6 Entanglement

Now we are ready to introduce features that are more specific to the study quantum entanglement. Included here are measures of the degree of purity and measures of the degree of entanglement. For a review of quantum entanglement, see Ref.[1].

6.1 ptrace, ptracen -- partial trace$.$

compute the partial trace of the density operator $\rho$ over the component spaces given by the indices. For ptrace( $\rho,i_1,\ldots$) the density matrix $\rho$ is assumed to represent an operator in $(H_2)^{\otimes m}$, with $n=2$, that is a tensor product of qubit spaces. For ptracen( $n,\rho,i_1,\ldots$) the component spaces are all $n$-state qudits.

In this example we create three arbitrary $3\times3$ matrices, and check that

$$\mbox{Tr}_{1,2}\left(m_1\otimes m_2\otimes m_3\right) = \left(\mbox{Tr}(m_1)\mbox{Tr}(m_2)\right) m_3$$

and that

$$\mbox{Tr}_{3}\left(m_1\otimes m_2\otimes m_3 \right) = \mbox{Tr}(m_3)(m_1\otimes m_2).$$

Notice that we use the Maxima function ratsimp^$\dagger$ to put both sides of the equation in the same canonical form. We also make use of Maxima's mat_trace^$\dagger$ function ( so named to avoid conflicting with the code-execution trace function) and Maxima's matrix^$\dagger$ function, which creates a matrix from a list of rows.

Here we trace over one component repeatedly and check that the result is equal to the full trace. Note that, each time, we are tracing over the new first component.

6.2 entropy -- von Neumann entropy$.$

entropy($\rho$) returns the von Neumann entropy of the density matrix $\rho$ defined by

$$S(\rho) = -\mbox{Tr}\left(\rho \log_2 \rho\right).$$ (5)

entropyf is the floating point version of entropy.

6.3 renyi_entropy$.$

renyi_entropy($alpha,rho$) gives the Rényi entropy, defined by

$$S_{\mbox{renyi}}(\alpha,\rho) = \frac{1}{1-\alpha}\log_2\left(\mbox{Tr}(\rho^\alpha)\right).$$ (6)

renyi_entropyf is the floating point version of renyi_entropy.

6.4 tsallis_entropy$.$

tsallis_entropy($q,rho$) gives the Tsallis entropy, defined by

$$S_{\mbox{tsallis}}(q,\rho) = \frac{1}{q-1}\left(1-\mbox{Tr}(\rho^q)\right).$$ (7)

tsallis_entropyf is the floating point version of tsallis_entropy.

6.5 purity$.$

purity($\rho$) returns the purity of the density matrix $\rho$ defined by $\mbox{Tr}(\rho^2)$. The purity is $1$ for a pure state and is less than $1$ for a mixed state.

6.6 fidelity$.$

fidelity($\rho_1,\rho_2$) returns the scalar valued fidelity of the density matrices $\rho_1$ and $\rho_2$ defined by

$$\mbox{Tr}\left(\sqrt{\sqrt{\rho_2}\rho_1\sqrt{\rho_2}}\right).$$

6.7 concurrence -- Wootter's concurrence$.$

The concurrence of a two-qubit state $\rho$ is defined by $\max(0,\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4}$), where the $\lambda_i$ are the eigenvaluse of

$$\rho (\sigma_y \otimes \sigma_y) \rho^{\mbox{*}} (\sigma_y \otimes \sigma_y),$$

in decreasing order[2]

concurrence(rho) returns the concurrence of the state rho. concurrence_vals(rho) returns a list of the square roots of the eigenvalues in decreasing order, to the extent that Maxima can determine the order.

6.8 separable -- test for separability$.$

This currently accepts only pure, bipartite states. separable($e$) attempts to determine if e is a separable state. separable returns a scalar value $r$. If $r=1$, then e is separable. If $r<1$, it is not. Explicitly, the value returned for the state with density operator $\rho_{AB}$ is $\mbox{Tr}(\rho_A^2)$

When other methods are added, the organization and naming of the tests may change.

6.9 An entangled pure global state has mixed reduced states

We examine a textbook example of entanglement-- the joint state of two qubits. The state of the whole system is pure, but the local states are mixed. We begin by creating a joint state of two qubits in Schmidt basis ${\lvert\alpha \rangle}=\sqrt{\alpha}{\lvert 00 \rangle}+ \sqrt{1-\alpha}{\lvert 11 \rangle}.$ In order to see the mixed character of the local states, we need to express the full state as a density operator (or equivalently as a projection operator.) Let's try to make ${\lvert\alpha \rangle}{\langle\alpha \rvert}$.

We see that Maxima is allowing that the quantities under the radicals may be negative. So we set some rules, and try again.

The entropy vanishes for a pure state, so that $S({\lvert\alpha \rangle}{\langle\alpha \rvert})$ is

The purity is equal to $1$ if and only if $\rho$ is a pure state.

Now we compute the reduced density matrix of the second qubit by tracing over the first-- $\rho_2=\mbox{Tr}_1 {\lvert\alpha \rangle}{\langle\alpha \rvert}.$

Tracing over the second qubit instead gives the same result

Computing the entropy of a local state shows that this state is, in general, mixed

Each eigenvalue $\lambda$ satisfies $0\le \lambda <1$, so that the sum of their squares is less than one

We can plot the results (the plot function plot2d is more common, depending on your user interface. wxplot2d has the same calling syntax, but inlines the resulting plot.) We see that the maximum entanglement occurs at $\alpha=1/2$ and decreases monotonically from there in both directions, with $\alpha=0,1$ giving pure joint states.