Quantifying the entanglement of quantum states under the geometric method

Quantum entanglement is an essential feature of quantum mechanics. It plays an important role in quantum information and quantum computation theory horodecki2009quantum , such as superdense coding bennett1992communication , teleportation bennett1993teleporting and the speedup of quantum algorithms shimoni2005entangled .
How to quantify the amount of entanglement for a multipartite system is important in quantum information theory. As it is linked with many areas of the entanglement theory, such as entanglement distillability bennett1996mixed , the transformation of quantum states vedral1997quantifying ; jonathan1999minimal ; rains1999bound , monogamy of entanglement coffman2000distributed ; bai2014general ; zhu2014entanglement ; de2014monogamy ; shi2021multilinear , quantum speed limit rudnicki2021quantum and so on. Due to the importance of this issue, it has been investigated almost since the end of the last century shimony1995degree ; bennett1996mixed ; vedral1997quantifying . In 1996, Bennett $et$ $al.$ proposed the distillable entanglement and entanglement cost, as well as their operational interpretations bennett1996mixed . In 1997, Vedral presented three necessary conditions that an entanglement measure should satisfy vedral1997quantifying . And Vidal presented a general method to quantify the entanglement for the bipartite entangled states vidal2000entanglement . This method is built on the function of pure states in the bipartite system and generalized to the mixed states under the convex roof extended methods on the bipartite system. Other than the generic conditions and the building methods of entanglement measures, some meaningful thoughts to bulid the entanglement measures are presented. The robustness of an entangled state tells us how many degrees of the separable states needed to make the state separable vidal1999robustness . The squashed entanglement was motived by the intrinsic information tucci2002entanglement ; christandl2004squashed , it is monogamous for arbitrary dimensional systems christandl2004squashed and faithful brandao2011faithful . Moreover, the way to measure the entanglement can also be defined based on the geometrical ways, such as, the geomeric measure of entanglement barnum2001monotones ; wei2003geometric , the quantum relative entropy vedral1997quantifying and the fidelity streltsov2010linking . However, there are few results of the measure based on the distance between an entangled state and the set of separable states under the trace norm.

Trace norm not only can quantify the entanglement for a multipartitie mixed state, but also it provides a way to quantify the discord paula2013geometric ; montealegre2013one ; ciccarello2014toward ; roga2016geometric , the measurement-induced nonlocality hu2015measurement , asymmetry marvian2014extending , steering sainz2018formalism and coherence baumgratz2014quantifying ; rana2016trace ; chen2018notes . It is also helpful in quantum communication pirandola2017fundamental ; pirandola2020advances and quantum algorithms gebhart2021quantifying ; bai2022quantum .

This article is organized as follows. In section II, first we present the preliminary knowledge needed. We also present some properties of the modified entanglement measure. In section \@slowromancapiii@, we consider the properties of the entanglement mesurement in terms of the trace norm under two methods. Specifically, in section III.1, we present the solutions of the modified measure for pure states, and then we generalize it to a measure for tripartite mixed states $D^{{}^{\prime}}_{fsep}(\cdot)$, which tells the distance between a tripartite state and the cone of full separable states. In section III.2, we consider the measure based on the trace norm under the method proposed in gour2020optimal ; shi2021extension . In section IV, we end with a conclusion.

In the following, we denote $\mathcal{D}_{A}$ as the set of states on $\mathcal{H}_{A}$ with finite dimensions. Next for a state $\rho_{AB}\in\mathcal{D}_{AB}$, if it can be written as $\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle_{A}\langle\psi_{i}|\otimes|\phi_{i}\rangle_{B}\langle\phi_{i}|$, then it is separable, otherwise, it is an entangled state. We denote $\mathop{\rm SEP}_{AB}$ as the set of separable states on $\mathcal{H}_{AB}$ with finite dimensions. And we may leave out the systems in the absence of ambiguity in the following.

