Schrödinger–HJW theorem explained

In quantum information theory and quantum optics, the Schrödinger–HJW theorem is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators. The theorem is named after physicists and mathematicians Erwin Schrödinger,^[1] Lane P. Hughston, Richard Jozsa and William Wootters.^[2] The result was also found independently (albeit partially) by Nicolas Gisin,^[3] and by Nicolas Hadjisavvas building upon work by Ed Jaynes,^{[4] [5]} while a significant part of it was likewise independently discovered by N. David Mermin.^[6] Thanks to its complicated history, it is also known by various other names such as the GHJW theorem,^[7] the HJW theorem, and the purification theorem.

Purification of a mixed quantum state

Let

l H _{S} be a finite-dimensional complex Hilbert space, and consider a generic (possibly mixed) quantum state \rho defined on l H _{S} and admitting a decomposition of the form$$\backslash rho\; =\; \backslash sum\_i\; p\_i|\backslash phi\_i\backslash rangle\backslash langle\backslash phi\_i|$$for a collection of (not necessarily mutually orthogonal) states |\phi _{i\rangle} \in l H _{S} and coefficients p _{i} \ge 0 such that $\backslash sum\_i\; p\_i\; =\; 1$. Note that any quantum state can be written in such a way for some \{|\phi _{i\rangle\}} _{i} and \{p _{i\}} _{i} .^[8]

Any such

\rho can be purified, that is, represented as the partial trace of a pure state defined in a larger Hilbert space. More precisely, it is always possible to find a (finite-dimensional) Hilbert space l H _{A} and a pure state |\Psi _{SA} \rangle \in l H _{S} ⊗ l H _{A} such that \rho = \operatorname{Tr} _{A(|\Psi} _{SA} \rangle\langle\Psi _{SA} | ) . Furthermore, the states |\Psi _{SA} \rangle satisfying this are all and only those of the form$$|\backslash Psi\_\backslash rangle\; =\; \backslash sum\_i\; \backslash sqrt\; |\backslash phi\_i\backslash rangle\; \backslash otimes\; |a\_i\backslash rangle$$for some orthonormal basis \{|a _{i\rangle\}} _{i} \subset l H _{A} . The state |\Psi _{SA} \rangle is then referred to as the "purification of \rho ". Since the auxiliary space and the basis can be chosen arbitrarily, the purification of a mixed state is not unique; in fact, there are infinitely many purifications of a given mixed state.^[9] Because all of them admit a decomposition in the form given above, given any pair of purifications |\Psi\rangle, |\Psi'\rangle \in l H _{S} ⊗ l H _{A} , there is always some unitary operation U : l H _{A} \to l H _{A} such that$$|\backslash Psi\text{'}\backslash rangle\; =\; (I\backslash otimes\; U)\; |\backslash Psi\backslash rangle.$$

Theorem

Consider a mixed quantum state

\rho with two different realizations as ensemble of pure states as $\backslash rho\; =\; \backslash sum\_i\; p\_i\; |\backslash phi\_i\backslash rangle\backslash langle\backslash phi\_i|$ and $\backslash rho\; =\; \backslash sum\_j\; q\_j\; |\backslash varphi\_j\backslash rangle\backslash langle\backslash varphi\_j|$. Here both |\phi _{i\rangle} and |\varphi _{j\rangle} are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state \rho reading as follows:

Purification 1:

|\Psi ^{1 \rangle = \sum} _{i\sqrt{p} _{i}|\phi} _{i\rangle} ⊗ |a _{i\rangle} ;

Purification 2:

|\Psi ^{2 \rangle = \sum} _{j\sqrt{q} _{j}|\varphi} _{j\rangle} ⊗ |b _{j\rangle} .

The sets

\{|a _{i\rangle\}} and \{|b _{j\rangle\}} are two collections of orthonormal bases of the respective auxiliary spaces. These two purifications only differ by a unitary transformation acting on the auxiliary space, namely, there exists a unitary matrix U _{A} such that |\Psi ^{1} _{SA} \rangle = (I ⊗ U ^{2} _{SA} \rangle .^[10] Therefore, $|\backslash Psi\_^1\backslash rangle\; =\; \backslash sum\_j\; \backslash sqrt|\backslash varphi\_j\backslash rangle\backslash otimes\; U\_A|b\_j\backslash rangle$, which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.

Notes and References

1. Schrödinger . Erwin . Probability relations between separated systems . 1936 . . 32 . 3 . 446–452 . 10.1017/S0305004100019137 . 1936PCPS...32..446S .
2. Hughston . Lane P. . Jozsa . Richard . Wootters . William K. . November 1993 . A complete classification of quantum ensembles having a given density matrix . . 183 . 1 . 14–18 . 10.1016/0375-9601(93)90880-9 . 0375-9601 . 1993PhLA..183...14H .
3. Gisin, N. (1989). “Stochastic quantum dynamics and relativity”, Helvetica Physica Acta 62, 363–371.
4. Hadjisavvas . Nicolas . 1981 . Properties of mixtures on non-orthogonal states . . 5 . 4 . 327–332 . 10.1007/BF00401481 . 1981LMaPh...5..327H .
5. Jaynes . E. T. . 1957 . Information theory and statistical mechanics. II . . 108 . 2 . 171–190 . 10.1103/PhysRev.108.171 . 1957PhRv..108..171J .
6. Book: Fuchs, Christopher A. . Coming of Age with Quantum Information: Notes on a Paulian Idea . 2011 . . 978-0-521-19926-1 . Cambridge . 535491156.
7. Mermin . N. David . N. David Mermin . 1999 . What Do These Correlations Know about Reality? Nonlocality and the Absurd . . 29 . 4 . 571–587 . quant-ph/9807055 . 1998quant.ph..7055M . 10.1023/A:1018864225930.
8. .
9. Book: Watrous, John . The Theory of Quantum Information . 2018. Cambridge University Press . 978-1-107-18056-7 . Cambridge . 10.1017/9781316848142.
10. Kirkpatrick . K. A. . February 2006 . The Schrödinger-HJW Theorem . . 19 . 1 . 95–102 . 10.1007/s10702-006-1852-1 . 0894-9875 . quant-ph/0305068 . 2006FoPhL..19...95K .