5.1: Composite Systems
- Page ID
- 56549
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Suppose we have two systems, described by Hilbert spaces \(\mathscr{H}_{1}\) and \(\mathscr{H}_{2}\), respectively. We can choose orthonormal bases for each system:
\[\mathscr{H}_{1}:\left\{\left|\phi_{1}\right\rangle,\left|\phi_{2}\right\rangle, \ldots,\left|\phi_{N}\right\rangle\right\} \quad \text { and } \mathscr{H}_{2}:\left\{\left|\psi_{1}\right\rangle,\left|\psi_{2}\right\rangle, \ldots,\left|\psi_{M}\right\rangle\right\}\tag{5.1}\]
The respective dimensions of \(\mathscr{H}_{1}\) and \(\mathscr{H}_{2}\) are \(N\) and \(M\). We can construct \(N \times M\) basis states for the composite system via \(\left|\phi_{j}\right\rangle\) and \(\left|\psi_{k}\right\rangle\). This implies that the total Hilbert space of the composite system can be spanned by the tensor product
\[\left\{\left|\phi_{j}\right\rangle \otimes\left|\psi_{k}\right\rangle\right\}_{j k} \quad \text { on } \quad \mathscr{H}_{1+2}=\mathscr{H}_{1} \otimes \mathscr{H}_{2}\tag{5.2}\]
An arbitrary pure state on \(\mathscr{H}_{1+2}\) can be written as
\[|\Psi\rangle=\sum_{j k} c_{j k}\left|\phi_{j}\right\rangle \otimes\left|\psi_{k}\right\rangle \equiv \sum_{j k} c_{j k}\left|\phi_{j}, \psi_{k}\right\rangle\tag{5.3}\]
For example, the system of two qubits can be written on the basis \(\{|0,0\rangle,|0,1\rangle,|1,0\rangle,|1,1\rangle\}\). If system 1 is in state \(|\phi\rangle\) and system 2 is in state \(|\psi\rangle\), the partial trace over system 2 yields
\[\operatorname{Tr}_{2}(|\phi, \psi\rangle\langle\phi, \psi|)=\operatorname{Tr}_{2}(|\phi\rangle\langle\phi|\otimes| \psi\rangle\langle\psi|)=|\phi\rangle\langle\phi|\operatorname{Tr}(|\psi\rangle\langle\psi|)=| \phi\rangle\langle\phi|,\tag{5.4}\]
since the trace over any density operator is 1. We have now lost system 2 from our description! Therefore, taking the partial trace without inserting any other operators is the mathematical version of forgetting about it. This is a very useful feature: you often do not want to deal with every possible system you are interested in. For example, if system 1 is a qubit, and system two is a very large environment the partial trace allows you to “trace out the environment”.
However, tracing out the environment will not always leave you with a pure state as in Eq. (5.4). If the system has interacted with the environment, taking the partial trace generally leaves you with a mixed state. This is due to entanglement between the system and its environment.