Assume $|\psi\rangle_{AB}$ is a bipartite pure state in $\mathcal{H}_{AB}$, $|\psi\rangle_{AB}=\sum_{i}\sqrt{\lambda_{i}}|ii\rangle$, then its concurrence is defined as

here $\rho_{A}$ is the reduced density matrix of $\rho_{AB}.$ When $\rho_{AB}$ is a mixed state, then

where the minimum takes over all the decompositions $\{p_{i},|\psi_{i}\rangle\}$ of $\rho_{AB}$.

Next we recall the definition of a distance measure in terms of trace norm for a bipartite mixed state. Assume $\rho\in\mathcal{D}_{AB}$, its distance measure in terms of trace norm is defined as,

here $\beta$ takes over all the separable states in the system $\mathcal{H}_{A}\otimes\mathcal{H}_{B}.$

Then we recall $E:\mathcal{D}(\mathcal{H}_{AB})\rightarrow\mathbb{R}^{+}$ is an entanglement measure vedral1997quantifying if it satifies:

• (I)

  $E(\rho_{AB})=0$ if $\rho_{AB}\in\mathcal{S}(\mathcal{H}_{AB})$.

• (II)

  $E$ does not increase under the local operation and classical communication (LOCC),

  $\displaystyle E(\Psi(\rho_{AB}))\leq E(\rho_{AB}),$

  here $\Psi$ is LOCC.

In vidal2000entanglement , the author presented that when $E$ satisfies the following two conditions, $E$ is an entanglement monotone,

• (III)

  $E(\rho)\geq\sum_{k}p_{k}E(\sigma_{k}),$ here $\sigma_{k}=\frac{\mathcal{E}_{i,k}(\rho_{AB})}{p_{k}},$ $p_{k}=\mathop{\rm Tr}\mathcal{E}_{i,k}(\rho_{AB})$, $\mathcal{E}_{i,k}$ is any unilocal quantum operation performed by any party $A$ or $B,$ and $i$ are on behave of the party $A$ or $B.$

• (IV)

  For any decomposition $\{p_{k},\rho_{k}\}$ of $\rho_{AB}$

  $\displaystyle E(\rho)\leq\sum_{k}p_{k}E(\rho_{k}).$

The condition (IV) can also be regarded as the convexity of an entanglement measure. Next we recall the simplified conditions when the function satisifies the LOCC monotone horodecki2005simplifying : For a convex function $f$ does not increase under LOCC if and only if

• 1

  . $f$ satisfies local unitary invariance (LUI)

  $\displaystyle f(U_{A}\otimes U_{B}\rho_{AB}U_{A}^{\dagger}\otimes U_{B}^{\dagger})=f(\rho_{AB})$

  (4)

• 2

  . $f$ satisfies

  $\displaystyle f(\sum_{i}p_{i}\rho_{AB}^{i}\otimes|i\rangle_{X}\langle i|)=\sum_{i}p_{i}f(\rho_{AB}^{i}),$

  (5)

  for $X=A^{{}^{\prime}},B^{{}^{\prime}}$, where $|i\rangle$ are local orthogonal flags.

The properties (I), (II) and (IV) are satisfied by $D_{sep}(\cdot)$ vedral1997quantifying ; chen2014comparison , however, the property (III) for the entanglement measure $D_{sep}(\cdot)$ is not valid qiao2017activation . Then we recall the modified version $D_{sep}^{{}^{\prime}}(\cdot)$ of the entanglement measure $D_{sep}(\cdot)$,

Due to the properties of $\lVert\cdot\rVert_{1}$ and the definition of $D_{sep}^{{}^{\prime}}(\cdot),$ if we could prove

here $\{\mathcal{E}_{k}\}$ is a quantum operation on the party $A$ at some stage ($k$ labels different outcomes if at some stage A performs a measurement), $D^{{}^{\prime}}_{sep}(\cdot)$ satisfies the property (III). Next we recall the Naimark theorem paulsen2002completely :

