Editing Quantum mechanics (section)

=== Composite systems and entanglement ===
When two different quantum systems are considered together, the Hilbert space of the combined system is the [[tensor product]] of the Hilbert spaces of the two components. For example, let {{mvar|A}} and {{mvar|B}} be two quantum systems, with Hilbert spaces <math> \mathcal H_A </math> and <math> \mathcal H_B </math>, respectively. The Hilbert space of the composite system is then
<math display=block> \mathcal H_{AB} = \mathcal H_A \otimes \mathcal H_B.</math>
If the state for the first system is the vector <math>\psi_A</math> and the state for the second system is <math>\psi_B</math>, then the state of the composite system is
<math display=block>\psi_A \otimes \psi_B.</math>
Not all states in the joint Hilbert space <math>\mathcal H_{AB}</math> can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if <math>\psi_A</math> and <math>\phi_A</math> are both possible states for system <math>A</math>, and likewise <math>\psi_B</math> and <math>\phi_B</math> are both possible states for system <math>B</math>, then
<math display=block>\tfrac{1}{\sqrt{2}} \left ( \psi_A \otimes \psi_B + \phi_A \otimes \phi_B \right )</math>
is a valid joint state that is not separable. States that are not separable are called [[quantum entanglement|entangled]].<ref name=":0">{{Cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010 |edition=2nd |oclc=844974180 |isbn=978-1-107-00217-3 |author-link1=Michael Nielsen |author-link2=Isaac Chuang}}</ref><ref name=":1">{{Cite book |title-link=Quantum Computing: A Gentle Introduction |title=Quantum Computing: A Gentle Introduction |last1=Rieffel |first1=Eleanor G. |last2=Polak |first2=Wolfgang H. |year=2011 |publisher=MIT Press |isbn=978-0-262-01506-6 |author-link=Eleanor Rieffel}}</ref>

If the state for a composite system is entangled, it is impossible to describe either component system {{mvar|A}} or system {{mvar|B}} by a state vector. One can instead define [[reduced density matrix|reduced density matrices]] that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.<ref name=":0" /><ref name=":1" /> Just as density matrices specify the state of a subsystem of a larger system, analogously, [[POVM|positive operator-valued measures]] (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.<ref name=":0" /><ref name="wilde">{{Cite book |last=Wilde |first=Mark M. |title=Quantum Information Theory |publisher=Cambridge University Press |year=2017 |isbn=978-1-107-17616-4 |edition=2nd |doi=10.1017/9781316809976.001 |arxiv=1106.1445 |s2cid=2515538 |oclc=973404322}}</ref>

As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as [[quantum decoherence]]. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.<ref>{{Cite journal |last=Schlosshauer |first=Maximilian |date=October 2019 |title=Quantum decoherence |journal=Physics Reports |volume=831 |pages=1–57 |arxiv=1911.06282 |bibcode=2019PhR...831....1S |doi=10.1016/j.physrep.2019.10.001 |s2cid=208006050}}</ref>