How Do You Compute the Density Matrix of a Bipartite State?

In summary, computing the density matrix of a bipartite state involves expressing the joint state of two subsystems in a mathematical framework. This is typically done by taking the outer product of the state vectors of the individual subsystems, or by using the Schmidt decomposition if the state is entangled. The density matrix can then be constructed by tracing out one subsystem to obtain the reduced density matrix for the other subsystem, allowing for the analysis of quantum correlations and entanglement between the two parts. Key mathematical techniques include tensor products and partial traces, which facilitate the computation of the overall state and its properties.
  • #1
Rayan
17
1
TL;DR Summary
What is the easiest way to compute a density matrix of bipartite states?
If we for example have such a bipartite state:

$$ | \phi > = \frac{1}{2} [ |0>|0> + |1>|0> + |0>|1> + |1>|1> ] $$

What is the easiest way to compute a density matrix of bipartite states? Should I just compute it as it is? i.e:

$$ \rho = | \phi > < \phi | $$

Or should I convert to matrix form first? Any advice appreciated!

I tried to convert it to matrix form and got the following:

$$ | \phi > = \frac{1}{2}
\begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} $$

and

$$ < \phi | = \frac{1}{2}
\begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix}
$$

But then I don't think it is possible to compute the following outer product?

$$ \rho = \frac{1}{4}
\begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} \cdot
\begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix}
$$
 
Last edited:
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  • #2
Things went wonky when you calculated the dual vector (bra) of the state. How did the complex conjugate transpose turn a column vector into a matrix?
 
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Likes PeterDonis
  • #3
Rayan said:
Should I just compute it as it is? i.e:
$$ \rho = | \phi > < \phi | $$
Yes.
 

FAQ: How Do You Compute the Density Matrix of a Bipartite State?

What is a bipartite quantum state?

A bipartite quantum state refers to a quantum system composed of two subsystems. It can be described by a combined state vector in the tensor product of the individual Hilbert spaces of the subsystems. Mathematically, if subsystem A has a Hilbert space \( \mathcal{H}_A \) and subsystem B has a Hilbert space \( \mathcal{H}_B \), the combined system lives in the Hilbert space \( \mathcal{H}_A \otimes \mathcal{H}_B \).

How do you represent a bipartite state in matrix form?

A bipartite state can be represented in matrix form using the density matrix formalism. For a pure state \( |\psi\rangle \) in a bipartite system, the density matrix \( \rho \) is given by \( \rho = |\psi\rangle \langle \psi| \). For a mixed state, the density matrix is a weighted sum of pure state density matrices: \( \rho = \sum_i p_i |\psi_i\rangle \langle \psi_i| \), where \( p_i \) are probabilities.

How do you construct the density matrix of a bipartite pure state?

To construct the density matrix of a bipartite pure state \( |\psi\rangle \), you first express the state in the form \( |\psi\rangle = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B \), where \( |i\rangle_A \) and \( |j\rangle_B \) are basis states of subsystems A and B, respectively. The density matrix \( \rho \) is then given by \( \rho = |\psi\rangle \langle \psi| \), which translates to \( \rho = \sum_{i,j,k,l} c_{ij} c_{kl}^* (|i\rangle_A \langle k|_A) \otimes (|j\rangle_B \langle l|_B) \).

What is the partial trace, and why is it important in computing the reduced density matrix?

The partial trace is an operation used to trace out one of the subsystems in a bipartite system, resulting in the reduced density matrix of the remaining subsystem. For a bipartite density matrix \( \rho \) in \( \mathcal{H}_A \otimes \mathcal{H}_B \), the reduced density matrix of subsystem A is obtained by tracing out subsystem B: \( \rho_A = \text{Tr}_B(\rho) \). This operation is important because it allows us to study the properties and behavior of a

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