Assume $\{F_{i}\}_{i=1}^{n}$ is a POVM acting on a Hilbert space $\mathcal{H}_{A}$ with dimension $d_{A}$, then there exists a projection-valued measures (PVM) $\{\Pi_{i}\}_{i=1}^{n}$ acting on a Hilbert space $\mathcal{H}_{A^{{}^{\prime}}}$ with dimension $d_{A^{{}^{\prime}}}$, such that

A method is via the direct sum, requiring

here $\rho$ is an arbitrary operator in $\mathcal{H}_{A}$, $0$ is the zero matrix with dimension $d_{A}^{{}^{\prime}}-d_{A}$. As the local operations $\mathcal{E}$, addition of an uncorrelated ancilla system and the dismissal of a local part of the whole system satisfy

the above inequality is due to the definition of $D_{sep}^{{}^{\prime}}(\cdot)$ and the monotonicity of the trace norm under $\mathcal{E}.$ Combing the Naimark theorem, we only need to prove (7) is valid for the PVM, that is, we need to prove that

here $p_{k}=\mathop{\rm Tr}\Pi_{k}\rho,$ $\rho_{k}=\frac{\Pi_{k}\rho\Pi_{k}}{p_{k}}$, $\{\Pi_{k}\}$ is a PVM on the $A$-party.

Here in the first equality, $\sigma$ is the optimal in terms of $D^{{}^{\prime}}_{sep}(\cdot)$ for $\rho$. In the first inequality, $\{\Pi_{k}\}$ is a PVM on the $A$-party of $\rho_{AB}$, $\sum_{k}\Pi_{k}=I$, then due to the contractive under the trace-preserving quantum operations, the first inequality is valid. In the third equality, $p_{k}=\mathop{\rm Tr}\Pi_{k}\rho\Pi_{k},$ $\sigma_{k}=\frac{\Pi_{k}\sigma\Pi_{k}}{p_{k}},$ here the trace of $\sigma_{k}$ may be not 1, that is, $\sigma_{k}$ may be not a state. As $\sigma_{k}$ a separable state, and due to the definition of $D_{sep}(\cdot),$ the last inequality is valid. We finish the proof of the inequality $(\ref{m2})$.

At last, we prove the convexity of $D_{sep}^{{}^{\prime}}(\cdot)$. Assume $\{p_{k},\rho_{k}\}$ is an arbitrary decompostion of $\rho_{AB}$, $\sigma_{k}$ is the optimal for the state $\rho_{k}$ in terms of $D^{{}^{\prime}}_{sep}(\cdot)$, then

the first equality is due to the definition of $(\ref{mem}),$ the first inequality is due to the convexity of the $\lVert\cdot\rVert_{1}$. In the second inequality, we denote $\sigma$ as the optimal in terms of $D_{sep}^{{}^{\prime}}(\cdot)$ for $\rho$, and the second inequality is due to the definition of $D_{sep}^{{}^{\prime}}(\cdot).$ Then we prove the property (IV) is satisfied by the entanglement measure $D_{sep}(\cdot)$.

In this section, we will consider the entanglement measures based on trace norm in terms of two methods, the modified distance measure and the extended distance measure. First we present the analytical formula of a class of entangled states in terms of the modified entanglement measure, we also present some properties of the modified measure. At last, we consider the extended measure generated under the method presented in gour2020optimal ; shi2021extension and show this measure is monogamous for $n$-qubit systems .

In this paper, we have considered the entanglement measures in terms of the trace norm under two methods. We have presented the modified distance measure satisfies convexity for arbitrary dimensional systems. And we also have presented the analytical expressions for some classes of states. Then we have shown the extended measure is an entanglement monotone for the 2-qubit systems and monogamous for $n$-qubit systems. We think the above results may be not valid for larger systems, although we havenot found a counterexample. At last, we believe that our results are helpful in the study of monogamy relations for multipartite entanglement systems. And we hope our work could shed some light on related studies.

This work was supported by the NNSF of China (Grant No. 11871089), and the Fundamental Research Funds for the Central Universities (Grant Nos. KG12080401 and ZG216S1902).

Here we present the proof of Example 1 and Theorem 1.