开放系统：量子纠缠的度量

A note on different measures of the mixed-state entanglement.

Entanglement is a property of bipartite systems—systems consisting of two parts A and B that are too far apart to interact, and whose state, pure or mixed, lies in a Hilbert space that is the tensor product of Hilbert spaces of these parts.

For pure states there exists many reliable ways to measure the entanglement, while for mixed states, the measures are not equivalent and sometimes ill-defined.

1. Necessary Conditions for Entanglement Measure

1.1 LOCC operations

Local Operations (LO): Performed in two seperate parts and described by two sets of operators $\sumi A_i^\dagger A_i = I\sum_j B_j^\dagger B_j = I\sum{ij} A_i \otimes B_j$ .

Classical Communication (CC): The action of A and B is classical correlated. A LOCC operation would be like:

$$\rho_{AB} \to \sum_i (A_i\otimes B_i) \rho_{AB} (A_i^\dagger \otimes B_i^\dagger)\tag{1.1}$$

Theorem: All entangled states can be purified into an ensemble of maximally entangled states using LOCC.

1.2 Necessary conditions¹

¹:V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight. (1997). Quantifying Entanglement. Phys. Rev. Lett. 78, 2275. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.2275

A good measure of entanglement must satisfy:

• iff is separable.
• Local unitary operations leave invariant, i.e., .
• LOCC cannot increase , i.e., , with a LOCC action. This is because any correlations achieved by LOCC is classical and therefore should not be included in a good measure.

Note: The third condition is not always true for most well-known measures of entanglement entropy because they also contain classical correlations.

2. Several Entanglement Measures for Mixed States

2.1 Schmidt Rank and Schmidt Coefficients

(Copied from the previous note:)

Theorem: For any vector in Hilbert space $\mathcal H1 \otimes \mathcal H_2{u_i^1},{u_i^2}1 \leq i \leq m\leq min{D_1,D_2}vv=\sum{i=1}^m a_i u_i^1\otimes u_i^2a_i$ non-negative.

The number of vectors needed for the decomposition is called the Schmidt rank and the are Schimdt coefficients. If the Schmidt rank of a vector is one, the associate state is separable.

So the Schmidt rank can be used to quantify the entanglement in pure states. Whether it is a useful measure of mixed states remains unknown.

Note: Schmidt coefficients are invariant under local unitary operations.
Proof: $v=\sum{i=1}^m a_i u_i^1\otimes u_i^2 \to \sum{i=1}^m a_i (U_A u_i^1)\otimes (U_B u_i^2)$. So it also describes the entangle of the state.

2.2 MPO Entanglement Entropy²

²:Noh, K., Jiang, L., & Fefferman, B. (2020). Efficient classical simulation of noisy random quantum circuits in one dimension. Quantum, 4. https://doi.org/10.22331/Q-2020-09-11-318

The MPO entanglement entropy is defined to describe the entanglement entropy of mixed states. In this paper², it is called MPO EE because it is defined in a Matrix Product Operator tensor network. For a general mixed state, its definition is based on the Schmidt decomposition of the density matrix.

Suppose we are interested in a system with 2-level particles, and want to get the correlation between the left particles and the rest of the system. We can reshape the density matrix to $Dl \otimes D{(n-l)}\lambda_i$ (Schimdt coefficients). MPO entanglement entropy is defined as:

$$\mathcal{S}_{l}(\rho) \equiv-\sum_{\alpha_{l}=0}^{\chi-1} \frac{\left(\lambda_{\alpha_{l}}^{[l]}\right)^{2}}{\sum_{\beta_{l}=0}^{\chi-1}\left(\lambda_{\beta_{l}}^{[l]} \right)^{2}} \log_{2}\left(\frac{\left(\lambda_{\alpha_{l}}^{[l]}\right)^{2}}{\sum_{\beta_{l}=0}^{\chi-1}\left(\lambda_{\beta_{l}}^{[l]}\right)^{2}}\right)$$

The SVD will give sigular values.

2.2 Von Neumann Entropy

The von Neumann entropy is a well-known entanglement measure for pure states. However for a mixed density matrix we can still define the von Neumann entropy:

$$S_{A} \equiv -\operatorname{tr} \rho_{A} \log \rho_{A}\tag{2.1}$$

From the Schmidt decomposition, we can easily get .

This quantity can be interpreted as the number of entangled bits between A and B, and counts the number of entangled states.

