Entanglement distance for an arbitrary state of M qubits

TL;DR

This paper introduces a novel entanglement measure for pure states of M-qubit systems based on a distance derived from the Fubini-Study metric, which is invariant under local unitaries and provides insights into entanglement robustness.

Contribution

It proposes a new entanglement measure as a distance derived from the Fubini-Study metric, applicable to any pure M-qubit state, and analyzes its eigenvalues for robustness insights.

Findings

The measure is invariant under local unitary transformations.

Eigenvalues of the entanglement metric relate to entanglement robustness.

The measure can be computed for any pure M-qubit state.

Abstract

We propose a measure of entanglement that can be computed for any pure state of an $M$-qubit system. The entanglement measure has the form of a distance that we derive from an adapted application of the Fubini-Study metric. This measure is invariant under local unitary transformations and defined as trace of a suitable metric that we derive, the entanglement metric $\tilde{g}$. Furthermore, the analysis of the eigenvalues of $\tilde{g}$ gives information about the robustness of entanglement.

Equations

$$⟨d\psi |d\psi ⟩-\frac{1}{4}|⟨\psi |d\psi ⟩-⟨d\psi |\psi ⟩{|}^2,$$

$$|U,s⟩=\prod _{j=0}^{M-1}{U}^j|s⟩$$

$$|dU,s⟩=\sum _{j=0}^{M-1}d{\tilde{U}}^j|U,s⟩,$$

$$d{\tilde{U}}^j=-i({\mathbf{n}}^j\cdot {\mathbf{\sigma }}^j)d{\xi }^j/2$$

$$g_{\mu \nu }({\mathbf{v}}^{\nu })d{\xi }^{\mu }d{\xi }^{\nu }$$

$$-⟨s|({\mathbf{v}}^{\mu }\cdot {\mathbf{\sigma }}^{\mu })|s⟩⟨s|({\mathbf{v}}^{\nu }\cdot {\mathbf{\sigma }}^{\nu })|s⟩)d{\xi }^{\mu }d{\xi }^{\nu }.$$

$${\mathbf{v}}^{\nu }\cdot {\mathbf{\sigma }}^{\nu }=U^{\nu †}{\mathbf{n}}^{\nu }\cdot {\mathbf{\sigma }}^{\nu }{U}^{\nu },$$

$$E(|s⟩)=\underset{\mathcal{M}\{{\mathbf{v}}^{\nu }{\}}_{\nu }}{\inf }\text{tr}(g),$$

$$\tilde{g}=g({\tilde{\mathbf{v}}}^{\nu })$$

$$|r,0⟩=\underset{j=0}{\overset{M-1}{⨂}}\frac{1}{\sqrt{2}}(|0{⟩}_j+|1{⟩}_j),$$

$$U_0(\varphi )=\exp (-i\varphi H_0)=\prod _{j=1}^{M-1}\left(\mathbb{I}+\alpha {\Pi }_0^j{\Pi }_1^{j+1}\right),$$

$${\lambda }_k=\sum _{j=0}^{n(k)}\left(\genfrac{}{}{0ex}{}{n(k)}{j}\right){\alpha }^j,$$

$$|r,0{⟩}_M=2^{-M/2}\sum _{k=0}^{2^M-1}|k⟩,$$

$$|r,\varphi {⟩}_M=2^{-M/2}\sum _{k=0}^{2^M-1}\sum _{j=0}^{n(k)}(\genfrac{}{}{0ex}{}{n(k)}{j}){\alpha }^j|k⟩.$$

$$\text{tr}(g)=\sum _{\nu =0}^1\left[1-c^2{\left(c{v}_1^{\nu }+{\left(-1\right)}^{\nu +1}s{v}_2^{\nu }\right)}^2\right]/4,$$

$$\tilde{g}=\dfrac{1}{4}\left(\begin{array}[]{cc}s^{2}&1\\ 1&s^{2}\end{array}\right)$$

$$E(|r,\varphi {⟩}_2)=\frac{s^2}{2}.$$

$$\text{tr}(g)=\left[3-c^2{\left(c(v_1^0+v_1^1+v_1^2)+s(v_2^2-v_2^0)\right)}^2\right]/4,$$

$$\tilde{g}=\dfrac{s^{2}}{4}\left(\begin{array}[]{ccc}1&c&-2s^{2}c^{2}\\ c&1+c^{2}&c\\ -2s^{2}c^{2}&c&1\end{array}\right)$$

