Comment on: Multipartite entanglement in four-qubit graph state

TL;DR

This paper comments on previous work calculating entanglement in four-qubit graph states, highlighting limitations of the Scott measure and clarifying the relationship between different entanglement measures.

Contribution

It reveals limitations of the Scott measure in four-qubit systems and confirms that Q_2 is not always greater than Q_3, clarifying entanglement measure relationships.

Findings

Scott measure limits prevent Q_3 calculation in four-qubit systems

Q_2 is not necessarily greater than Q_3 in four-qubit graph states

Counterexample confirms previous conclusions about entanglement measures

Abstract

The following comment is based on an article by M. Jafarpour and L. Assadi [Eur. Phys. J. D 70, 62 (2016), doi:10.1140/epjd/e2016-60555-5] which with an exploitation of Scott measure (or generalized Meyer-Wallach measure) the entanglement quantity of four-qubit graph states has been calculated. We are to reveal that the Scott measure (Q_m) nominates limits for m which would prevent us from calculating Q_3 in four-qubit system. Incidentally in a counterexample we will confirm as it was recently concluded in the mentioned article, the Q_2 quantity is not necessarily always greater than Q_3.

Equations

$$Q_m(|\psi ⟩)={\left(\genfrac{}{}{0ex}{}{N}{m}\right)}^{-1}\sum _{|S|=m}\frac{2^m}{2^m-1}(1-\mathrm{Tr}[{\rho }_S^2]),$$

$$|G_{\star }⟩$$

$$+|100⟩|{\varphi }_5⟩+|101⟩|{\varphi }_6⟩+|110⟩|{\varphi }_7⟩+|111⟩|{\varphi }_8⟩).$$

$$|{\varphi }_1⟩$$

$$+|100⟩-|101⟩-|110⟩-|111⟩\},$$

$$|{\varphi }_2⟩$$

$$-|100⟩+|101⟩+|110⟩+|111⟩\},$$

$$|{\varphi }_3⟩$$

$$+|100⟩-|101⟩+|110⟩+|111⟩\},$$

$$|{\varphi }_4⟩$$

$$+|100⟩-|101⟩+|110⟩+|111⟩\},$$

$$|{\varphi }_5⟩$$

$$+|100⟩+|101⟩-|110⟩+|111⟩\},$$

$$|{\varphi }_6⟩$$

$$+|100⟩+|101⟩-|110⟩+|111⟩\},$$

$$|{\varphi }_7⟩$$

$$-|100⟩-|101⟩-|110⟩+|111⟩\},$$

$$|{\varphi }_8⟩$$

$$+|100⟩+|101⟩+|110⟩-|111⟩\}.$$

$$\mathrm{Tr}[{\rho }_i^2]=\frac{1}{2},i\in S,$$

$$\mathrm{Tr}[{\rho }_{ij}^2]=\frac{1}{4},i<j\in S,$$

$$\mathrm{Tr}[{\rho }_{ijk}^2]=\frac{1}{8},i<j<k\in S.$$

$$Q_1(|G_{\star }⟩)=Q_2(|G_{\star }⟩)=Q_3(|G_{\star }⟩)=1.$$

Full text

PACS numbers

03.65. Ud, 03.65. Mn, 03.67.-a.

Recently, M. Jafarpour and L. Assadi Nielsen01 based on Scott measure have calculated the entanglement quantity in non-trivial four-qubit graphs. Scott studied various interesting aspects of $N$-qubit entanglement measures given by Love02 ; Scott03

[TABLE]

where $S\subset\{1,\cdots,N\}$ and $\rho_{S}$ = $\mathrm{Tr}_{\acute{S}}(|\psi\rangle\langle\psi|)$ is the reduced density matrix for $S$ qubits after tracing out the rest. Also $m=1,\cdots,\lfloor\frac{N}{2}\rfloor$ and $\lfloor\frac{N}{2}\rfloor$ is the integer part of $\frac{N}{2}$. The $Q_{m}$ quantities $(0\leq Q_{m}\leq 1)$ correspond to the average entanglement between subsystems that consists $m$ qubits and the remaining $N-m$ qubits Borras04 . Meanwhile, $Q_{m}$ is invariant under local unitary transformations (LU), non-incremental on average under local operations and classical communication (LOCC). Hence on account of four-qubit system, we are only authorized to merely calculate $Q_{1}$ and $Q_{2}$. We have obtained $Q_{1}=1$ for all non-trivial four-qubit graphs (No. 1-41). Whereas the authors have calculated $Q_{3}$ in Table 1, leading to an incorrect result. Thus Section $6-d$ (Conclusions and discussion) leads to $Q_{2}$ being always greater than $Q_{3}$ in all the graph states. We will rectify in a counterexample their achieved result is incorrect in general. To clarify, take graph $G_{\star}$ for example, which is plotted in Fig. 1. The graph state corresponding to graph $G_{\star}$ is as followed

[TABLE]

Where

[TABLE]

For six-qubit graphs the authorized $m$ is equivalent to 1, 2 and 3 $(m=1,2,3)$. Therefore $\mathrm{Tr}\rho_{i}^{2}$, $\mathrm{Tr}\rho_{ij}^{2}$ and $\mathrm{Tr}\rho_{ijk}^{2}$ are calculated as following

[TABLE]

[TABLE]

[TABLE]

In this calculation the final result will be

[TABLE]

In conclusion, the analysis above shows that we are only authorized to merely calculate $Q_{1}$ and $Q_{2}$ for four-qubit system. Accordingly, the calculation of $Q_{3}$ given by M. Jafarpour and L. Assadi Nielsen01 is unauthorized and ineffective. Moreover, we note that $Q_{2}$ is not necessarily greater than $Q_{3}$ in all the graph states (or in general) but in some cases $Q_{2}$ is equal to $Q_{3}$. M. Jafarpour and L. Assadi study four-qubit graph states for which they can choose to study ($Q_{1}$ and $Q_{2}$) or ($Q_{2}$ and $Q_{3}$), since in fact $Q_{1}$ and $Q_{3}$ are proportional to each other by a numerical factor (as seen from Eq. (1)). In fact for four-qubit system $Q_{2}$ refers to 2-2 partitions in the graph state and $Q_{1}$ or $Q_{3}$ both refer to 3-1 partitions of the same graph. If we consider the case $Q_{1}$, 3-1 partition presents a stronger entanglement than a 2-2 partition in non-trivial four-qubit graphs (Unlike a result of the aforementioned article).