From the view of open quantum system (which is what I am concerned so far), we can say that:

Given a state with entanglement entropy , the quantity is the minimal number of auxiliary states that we would need to entangle with A in order to obtain from a pure state of the enlarged system.

The von Neumann entropy works well for pure states, but for mixed states it contains both quantum and classical correlations.

Note: Maximally entangled state gives a maximally mixed state after partial trace. You can try.

Also Note: The outcome of partial trace is irrelevant to the basis you choose. You can also try.

2.2.1 Renyi Entropy

The Renyi entropy is defined as:

$$S_\alpha(\rho) = (1-\alpha)^{-1}\log \text{Tr}\rho^\alpha \tag{3.1}$$

Here .

• If , the zeroth Renyi entropy counts the number N of nonzero eigenvalues of the reduced density matrix: , with the number of nonzero eigenvalues of the reduced density matrix.
• If , Renyi entropy goes to the von Neumann entropy.
• , Renyi entropy is related to the purity: .
• For , Renyi entropy can be bounded by :

$$S_\infty \leq S_n \leq \frac{n}{n-1}S_\infty$$

2.3 Entanglement of Formation³

³:Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., & Wootters, W. K. (1996). Mixed-state entanglement and quantum error correction. Physical Review A - Atomic, Molecular, and Optical Physics, 54(5), 3824–3851. https://doi.org/10.1103/PhysRevA.54.3824

For a mixed state, there are infinite ways to decompose it into pure state ensemble. One of the way to evaluate the entangle in mixed states is defined as the least expected entanglement of any ensemble of pure states realizing.

$$E(\rho):=\min \sum_{i} p_{i} S\left(\rho_{A}^{i}\right)$$

When a particular ensemble reaches the minimum, we call such ensemble optimal.

It can be proved that the entanglement of formation is non-increasing under LOCC.

It it not easy to find the ensemble with smallest entangle in general cases. In 2 spin1/2 particles, such ensembles can be found for diagonal states in Bell basis.

This quantity also has relation to the quantum discord, which is another metric for the non-classical correlation.

2.4 Negativity⁴

⁴. Shapourian, H., Liu, S., Kudler-Flam, J., & Vishwanath, A. (2021). Entanglement Negativity Spectrum of Random Mixed States: A Diagrammatic Approach. PRX Quantum, 2(3), 1. https://doi.org/10.1103/prxquantum.2.030347 ↩

It is a simple computable measure for mixed state entanglement. It is defined as:

$$\mathcal N = \frac{\|\rho^\Gamma\|_1 -1}{2}$$

where is the trace norm. Since is Hermitian, the trace norm is the sum of the absolute value of its eigenvalues. And the logarithmic negativity (LN) is defined as:

$$\mathcal E_{\mathcal N}(\rho) = \text{log}_2 \|\rho^\Gamma\|_1$$

LN and the negativity are related by .

LN is the upper bound for distillable entanglement.

2.5 Quantum Discord⁵

⁵. Ollivier, H., & Zurek, W. H. (2002). Quantum Discord: A Measure of the Quantumness of Correlations. Physical Review Letters, 88(1), 4. https://doi.org/10.1103/PhysRevLett.88.017901 ↩

Two classically identical expressions for the mutual information generally differ when the systems involved are quantum. This difference defines the quantum discord.

There are two different ways to express the mutual infomation classically:

$$J(X: Y)=H(X)-H(X \mid Y)$$ $$I(X: Y)=H(X)+H(Y)-H(X, Y)$$

Here is the Von Neumann entropy: $H(X) = -\sumi p{Xi} \log p{X_i}X$ is classical. Then the quantum discord is defined as:

$$\delta(S:A)_{\{\Pi_j^{\lambda}\}} = I(S:A) - J(S:A)_{\{\Pi_j^{\lambda}\}}$$

With a complete set of projectors on A.

Separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord, thus showing that absence of entanglement does not imply classicality.

The states with zero discord is the preferred effectively classical states, i.e., the pointer states.

QC is defined as the difference between two classically equal expressions of the mutual information $I{AB}J{AB}$:

$$I_{A B}=S\left(\rho_{A}\right)+S\left(\rho_{B}\right)-S\left(\rho_{A B}\right)$$ $$J_{A \mid B}=S\left(\rho_{A}\right)-S_{A \mid B}$$ $$S_{A \mid B}=\min_{\left\{\Pi_{k}^{B}\right\} \in \mathcal{M}^{B}} \sum_{k} p_{k} S\left(\rho_{A \mid k}\right)$$

where the minimum is taken over all possible measurement basis of subsystem .