$$E(|r,\varphi {⟩}_3)=\frac{s^2}{4}\left(3+c^2\right),$$

$$\text{tr}(g)$$

$$\begin{array}[]{l}w^{\nu}_{-}=\sum^{2^{M}-1}_{k=0}\delta_{n_{\nu}^{k},0}c^{*}_{k+2^{\nu}}c_{k}\,,\\ w^{\nu}_{+}=\sum^{2^{M}-1}_{k=0}\delta_{n_{\nu}^{k},1}c^{*}_{k-2^{\nu}}c_{k}\,,\\ w^{\nu}_{3}=\sum^{2^{M}-1}_{k=0}(-1)^{n_{\nu}^{k}}|c_{k}|^{2}\,.\end{array}$$

$$E(|r,\varphi {⟩}_M)=\frac{1}{4}\left(M-\sum _{\nu =0}^{M-1}\Vert {\mathbf{w}}^{\nu }{\Vert }^2\right).$$

$$|GHZ,\theta {⟩}_M=\cos (\theta )|0⟩+\sin (\theta )e^{i\phi }|2^M-1⟩.$$

$$\text{tr}(g)$$

$$\tilde{g}=\frac{1}{4}{\sin }^2(2\theta )J_M$$

$$E(|GHZ,\theta {⟩}_M)=\frac{M}{4}{\sin }^2(2\theta ).$$

$$|\phi ,\gamma ,\tau {⟩}_3=$$

$$+$$

$$\text{tr}(g)$$

$$-[\sin (2\gamma )\sin (2\tau )v_1^2+\cos (2\gamma )v_3^2{]}^2\}$$

$${\tilde{v}}_1^3$$

Full text

I Introduction

Entanglement is an essential resource for progressing in the field of quantum-based technologies. Quantum information has confirmed its importance in quantum cryptography and computation, in teleportation, in the frequency standard improvement problem and metrology based on quantum phase estimation Gühne and Toth (2009). The rapid experimental progress on quantum control is driving the interest in entanglement theory. Nevertheless, despite its key role, entanglement remains elusive and the problem of its characterisation and quantification is still open (Sperling and Walmsley, 2017; Giovannetti et al., 2003). We propose a measure of entanglement that can be computed on any pure state of M-qubit systems. The measure is derived from a tailored form of the Fubini-Study metric that we verify to correspond to the quantum Fisher information but which allows for a deeper understanding thanks to its eigenvalues’ analysis. The measure that we propose: i) is invariant under local unitary transformations; ii) has the structure of a distance such that the higher is the entanglement of a given state the greater is its minimum distance from infinitesimally close states; iii) distinguishes between the case $M=2$ and the case $M>2$ which seems to be consistent with the fact that most of the propositions which are necessary and sufficient in the case $M=2$ lose the sufficient condition in the $M>2$ case.

II Distance entanglement

The Hilbert space of an $M$-qubit system carries the Fubini-Study metric Gibbons (1992)

[TABLE]

where $|\psi\rangle$ is a generic normalised state and $|d\psi\rangle$ is an infinitesimal variation of such state. The present study is aimed to endow the Hilbert space with a Fubini Study-like metric that has the desirable property of making it an attractive definition for entanglement measure. For this reason, such distance should not be affected by local operations on single qubits. As a matter of fact, the action of $M$ arbitrary $SU(2)$ local unitary operators $U^{j}$ ($j=0,\ldots,M-1$) on a given state $|s\rangle$, generates a class of states

[TABLE]

that share the same degree of entanglement. For each $j$, $U^{j}$ operates on the $j$th qubit. We define an infinitesimal variation of state (2) as

[TABLE]

where

[TABLE]

rotates the $j$th qubit by an infinitesimal angle $d\xi^{j}$ around the unitary vector ${\bf n}^{j}$. We denote by $\sigma^{j}_{1}$, $\sigma^{j}_{2}$ and $\sigma^{j}_{3}$ the Pauli matrices operating on the $j$-th qubit ($j=0,\ldots,M-1$) where the index $j$ numerates the spins from right to left. From Eq. (1), with this choice, we get the following expression for the Fubini-Study metric $g$

[TABLE]

The unitary vectors ${\bf v}^{j}$ in the latter equation are derived by a rotation of the original ones according to

[TABLE]

where there is no summation on the index $\nu$. *The proposed entanglement measure of the state $|s\rangle$ is *

[TABLE]

where $\tr$ is the trace operator and where the $\inf$ is taken “in measure” over all the possible orientations of the unitary vectors ${\bf v}^{\nu}$. With the term “in measure”, we mean that possible pathologies, similar to the one of the Dirichlet function, are eliminated. The $\inf$ operation, makes the measure (7) independent from the operators $U_{j}$ hence, its numerical value is associated to the class of states generated by local unitary transformations and not to the specific element chosen inside the class. This is a necessary condition for a good entanglement measure definition. The unitary vectors $\tilde{\bf v}^{\nu}$ corresponding to the $\inf$ of $\tr(g)$, identify a metric

