A question arose to me while reading the first chapter of Sakurai's Modern Quantum Mechanics. Given a Hilbert space, is the outer product [itex] \mathcal{H}\times \mathcal{H}^\ast \to End(\mathcal{H}); (| \alpha\rangle,\langle \beta|)\mapsto | \alpha\rangle\langle \beta|[/itex] a surjection? Ie, can any linear self-map of H be formed by tacking together a suitable ket and bra?(adsbygoogle = window.adsbygoogle || []).push({});

After thinking about this a bit longer I realise the answer is no. If we think about a n-dimensional Hilbert space (n < oo), then the outer product operation corresponds to matrix multiplication of a column vector with a row vector. Clearly not all n x n matrices can be formed in this way. I'm not sure quite how many matrices you can cover in this manner, however.

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# Outer product in Hilbert space

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