# How do I compute the partial trace of $\sqrt{p}|0\rangle_a|0\rangle_b+\sqrt{1-p}|1\rangle_a |1\rangle_b$?

Here I have a two pure state system composed from systems A and B: $$\Psi_{ab} = \sqrt{p}\, |0\rangle_a |0\rangle_b + \sqrt{1-p}\, |1\rangle_a |1\rangle_b$$

How do I extract system A in matrix form using trace?

My attempt: $$\hat A = \mathrm{Tr}_b\,( |\Psi_{ab} \rangle \langle \Psi_{ab}|) = \sum_N \langle n_b| \Psi_{ab} \rangle \langle \Psi_{ab}| n_b \rangle$$

which makes

$$\hat A = \left(\begin{matrix} p & 0 \\ 0 & 1-p \end{matrix}\right)$$

however, my confusion is that if I were to look for the operator of B then I would also get the same answer. Is my understanding of Trace for a Hilbert space (A or B) incorrect?

Any help would be most appreciated. Thanks.

• Since your system is unchanged under $a\leftrightarrow b$, isn’t it expected that $Tr_a=Tr_b$? Apr 22, 2021 at 3:53
• @ZeroTheHero I think so, my understanding of this is just fairly shaky. But that is my initial intuition, I am just not sure if that is the correct one. Apr 22, 2021 at 5:50

Let me illustrate this from a more general perspective: For a pure state of a bipartite system, the corresponding reduced density matrices have the same non-zero eigenvalues.

To see this, consider a Hilbert space $$\mathscr{H} = \mathscr{H}_1 \otimes \mathscr{H}_2$$ and let $$\{|a_i\rangle\}_i$$ and $$\{|b_j\rangle\}_j$$ denote complete and orthonormal bases sets in $$\mathscr{H}_1$$ and $$\mathscr{H}_2$$, respectively. Then a generic state $$|\psi\rangle \in \mathscr{H}$$ can be expanded as follows:

$$|\psi\rangle = \sum\limits_{ij}c_{ij}\,|a_i\rangle \otimes |b_j\rangle \quad .$$

However, using the Schmidt decomposition, we can also write this state as

$$|\psi\rangle = \sum\limits_{\alpha} \sigma_{\alpha} \,|A_\alpha\rangle \otimes |B_\alpha\rangle \quad,$$

with $$\sigma_\alpha >0$$, $$\sum\limits_{\alpha} \sigma_{\alpha}^2 = 1$$ and $$\{|A_{\alpha}\rangle\}_{\alpha}$$ as well as $$\{|B_{\alpha}\rangle\}_{\alpha}$$ denoting complete orthonormal sets in $$\mathscr{H}_1$$ and $$\mathscr{H}_2$$, respectively.

Defining the density operator $$\rho_{\psi} \equiv |\psi\rangle\langle\psi|$$, we can compute its reduced density matrices $$\rho_1\equiv \mathrm{Tr}_2 \,\rho_{\psi}$$ and $$\rho_2\equiv \mathrm{Tr}_1\,\rho_{\psi}$$ in a straightforward manner and find that they have the same non-zero eigenvalues, namely $$\sigma_{\alpha}^2$$:

\begin{align} \rho_1 &= \sum\limits_\alpha \sigma_{\alpha}^2 \, |A_{\alpha}\rangle \langle A_{\alpha}| \\ \rho_2 &= \sum\limits_\alpha \sigma_{\alpha}^2 \, |B_{\alpha}\rangle \langle B_{\alpha}| \quad . \end{align}

In conclusion, your result perfectly fits in this scheme, as the wave function you have given is already in its Schmidt decomposition.

A more detailed treatment of this should be given in any textbook on quantum information theory.

• Thank you! I didn't even know that this fits into quantum information theory, this is material from my quantum computing class. I now know where to look in the future. Apr 23, 2021 at 1:28