Consider pure or nearly pure quantum state, is it usually low-rank? Can you give a example of a concrete state and its rank?
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2$\begingroup$ Pure states by definition have rank one $\endgroup$– Quantum MechanicCommented Aug 10, 2022 at 12:39
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$\begingroup$ Thank you, but why? Does the N&C book contain this ? $\endgroup$– karryCommented Aug 10, 2022 at 12:47
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$\begingroup$ Sure it does. It's by definition a projector onto a one-dimensional subspace ... $\endgroup$– Markus HeinrichCommented Aug 10, 2022 at 12:52
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$\begingroup$ I find only from Exercise 2.73: A minimal ensemble for $\rho$ is an ensemble containing a number of elements equal to the rank of $\rho$. $\endgroup$– karryCommented Aug 10, 2022 at 13:16
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$\begingroup$ The rank is the dimension of the image. A pure state is a state of the form $\rho = |\psi\rangle\langle\psi|$, in particular it is an orthogonal projection onto the one-dimensional subspace spanned by the vector $\psi$. In formula, $\mathrm{im}(\rho) = \mathrm{span}(\psi)$, thus $\mathrm{rk}(\rho) = 1$. Or in the language of that exercise: a pure state is an ensemble containing exactly one element. $\endgroup$– Markus HeinrichCommented Aug 10, 2022 at 15:55
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A pure state is by definition rank one, $\rho = | \psi \rangle\langle \psi |$.
A state can have maximal rank and be arbitrarily close to a pure state. Just mix a pure state with the maximally mixed state, i.e. $\rho := (1-\varepsilon)| \psi \rangle\langle \psi | + \varepsilon \mathrm{I}/d$. This state is of maximal rank $d$ for any $\varepsilon > 0$.
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$\begingroup$ So only pure states have the rank one, the rank of mixed states will larger than 1? $\endgroup$– karryCommented Aug 10, 2022 at 13:17
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$\begingroup$ @KarryMa Exactly, pure states are the quantum states with rank one. Consequently, mixed (non-pure) states have higher rank. $\endgroup$ Commented Aug 10, 2022 at 15:51