The Separability Problem in Quantum Mechanics for Mixed States - Time Travel, Quantum Entanglement and Quantum Computing

Separability Problem in Quantum Mechanics for Mixed States

The **separability problem** in quantum mechanics refers to the task of determining whether a given **mixed quantum state** is **separable** or **entangled**. A separable state can be written as a mixture of product states, meaning that no quantum entanglement exists between subsystems. For mixed states, this becomes more complex than for pure states due to the probabilistic mixture of different quantum states.

Definition of Separable Mixed States

A **mixed state** is represented by a density matrix $\rho$ . A mixed state is **separable** if it can be written as a convex combination (a weighted sum where the weights are non-negative and sum to 1) of product states. For a bipartite system $A$ and $B$ , the state is separable if:

$\rho_{AB} = \sum_i p_i \, \rho_A^i \otimes \rho_B^i$

where:
– $p_i \geq 0$ are probabilities such that $\sum_i p_i = 1$ ,
– $\rho_A^i$ are density matrices of system $A$ ,
– $\rho_B^i$ are density matrices of system $B$ ,
– $\otimes$ denotes the tensor product, which describes the combination of states of subsystems $A$ and $B$ .

If $\rho_{AB}$ **cannot** be written in this form, it is **entangled**.

Definition of Mixed States and Density Matrices

In quantum mechanics, **mixed states** are described by a **density matrix** $\rho$ , which is a positive semi-definite, Hermitian matrix with trace equal to 1:

$\text{Tr}(\rho) = 1$

A mixed state represents a statistical ensemble of different pure states $\{|\psi_i\rangle\}$ , each occurring with probability $p_i$ . The density matrix for a mixed state is:

$\rho = \sum_i p_i |\psi_i\rangle \langle \psi_i|$

where $|\psi_i\rangle$ are pure states, and $p_i$ are probabilities such that $\sum_i p_i = 1$ .

The Separability Criterion for Mixed States

Determining whether a mixed state is separable is a non-trivial problem. There are several criteria that help in identifying whether a given state is separable or entangled:

#### 1. **Peres-Horodecki Criterion (Positive Partial Transpose – PPT Test)**
One of the most widely used methods to test separability for bipartite systems is the **Peres-Horodecki criterion**, which involves **partial transposition** of the density matrix.

For a bipartite system described by a density matrix $\rho_{AB}$ , the partial transpose with respect to subsystem $B$ is defined as:

$\rho_{AB}^{T_B} = \sum_{ij} \sum_{kl} \langle i,j | \rho_{AB} | k,l \rangle |i,l\rangle \langle k,j|$

where the transpose is taken only with respect to the indices of subsystem $B$ .

– If the matrix $\rho_{AB}^{T_B}$ has **negative eigenvalues**, the state is **entangled**.
– If all eigenvalues are non-negative, the state is **separable** (for 2×2 or 2×3 systems). However, for higher-dimensional systems, this criterion is necessary but not sufficient; there could be **bound entanglement** (states that are entangled but have a positive partial transpose).

Schmidt Decomposition for Pure States

For **pure states**, separability is simpler to check using the **Schmidt decomposition**. A bipartite pure state $|\psi\rangle$ can always be written as:

$|\psi\rangle = \sum_i \lambda_i |u_i\rangle_A \otimes |v_i\rangle_B$

where $\lambda_i \geq 0$ are the **Schmidt coefficients**, and $|u_i\rangle_A$ , $|v_i\rangle_B$ are orthonormal states of systems $A$ and $B$ .

– If only **one** Schmidt coefficient is non-zero, the state is **separable**.
– If **more than one** Schmidt coefficient is non-zero, the state is **entangled**.

For mixed states, this method doesn’t directly apply, as they are probabilistic mixtures of pure states.

Example of the Peres-Horodecki Criterion

Consider a **two-qubit state** given by the density matrix $\rho$ :

$\rho = \frac{1}{4} \left( I_4 + c (\sigma_x \otimes \sigma_x) + c (\sigma_y \otimes \sigma_y) + c (\sigma_z \otimes \sigma_z) \right)$

where $I_4$ is the identity matrix, $\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices, and $c$ is a parameter between -1 and 1.

The **partial transpose** with respect to the second qubit $B$ changes the signs of some off-diagonal terms. If the resulting matrix has any negative eigenvalues, the state is entangled. Otherwise, the state is separable (at least for this two-qubit system).

Entanglement Witnesses

An **entanglement witness** is an observable $W$ such that:
– For all separable states $\rho$ , $\text{Tr}(W \rho) \geq 0$ .
– For at least one entangled state $\rho_e$ , $\text{Tr}(W \rho_e) < 0$ .

Entanglement witnesses provide a practical method for detecting entanglement in mixed states. Given a state $\rho$ , if there exists an observable $W$ such that $\text{Tr}(W \rho) < 0$ , the state is entangled.

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### Mathematical Summary:

– **Mixed State**:

$\rho = \sum_i p_i \, |\psi_i\rangle \langle \psi_i|$

A separable mixed state can be written as:

$\rho = \sum_i p_i \, \rho_A^i \otimes \rho_B^i$

– **Peres-Horodecki Criterion**:
– Partial transpose of $\rho$ with respect to one subsystem.
– If any eigenvalues of the partial transpose are negative, the state is entangled.

– **Entanglement Witness**:
– Find an observable $W$ such that $\text{Tr}(W \rho) < 0$ for entangled states.

### Conclusion
The separability problem for mixed states is challenging because mixed states are probabilistic combinations of pure states, making it difficult to determine if their correlations are classical or quantum. Several mathematical tools, like the Peres-Horodecki criterion and entanglement witnesses, are used to address this problem.