[TABLE]

that we name entanglement metric (EM). The off-diagonal elements of $\tilde{g}$ provide the quantum correlations between qubits. In addition, states that differ one another for local unitary transformations, have the same form of $\tilde{g}$. In this way, the expression of EM identifies the classes of equivalence. Remarkably, the analysis of the eigenvalues and eigenvectors of $\tilde{g}$ allows one to check the existence of states with super-Heisenberg sensitivity, i.e. beyond Heisenberg limit.

III Examples

In order to verify the efficacy of the proposed entanglement measure, we have first considered two families of one-parameter states depending on a real parameter. The degree of entanglement of each state depends on this parameter and the configuration corresponding to the maximally entangled states for each of the families is known. The first family of states we consider has been introduced by Briegel and Raussendorf in Ref. Briegel and Raussendorf (2001). For this reason, we will name the elements in this family Briegel-Raussendorf states (BRS). The second family of states is related to the Greenberger-Horne-Zeilinger states Greenberger et al. (1989). We will name the elements of such family Greenberger-Horne-Zeilinger–like states (GHZLS). It is worth emphasizing that in Ref. Briegel and Raussendorf (2001) it has been shown that the maximally entangled states of these two families are not equivalent if $M\geq 4$, whereas they are equivalent if $M\leq 3$. This fact offers us a further test for our approach to entanglement estimation. In fact, we have found that i) the entanglement measure (7) provides the same value for the maximally entangled states of both the families; ii) in the case $M\leq 3$ the entanglement metric (8) has the same form for the maximally entangled states of the two families, whereas, if $M\geq 4$, the EMs of the maximally entangled states of the two families are not equivalent.

The last case we have considered is a family of three-qubit states depending on two real parameters. With a suitable choice of these parameters, the state can be fully separable or bi-separable, whereas in the generic case it is a genuine tripartite entangled state. We will show that the proposed entanglement measure provides an accurate description of all these cases.

III.1 Briegel Raussendorf states

We denote with $\Pi^{j}_{0}=(\mathbb{I}+\sigma^{j}_{3})/2$ and $\Pi^{j}_{1}=(\mathbb{I}-\sigma^{j}_{3})/2$ the projector operators onto the eigenstates of $\sigma^{j}_{3}$, $|0\rangle_{j}$ (with eigenvalue $+1$) and $|1\rangle_{j}$ (with eigenvalue $-1$), respectively. Each $M$ qubit state of the BRS class is derived by applying to the fully separable state

[TABLE]

the non local unitary operator

[TABLE]

where $H_{0}=\sum^{M-1}_{j=1}\Pi^{j}_{0}\Pi^{j+1}_{1}$ and $\alpha=(e^{-i\phi}-1)\,.$

The full operator (10) is diagonal on the states of the standard basis $\{|0\cdots 0\rangle\,,\,\,|0\cdots 01\rangle,\ldots,|1\cdots 1\rangle\}$. In fact, each vector of the latter basis is identified by $M$ integers $n_{0},\ldots,n_{M-1}=0,1$ as $\ket{\{n\}}=|n_{M-1}\ n_{M-2}\quad n_{0}\rangle\,$ and we can enumerate such vectors according to the binary integers representation $|k\rangle=\ket{\{n^{k}\}}$, with $k=\sum^{M-1}_{j=0}n^{k}_{j}2^{j}$, where $n^{k}_{\nu}$ is the $\nu$-th digit of the number $k$ in binary representation and $k=0,\ldots,2^{M}-1$. Then, the eigenvalue $\lambda_{k}$ of operator (10), corresponding to a given eigenstate $|k\rangle$ of this basis, results

[TABLE]

where $n(k)$ is the number ordered couples $01$ inside the sequence of the base vector $|k\rangle$. For the initial state (9) we consistently get

[TABLE]

and, under the action of $U_{0}(\phi)$ one obtains

[TABLE]

For $\phi=2\pi k$, with $k\in\mathbb{Z}$, this state is separable, whereas, for all the other choices of the value $\phi$, it is entangled. In particular, in Briegel and Raussendorf (2001) it is argued that the values $\phi=(2k+1)\pi$, where $k\in\mathbb{Z}$, give the maximally entangled states.

