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| In [[quantum mechanics]], in particular [[quantum information]], the '''Range criterion''' is a necessary condition that a state must satisfy in order to be [[separable states|separable]]. In other words, it is a ''separability criterion''.
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| == The result ==
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| Consider a quantum mechanical system composed of ''n'' subsystems. The state space ''H'' of such a system is the tensor product of those of the subsystems, i.e. <math>H = H_1 \otimes \cdots \otimes H_n</math>.
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| For simplicity we will assume throughout that all relevant state spaces are finite dimensional.
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| The criterion reads as follows: If ρ is a separable mixed state acting on ''H'', then the range of ρ is spanned by a set of product vectors.
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| === Proof ===
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| In general, if a matrix ''M'' is of the form <math>M = \sum_i v_i v_i^*</math>, it is obvious that the range of ''M'', ''Ran(M)'', is contained in the linear span of <math>\; \{ v_i \}</math>. On the other hand, we can also show <math>v_i</math> lies in ''Ran(M)'', for all ''i''. Assume without loss of generality ''i = 1''. We can write
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| <math>M = v_1 v_1 ^* + T</math>, where ''T'' is Hermitian and positive semidefinite. There are two possibilities:
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| 1) ''span''<math>\{ v_1 \} \subset</math>''Ker(T)''. Clearly, in this case, <math>v_1 \in</math> ''Ran(M)''.
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| 2) Notice 1) is true if and only if ''Ker(T)''<math>\;^{\perp} \subset</math> ''span''<math>\{ v_1 \}^{\perp}</math>, where <math>\perp</math> denotes orthogonal compliment. By Hermiticity of ''T'', this is the same as ''Ran(T)''<math>\subset</math> ''span''<math>\{ v_1 \}^{\perp}</math>. So if 1) does not hold, the intersection ''Ran(T)'' <math>\cap</math> ''span''<math>\{ v_1 \}</math> is nonempty, i.e. there exists some complex number α such that <math>\; T w = \alpha v_1</math>. So
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| :<math>M w = \langle w, v_1 \rangle v_1 + T w = ( \langle w, v_1 \rangle + \alpha ) v_1.</math> | |
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| Therefore <math>v_1</math> lies in ''Ran(M)''.
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| Thus ''Ran(M)'' coincides with the linear span of <math>\; \{ v_i \}</math>. The range criterion is a special case of this fact.
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| A density matrix ρ acting on ''H'' is separable if and only if it can be written as
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| :<math>\rho = \sum_i \psi_{1,i} \psi_{1,i}^* \otimes \cdots \otimes \psi_{n,i} \psi_{n,i}^*</math>
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| where <math>\psi_{j,i} \psi_{j,i}^*</math> is a (un-normalized) pure state on the ''j''-th subsystem. This is also
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| :<math>
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| \rho = \sum_i ( \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} ) ( \psi_{1,i} ^* \otimes \cdots \otimes \psi_{n,i} ^* ).
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| </math>
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| But this is exactly the same form as ''M'' from above, with the vectorial product state <math>\psi_{1,i} \otimes \cdots \otimes \psi_{n,i}</math> replacing <math>v_i</math>. It then immediately follows that the range of ρ is the linear span of these product states. This proves the criterion.
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| == References ==
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| * P. Horodecki, "Separability Criterion and Inseparable Mixed States with Positive Partial Transposition", ''Physics Letters'' '''A 232''', (1997).
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| [[Category:Quantum information science]]
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