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This question is a follow up to this question on the derivation of the reduced density matrix.

What this question is about:

According to Schlosshauer (ISBN: 978-3540357735), the reduced density matrix is a mathematical object that contains all information that an observer could learn about a subsystem $\mathcal{A}$ of a system $\mathcal{AB}$ (so it contains all measurement-statistics). In his derivation of it, he proves that it allows the calculation of the expected value of an observable $O_{\mathcal{A}}\otimes I$ acting on $\mathcal{AB}$ - Indeed, he is looking for an object to do just that in order to find the reduced density matrix which (repeating myself) "contains all information that an observer could learn about a subsystem $\mathcal{A}$ of a system $\mathcal{AB}$ (so it contains all measurement-statistics)."

The question:

Is proving

that the reduced density matrix allows the calculation of above mentioned expectation value

enough to state that it contains all information an observer could learn from $\mathcal{A}$ ("all measurement statistics"? What I mean is that, while it is now proven that the expected value (above) can be calculated with the reduced density matrix, it is (in my opinion) not clear that one could calculate the state of $\mathcal{A}$ after the measurement - for either of the measurement results, which are defined by $O_{\mathcal{A}}\otimes I$.

manuel459
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  • Could you elaborate/ rephrase the question a little bit? Suppose you have the definition of the reduced density operator. What do you think is not possible to compute (using only the reduced density matrix) regarding the subsystem? – Tobias Fünke Jun 21 '22 at 12:41
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    I rephrased the last part of the question. I don't see e.g. why the reduced density matrix allows to calculate the state of the subsystem after the measurement, when one only proved that the expected value (in consideration) can be calculated. – manuel459 Jun 21 '22 at 12:48

1 Answers1

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The reduced density matrix allows to compute all expectation values for the subsystem alone, in other words: For all hermitian operators $O_A$ on $H_A$ we have that

$$ \mathrm{Tr}\, \rho \,O_A \otimes \mathbb I_B= \mathrm{Tr}^{(A)}\rho_A O_A \tag{1} \quad .$$

This, in particular, includes the calculation of probabilities of measurement results of local observables: Write the observable $O_A$ in its spectral representation:

$$O_A=\sum\limits_j o_j\, P_j \tag{2} \quad .$$

The probability to measure $o_j$ is then given by $$\mathrm{Tr}\,\rho\, P_j \otimes \mathbb I_B = \mathrm{Tr}^{(A)}\rho_A P_j \tag{3} \quad . $$ To emphasize: The left-hand side is a postulate, the right-hand side a trivial consequence of $(1)$ (which itself is a consequence of the definition of $\rho_A$). The state of the bipartite system after the measurement is $$ \rho^\prime \propto P_j \otimes \mathbb I_B \, \rho\,P_j \otimes \mathbb I_B \tag{4}$$ with the associated reduced density matrix

$$\rho_A^\prime= \mathrm{Tr}_B \,\rho^\prime \propto P_j\, \rho_A\, P_j \quad . \tag{5} $$ Again: Equation $(4)$ is a postulate, equation $(5)$ is a consequence of it. Indeed, the above follows from the observation that for $|\psi\rangle \in H_B$ it holds that

$$\left(P_j \otimes \mathbb I_B\right) \left(\mathbb I_A\otimes |\psi\rangle\right) =P_j \otimes |\psi\rangle = \left(\mathbb I_A \otimes |\psi\rangle\right)\, P_j \tag{6}$$

and $$ \left(\mathbb I_A\otimes \langle \psi| \right) \left(P_j \otimes \mathbb I_B\right) = P_j \otimes \langle \psi| = P_j\, \left(\mathbb I_A\otimes \langle \psi|\right) \quad . \tag{7} $$

Hence, the knowledge of $\rho_A$ suffices to compute all observable properties of the subsystem corresponding to $H_A$.

Source and further reading: J. Audretsch. Entangled Systems: new directions in quantum physics. Wiley, especially chapter 7. A pdf of the relevant chapter can be found here.

Tobias Fünke
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  • I understand what you are saying, but still have a little worry. Let me try to put it into words: Since the reduced density matrix is a density matrix and the above expected value can be calculated - how can I infer that one is allowed to calculate state after measurement and measure-probability as it is done when considering density matrices? This could already be done once you know the reduced density matrix is a density matrix - no need for proving that the expected value can be calculated? It seems there is a non mathematical, rather physical thought still to be seen by myself. – manuel459 Jun 21 '22 at 13:31
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    Well, in my opinion, the above considerations prove this: All predictions you can make regarding observable properties of $H_A$ require only knowledge of $\rho_A$. Put differently: Suppose you know $\rho$. Then you can calculate all this stuff (this is axiomatic). Now if you are interested in the statistics of the subsystem only, then the above calculations show that the knowledge of $\rho_A$ suffices. I did nowhere postulate that we compute the probabilities etc. from $\rho_A$; I derived that. Do I miss or misunderstand something? – Tobias Fünke Jun 21 '22 at 13:35
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    To summarize: The right-hand sides of $(3)$ and $(5)$ are results, derived from the postulates of quantum mechanics, and not postulates themselves. – Tobias Fünke Jun 21 '22 at 13:36
  • I think this hits the nail on his head: Why do I know I can calculate it this way? this foralisms are used for density matrices and I wonder (while reduced density matrices are mathematically identical to density matrices) if this is true for reduced density matrices. It seems to me one might need to prove that using (3), (4) and so on give the same solutions one gets when considering probabilities and states after measurement when considering $\rho$ and $O_{\mathcal{A}}\otimes\mathbb{I}$ using the respective formalisms. – manuel459 Jun 21 '22 at 13:44
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    @manuel459 But this is what I did, at least tried to. I've added a few sentences... Let me know if something is still unclear. I've started with the rules of quantum mechanics (for bipartite systems) and then, using the properties of the reduced density matrix, derived the equations $(3)$ and $(5)$. – Tobias Fünke Jun 21 '22 at 13:48
  • In other words: The postulates work with densitry matrices that are descriptions of systems (which is based on physical thoughts). But reduced density matrices (by now) only formal identical objects that (in my opinion) not necessarily are a descritpion of the subsystem - this has to be proven using postulates and the densitry matrix $\rho$. – manuel459 Jun 21 '22 at 13:48
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    As a side note: Every density matrix of a system can be seen as a reduced density matrix of a composite system (see purification) – Tobias Fünke Jun 21 '22 at 13:50
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    Forget my last comment. I can fully follow your thoughts now! Now it totally makes sense to me. Thank you for the dialogue! – manuel459 Jun 21 '22 at 13:50