Action-angle coordinates: Difference between revisions

Revision as of 18:57, 4 April 2013

In quantum mechanics, in particular quantum information, the Range criterion is a necessary condition that a state must satisfy in order to be separable. In other words, it is a separability criterion.

The result

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. $H = H_{1} \otimes \dots \otimes H_{n}$ .

For simplicity we will assume throughout that all relevant state spaces are finite dimensional.

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.

Proof

In general, if a matrix M is of the form $M = \sum_{i} v_{i} v_{i}^{*}$ , it is obvious that the range of M, Ran(M), is contained in the linear span of ${v_{i}}$ . On the other hand, we can also show $v_{i}$ lies in Ran(M), for all i. Assume without loss of generality i = 1. We can write $M = v_{1} v_{1}^{*} + T$ , where T is Hermitian and positive semidefinite. There are two possibilities:

1) span ${v_{1}} \subset$ Ker(T). Clearly, in this case, $v_{1} \in$ Ran(M).

2) Notice 1) is true if and only if Ker(T) $^{⊥} \subset$ span ${v_{1}}^{⊥}$ , where $⊥$ denotes orthogonal compliment. By Hermiticity of T, this is the same as Ran(T) $\subset$ span ${v_{1}}^{⊥}$ . So if 1) does not hold, the intersection Ran(T) $\cap$ span ${v_{1}}$ is nonempty, i.e. there exists some complex number α such that $T w = α v_{1}$ . So

M w = ⟨ w, v_{1} ⟩ v_{1} + T w = (⟨ w, v_{1} ⟩ + α) v_{1} .

Therefore $v_{1}$ lies in Ran(M).

Thus Ran(M) coincides with the linear span of ${v_{i}}$ . The range criterion is a special case of this fact.

A density matrix ρ acting on H is separable if and only if it can be written as

ρ = \sum_{i} ψ_{1, i} ψ_{1, i}^{*} \otimes \dots \otimes ψ_{n, i} ψ_{n, i}^{*}

where $ψ_{j, i} ψ_{j, i}^{*}$ is a (un-normalized) pure state on the j-th subsystem. This is also

ρ = \sum_{i} (ψ_{1, i} \otimes \dots \otimes ψ_{n, i}) (ψ_{1, i}^{*} \otimes \dots \otimes ψ_{n, i}^{*}) .

But this is exactly the same form as M from above, with the vectorial product state $ψ_{1, i} \otimes \dots \otimes ψ_{n, i}$ replacing $v_{i}$ . It then immediately follows that the range of ρ is the linear span of these product states. This proves the criterion.

References

P. Horodecki, "Separability Criterion and Inseparable Mixed States with Positive Partial Transposition", Physics Letters A 232, (1997).

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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''.
+== The result ==
+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>.
+For simplicity we will assume throughout that all relevant state spaces are finite dimensional.
+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.
+=== Proof ===
+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
+<math>M = v_1 v_1 ^* + T</math>, where ''T'' is Hermitian and positive semidefinite. There are two possibilities:
+) ''span''<math>\{ v_1 \} \subset</math>''Ker(T)''. Clearly, in this case, <math>v_1 \in</math> ''Ran(M)''.
+) 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
+:<math>M w = \langle w, v_1 \rangle v_1 + T w = ( \langle w, v_1 \rangle + \alpha ) v_1.</math>
+Therefore <math>v_1</math> lies in ''Ran(M)''.
+Thus ''Ran(M)'' coincides with the linear span of <math>\; \{ v_i \}</math>. The range criterion is a special case of this fact.
+A density matrix ρ acting on ''H'' is separable if and only if it can be written as
+:<math>\rho = \sum_i \psi_{1,i} \psi_{1,i}^* \otimes \cdots \otimes \psi_{n,i} \psi_{n,i}^*</math>
+where <math>\psi_{j,i} \psi_{j,i}^*</math> is a (un-normalized) pure state on the ''j''-th subsystem. This is also
+:<math>
+\rho = \sum_i ( \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} ) ( \psi_{1,i} ^* \otimes \cdots \otimes \psi_{n,i} ^* ).
+</math>
+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.
+== References ==
+* P. Horodecki, "Separability Criterion and Inseparable Mixed States with Positive Partial Transposition", ''Physics Letters'' '''A 232''', (1997).
+[[Category:Quantum information science]]

Action-angle coordinates: Difference between revisions

Revision as of 18:57, 4 April 2013

The result

Proof

References

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