the partial trace found and defined in terms of the matrix representation of the linear operator. … Does the input and output basis have to be the same to define partial trace similar to definition of trace ? …

Why I am having trouble is perhaps because of the partial trace. … trace. …

I mean an explanation which is less technical than that decoherence is the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix, i.e. the … trace, with respect to any environmental basis, of the density matrix of the combined system and its environment [...] …

.$$ A partial trace over the second system yields the first state. This is the purification. A mixed state is always a partial trace of some pure state in a composite system. …

Why should taking the partial trace give the correct description of the system? … Taking the partial trace turns a pure density matrix into a mixed density matrix, but a pure state is always a pure state. …

I want to know if a partial trace keeps the cyclic property of the trace. … $$ When is this partial trace cyclic? …

A way out of this is to introduce density matrices in order to relax the "state"-notion just enough that the "reducing to subsystem" is well-defined by taking the partial trace. …

I saw the following definition for the partial trace operator: $\rho_A=\sum_k \langle e_k|\rho_{AB}|e_k\rangle$, where $e_k$ is basis for the state space of system $B$. … trace. …

For $d<7$ in a reasonable time (few seconds) I able to perform the partial trace $\rho_{AD}$ using the code proposed in http://www3.imperial.ac.uk/people/m.tame/research. … Do you know an efficient algorithm for calculating the partial trace of qudit which uses the fact that the matrix is sparse? …

We then take the partial trace to get \begin{align} \rho_A(t) &= \text{Tr}_B[\rho(t)]=\text{Tr}_B\left[e^{\mathcal{L}t}\rho(0)\right]\\ \dot{\rho}_A(t) &= \frac{d}{dt}\text{Tr}_B\left[\rho(t)\right] = …

Do we just take the partial trace over the ghost sector? But that's not gauge invariant except asymptotically, is it? …

I know that the partial trace of a pure entangled state must be mixed and that of a product pure state must be pure; but I couldn't find an answer to my above question. …

The first one is the definition of the reduced density operator $$\rho^{A} =\mathrm{tr}_B(\rho^{AB})$$ The second one is the definition of the partial trace, which is defined (completely independently … I found the general definition of the partial trace here. So, thanks to this and @CraigGidney, @WetSavannaAnimalakaRodVance, @ACuriousMind answers I now know that the answer is no. …