Come faccio a dimostrare che uno stato a due qubit è uno stato impigliato?

Lo stato di Bell è uno stato entangled. Ma perché è così? Come posso dimostrarlo matematicamente?$|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle )$

Definizione

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Uno stato a due qubit è uno stato entangled se e solo se non ci non esistono due stati uno-qubit | un ⟩ = alfa | 0 ⟩ + ß | 1 ⟩ ∈ C 2 e | b ⟩ = y | 0 ⟩ + À | 1 ⟩ ∈ C 2 tali che | un ⟩ ⊗ | b ⟩ = | $|\psi\rangle \in \mathbb{C}^4$$|a\rangle = \alpha |0\rangle + \beta |1\rangle \in \mathbb{C}^2$$|b\rangle = \gamma |0\rangle + \lambda |1\rangle \in \mathbb{C}^2$ , Dove ⊗ indica ilprodotto tensorialee α , β , γ , À ∈ C .$|a\rangle \otimes |b\rangle = |\psi\rangle$$\otimes$$\alpha, \beta, \gamma, \lambda \in \mathbb{C}$

Quindi, per dimostrare che lo stato di Bell è uno stato di entanglement, dobbiamo semplicemente dimostrare che non esistono due stati uno-qubit| un⟩e| b⟩tali che| Φ+⟩=| un⟩⊗| b⟩.$|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle )$$|a\rangle$$|b\rangle$$|\Phi^{+}\rangle = |a\rangle \otimes |b\rangle$

Prova

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Supporre che

$$\begin{align} |\Phi^{+}\rangle &= |a\rangle \otimes |b\rangle \\ &= \left( \alpha |0\rangle + \beta |1\rangle \right) \otimes \left( \gamma |0\rangle + \lambda |1\rangle \right) \end{align}$$

Ora possiamo semplicemente applicare la proprietà distributiva per ottenere

$$\begin{align} |\Phi^{+}\rangle &= \cdots \\ &= \left( \alpha \gamma |00\rangle + \alpha \lambda |01\rangle + \beta \gamma |10\rangle + \beta \lambda |11\rangle \right) \end{align}$$

Questo deve essere uguale a , che è, dobbiamo trovare i coefficientialfa,β,γeλ, tali che$\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle )$$\alpha$$\beta$$\gamma$$\lambda$

$$\begin{align} \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle ) &= \left( \alpha \gamma |00\rangle + \alpha \lambda |01\rangle + \beta \gamma |10\rangle + \beta \lambda |11\rangle \right) \end{align}$$

Osservare che, nell'espressione , vogliamo mantenere sia | 00 ⟩ e | 11 ⟩ . Quindi, α e γ , che sono i coefficienti di | 00 ⟩ , non può essere pari a zero; in altre parole, dobbiamo avere α ≠ 0 e γ ≠ 0 . Allo stesso modo,$\alpha \gamma |00\rangle + \alpha \lambda |01\rangle + \beta \gamma |10\rangle + \beta \lambda |11\rangle$$|00\rangle$$|11\rangle$$\alpha$$\gamma$$|00\rangle$$\alpha \neq 0$$\gamma \neq 0$$\beta$ and $\lambda$, which are the complex numbers multiplying $|11\rangle$ cannot be zero, i.e. $\beta \neq 0$ and $\lambda \neq 0$. So, all complex numbers $\alpha$, $\beta$, $\gamma$ and $\lambda$ must be different from zero.

$|\Phi^{+}\rangle$, we want to get rid of $|01\rangle$ and $|10\rangle$. So, one of the numbers (or both) multiplying $|01\rangle$ (and $|10\rangle$) in the expression $\alpha \gamma |00\rangle + \alpha \lambda |01\rangle + \beta \gamma |10\rangle + \beta \lambda |11\rangle$, i.e. $\alpha$ and $\lambda$ (and, respectively, $\beta$ and $\gamma$), must be equal to zero. But we have just seen that $\alpha$, $\beta$, $\gamma$ and $\lambda$ must all be different from zero. So, we cannot find a combination of complex numbers $\alpha$, $\beta$, $\gamma$ and $\lambda$ such that

$$\begin{align} \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle ) &= \left( \alpha \gamma |00\rangle + \alpha \lambda |01\rangle + \beta \gamma |10\rangle + \beta \lambda |11\rangle \right) \end{align}$$

In other words, we are not able to express $|\Phi^{+}\rangle$ as a tensor product of two one-qubit states. Therefore, $|\Phi^{+}\rangle$ is a entangled state.

We can perform a similar proof for other Bell states or, in general, if we want to prove that a state is entangled.

A two qudit pure state is separable if and only if it can be written in the form

$$|\Psi\rangle=|\psi\rangle|\phi\rangle$$ for arbitrary single qudit states $|\psi\rangle$ and $|\phi\rangle$. Otherwise, it is entangled.

To determine if the pure state is entangled, one could try a brute force method of attempting to find satisfying states $|\psi\rangle$ and $|\phi\rangle$, as in this answer. This is inelegant, and hard work in the general case. A more straightforward way to prove whether this pure state is entangled is the calculate the reduced density matrix $\rho$ for one of the qudits, i.e. by tracing out the other. The state is separable if and only if $\rho$ has rank 1. Otherwise it is entangled. Mathematically, you can test the rank condition simply by evaluating $\text{Tr}(\rho^2)$. The original state is separable if and only if this value is 1. Otherwise the state is entangled.

For example, imagine one has a pure separable state $|\Psi\rangle=|\psi\rangle|\phi\rangle$. The reduced density matrix on $A$ is

$$\rho_A=\text{Tr}_B(|\Psi\rangle\langle\Psi|)=|\psi\rangle\langle\psi|,$$ and $$\text{Tr}(\rho_A^2)=\text{Tr}(|\psi\rangle\langle\psi|\cdot |\psi\rangle\langle\psi|)=\text{Tr}(|\psi\rangle\langle\psi|)=1.$$ Thus, we have a separable state.

Meanwhile, if we take $|\Psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, then

$$\rho_A=\text{Tr}_B(|\Psi\rangle\langle\Psi|)=\frac12\left(|0\rangle\langle 0|+|1\rangle\langle 1|\right)=\frac12\mathbb{I}$$ and $$\text{Tr}(\rho_A^2)=\frac14\text{Tr}(\mathbb{I}\cdot\mathbb{I})=\frac12$$ Since this value is not 1, we have an entangled state.

If you wish to know about detecting entanglement in mixed states (not pure states), this is less straightforward, but for two qubits there is a necessary and sufficient condition for separability: positivity under the partial transpose operation.