III.2 Fubini-Study metric for the Briegel Raussendorf states $M=2,3$

In the case of two-qubit BRS the trace of the Fubini-Study metric is

[TABLE]

where $c=\cos\left({\phi}/{2}\right)$ and $s=\sin\left({\phi}/{2}\right)$. (14) is minimised with the choice $\tilde{\bf v}^{\nu}=\pm(c,(-1)^{\nu+1}s,0)$. Consistently, the EM results in

[TABLE]

and

[TABLE]

We have already mentioned that in the case $M=2,3$, the maximally-entangled BRS $|r,2\pi k+\pi\rangle$, where $k\in\mathbb{Z}$, and the maximally entangled GHZLS are equivalent because differing just for local unitary transformations. In the following, we will show that the EM for these states have the same forms in the case $M=2,3$ in accordance to the results of Ref. Briegel and Raussendorf (2001). In the case $M=3$ and $\phi\neq(2k+1)\pi$, with $k\in\mathbb{Z}$, the trace of $g$,

[TABLE]

is minimised with the choices $\tilde{\bf v}^{0}=(c,-s,0)$, $\tilde{\bf v}^{1}=(1,0,0)$ and $\tilde{\bf v}^{2}=(c,s,0)$. The EM and the entanglement measure in this case results to be

[TABLE]

and

[TABLE]

respectively. By direct calculation, one can verify that in the case of the maximally entangled BRS ($M=3$), the choice ${\bf v}^{0}=(-1,0,0)$, ${\bf v}^{1}=(0,0,1)$ and ${\bf v}^{2}=(1,0,0)$ makes the EM equivalent to the one of the three-qubit Greenberger-Horne-Zeilinger state. This agrees with the results of Ref. Briegel and Raussendorf (2001).

III.3 Fubini-Study metric for the Briegel Raussendorf states $M>3$

In the general case, the trace of $g$ results

[TABLE]

where $v^{\nu}_{\pm}=v^{\nu}_{1}\pm iv^{\nu}_{2}$, $c_{k}=2^{-M/2}\lambda_{k}$, and

[TABLE]

The trace is minimised by setting $\tilde{v}^{\nu}_{+}={{w}^{\nu\star}_{-}}/{\|{\bf w}^{\nu}\|}$, $\tilde{v}^{\nu}_{-}={{w}^{\nu\star}_{+}}/{\|{\bf w}^{\nu}\|}$ and $\tilde{v}^{\nu}_{3}=\frac{{w}^{\nu}_{3}}{\|{\bf w}^{\nu}\|}$. From the latter we get the entanglement measure for the BRS that is

[TABLE]

III.4 Greenberger-Horne-Zeilinger–like states

Now, we consider a second class of states (GHZLS) defined according to

[TABLE]

For $\theta=k\pi/2$, where $k\in\mathbb{Z}$, these states are fully separable, whereas $\theta=k\pi/2+\pi/4$ selects the maximally entangled states. In this case, the trace for the Fubini-Study metric,

[TABLE]

is minimised by the values $v^{\nu}_{3}=1$. Consistently, we have

[TABLE]

where $J_{M}$ is the $M\times M$ matrix of ones. The entanglement measure for the GHZLS results

[TABLE]

III.5 Three-qubit states depending on two parameters

The last class of states we consider is

[TABLE]

These states are fully separable for $\gamma=0,\pi/2$ and $\tau=0,\pi/2$ whereas they are bi-separable for $\tau=\pi/4$. In this case, the trace of the Fubini-Study metric is

[TABLE]

and it is minimised by the values $\tilde{\bf v}^{\nu}_{3}=(0,0,1)$, $\nu=0,1$ and

[TABLE]

Consistently, the entanglement measure for these states results to be

[TABLE]

IV Results

IV.1 Entanglement measure

In Fig. 1 we plot the measure $E(|r,\phi\rangle_{M})/M$ vs $\phi/(2\pi)$ according to Eq. (22), for the states (13) in the case $M=3,4,7,9$. Figure 1 show that the proposed entanglement measure provides in all these cases a correct estimation of the degree of entanglement for the BRS. In particular, for the fully separable states ($\phi=0$) it gives a vanishing value, whereas for the maximally entangled states ($\phi=\pi$) it provides the maximum possible value for the trace, that is $E(|r,\pi\rangle_{M})/M=1/4$. This implicitly indicates that on the maximally entangled states the expectation values for all $\tilde{\bf v}^{\nu}\cdot\boldsymbol{\sigma}^{\nu}$ ($\nu=0,\ldots,M-1$) vanish.

The entanglement measure (7) successfully passes also the second test of the GHZLS for which it provides zero in the case of fully separable states ($\theta=0$) and the maximum value ($1/4$) in the case of the maximally entangled state ($\theta=\pi/2$). In figure 2 we compare the curves $E(|r,\phi\rangle_{M})/M$ vs $\phi/(2\pi)$ in continuous line and $E(|GHZ,\theta\rangle_{M})/M$ vs $2\theta/\pi$ in dashed line for the case $M=3$. Also in this case, for the maximally entangled states the expectation value for the operators $\tilde{\bf v}^{\nu}\cdot\boldsymbol{\sigma}^{\nu}$ ($\nu=0,\ldots,M-1$) is zero.

In Fig. 3, we report, with a 3D plot, the measure $E(|\varphi,\gamma,\tau\rangle_{3})/3$ as a function of $\gamma/(2\pi)$ and $\tau/(2\pi)$ according to Eq. (30), for the states (27).

The measure (7) catches, in a surprisingly clear way, the entanglement properties of this family of states. In particular, $E(|\varphi,\gamma,\tau\rangle_{3})/3$ is null in the case of fully separable states ($\gamma=0,\pi/2,\pi$ and $\tau=0,\pi/2,\pi$) and it is maximum (with value $1/4$) in the case of maximally entangled states ($\gamma=\pi/4,3\pi/4$ and $\tau=0,\pi/2,\pi$). In addition, the case of bi-separable states ($\tau=\pi/4$) results in $0<E(|\varphi,\gamma,\tau\rangle_{3})/3<1/4$.

IV.2 Eigenvalues analysis

Other interesting characteristics of the entanglement measure come from the analysis of the metric’s eigenvalues. In fig. 4, we plot the eigenvalues of $\tilde{g}$ for the state $|r,\phi\rangle_{M}$ vs $\phi/(2\pi)$ for the case $M=7$. In the general case $\phi\neq 0,2\pi$ the BRS $\tilde{g}$ have $M$ not null eigenvalues.

This fact makes the class of the BRS robust, concerning entanglement, inasmuch the minimum distance between states in a direction randomly chosen is greater than the minimum eigenvalue. On the contrary, the GHZLS have only one non-vanishing eigenvalue. Although the value of the latter is greater than the eigenvalues of the BRS (see Fig. 5), the GHZLS appear weak, in the sense of entanglement, since there exist $M-1$ directions with null minimum distance between states. In fig. 5, we compare the plots of the eigenvalues of $\tilde{g}$ for $|r,\phi\rangle_{M}$ vs $\phi/(2\pi)$ (dotted lines), with the plot of the unique not vanishing eigenvalue of $\tilde{g}$ for GHZLS vs $2\theta/\pi$ (continuous line), in the case $M=7$.

Within the scenario that we have proposed, the entanglement has the physical interpretation of an obstacle to the minimum distance between infinitesimally close states. In fact, by defining the distance between a given state represented by the vector $|U,s\rangle$ and an infinitesimally close state associated with the vector $|dU,s\rangle$ as $ds^{2}=\tr(g({\bf v}))dr^{2}$ where $\sum_{\mu}(d\xi^{\mu})^{2}=dr^{2}$, it results

[TABLE]

This shows that the minimum density distance $ds^{2}/dr^{2}$, obtained by varying the vectors ${\bf v}$, is bounded from below by the entanglement measure $E(|s\rangle)$. For fully separable states, the minimum density distance is zero whereas for maximally entangled states, it results $M/4$ at the very best. It is worth emphasizing that $ds^{2}$ can overcome the value of $E(|s\rangle)dr^{2}$.

V Concluding remarks

In this paper, we have introduced a new measure of entanglement for the case of an arbitrary pure state of M qubits (7). We verified the invariance under local unitary transformations identifying classes of equivalence of states, a demanded property of a good entanglement measure. Furthermore, the measure has the characteristics of a distance and assumes the intuitive physical interpretation of an obstacle to the minimum distance between infinitesimally close states. Finally, the analysis of the eigenvalues allows one to determine if there are any states which are more sensitive to small variations than others. For instance, Fig. 4 shows that, in the case of $|r,\pi/2\rangle_{7}$ state, a small variation along the eigenvector’s direction of the maximum eigenvalue of $\tilde{g}$ brings a greater distance than the one derived in the case of the maximally entangled state $|r,\pi\rangle_{7}$. This analysis is a possible useful mean in the task of determining states with super-Heisenberg sensitivity.

Acknowledgements.

We are grateful to A. Smerzi and L. Pezzé for useful discussions.

R. F. thanks the support by the QuantERA project “Q-Clocks” and the European Commission.