Strict entanglement monotonicity under local operations and classical communication

TL;DR

This paper establishes conditions under which certain entanglement measures strictly decrease on average under LOCC, highlighting the fundamental nature of entanglement reduction in quantum systems.

Contribution

It proves that convex roof extended entanglement monotones with strictly concave functions, as well as negativity and relative entropy of entanglement, are strict entanglement monotones.

Findings

Convex roof extended entanglement monotones with strictly concave functions are SEMs.

Negativity and relative entropy of entanglement are SEMs.

If squashed entanglement can be obtained by an optimal extension, it is a SEM.

Abstract

Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.

Tables

Table 1. Table 1: A list of entanglement measures. We denote the distillable entanglement  Plenio2006qic , entanglement cost  Bennett1996pra , entanglement of formation (the original one defined in Ref.  HillWotters ), concurrence  Rungta2003pra , G 𝐺 G concurrence  Gour2005 , negativity, convex roof extended negativity  Lee , the logarithmic negativity, tangle, squashed entanglement, Tsallis- q 𝑞 q entanglement  Kim2010pra , Rényi- α 𝛼 \alpha entanglement  Gour2007jmp ; Kimand , the relative entropy of entanglement and the conditional entanglement of mutual information  Yang2008prl by E d subscript 𝐸 𝑑 E_{d} , E c subscript 𝐸 𝑐 E_{c} , E f subscript 𝐸 𝑓 E_{f} , C 𝐶 C , G 𝐺 G , N 𝑁 N , N F subscript 𝑁 𝐹 N_{F} , E N subscript 𝐸 𝑁 E_{N} , τ 𝜏 \tau , E sq subscript 𝐸 sq E_{\rm sq} , T q subscript 𝑇 𝑞 T_{q} , R α subscript 𝑅 𝛼 R_{\alpha} , E r subscript 𝐸 𝑟 E_{r} and E I subscript 𝐸 𝐼 E_{I} , respectively.

$E$	Continuity	Additivity	Convex	Faithfull	Relation	$h$	Monogamy	Strict decreasing111Here strict decreasing refers to the strict decreasing property of the measure under LOCC on average.
$E_d$	?222? means it is unknown.	$\times$Shor2001	$\times$Shor2001	$\times$	$\le E_c$	Strict concave	All states333The one-way distillable entanglement is monogamous Koashi	Pure states
$E_c$	?	$✓$	$✓$Donald2002jmp	?	$\le E_f$Christandl2003 ; Hayden2001jpa	Strict concave	Pure states	Pure states
$E_f$	$✓$	?	$✓$	$✓$		Strict concave	All states	$✓$
$C$	$✓$		$✓$	$✓$		Strict concave	All states	$✓$
$G$	$✓$		$✓$	$✓$		Strict concave	All states	$✓$
$N$	$✓$		$✓$	$\times$		Strict concave	Pure states	$✓$
$N_F$	$✓$		$✓$	$✓$		Strict concave	All states	$✓$
$E_N$	$✓$		$\times$	$✓$		Strict concave	All states	$✓$
$\tau$	$✓$		$✓$	$✓$		Strict concave	All states	$✓$
$E_{sq}$Christandl2004jmp	$✓$Alicki	$✓$	$✓$	$✓$	$E_d\le E_{sq}\le E_c$	Strict concave	All states	Pure states444For mixed states, see Theorem 7.
$T_q$, $q>0$	$✓$		$✓$	$✓$		Strict concave	All states	$✓$
$R_{\alpha }$, $0\le \alpha \le 1$	$✓$		$✓$	$✓$		Strict concave	All states	$✓$
$E_r$	$✓$	$\times$Vollbrecht	$✓$	$✓$	$\le E_f$Vedral1998pra	Strict concave	Pure states	$✓$
$E_I$  Yang2008prl	$✓$	$✓$	$✓$	?	$\le E_f$,$E_c$	Strict concave555It is easy to check that $E_I\left(|\psi ⟩{⟨\psi |}^{AB}\right)=S\left({\rho }^A\right)$ for any pure state ${|\psi ⟩}^{AB}$, ${\rho }^A={\mathrm{Tr}}_B|\psi ⟩{⟨\psi |}^{AB}$.	Pure states	Pure states

Equations

$$E(\rho )\ge \sum _j{p}_{j}E\left({\sigma }_j\right),\forall \rho \in {\mathcal{S}}^{AB}.$$

$$E_F\left(\rho \right)\equiv \min \sum _{j=1}^n{p}_{j}E\left(|{\psi }_j⟩⟨{\psi }_j|\right),$$

$$h\left({\rho }^A\right)\equiv E\left(|\psi ⟩⟨\psi |\right).$$

$$h[\lambda {\rho }_1+(1-\lambda ){\rho }_2]\ge \lambda h({\rho }_1)+(1-\lambda )h({\rho }_2)$$

$$\{{\Phi }_j:\mathrm{Tr}{\Phi }_j(\rho )=p_j,\sum _j{p}_j=1,$$

$${\Phi }_j(\rho )\ne p_j{U}_j^A\otimes U_j^B\rho {(U_j^A)}^{†}\otimes {(U_j^B)}^{†},$$

$${(U_j^X)}^{†}{U}_j^X=I_j^X,j=1,2,\dots ,d.\}$$

$$E\left(\rho \right)>\sum _j{p}_{j}E\left({\sigma }_j\right),$$

$$E\left(\rho \right)=\sum _j{p}_{j}E\left({\sigma }_j\right)$$

$${\sigma }_k={\Phi }_k(\rho )/p_k$$

$$E(|\psi ⟩⟨\psi |)\ge \sum _k{p}_{k}E\left({\sigma }_k\right)=\sum _k{p}_k{E}_F\left({\sigma }_k\right).$$

$$E_F(\rho )=\sum _j{t}_{j}E\left(|{\eta }_j⟩\right).$$

$$E_F(\rho )=\sum _j{t}_{j}E\left(|{\eta }_j⟩\right)$$

$$h({\sigma }^A)=h\left(\sum _k{p}_k{\sigma }_k^A\right)=\sum _k{p}_{k}h\left({\sigma }_k^A\right)$$

$${\sigma }_{j_0}^A\ne {\rho }^A$$

$$|\psi {⟩}^{ABC}=|\varphi {⟩}^{A{B}_1}|\eta {⟩}^{B_{2}C},$$

$$V_j|\psi ⟩\equiv \sum _i⟨i_b|⟨j_c|\psi ⟩|i_b⟩,\forall |\psi ⟩\in {\mathcal{H}}^{BC}.$$

$${\mathrm{Tr}}_C{\rho }^{ABC}=\sum _j{I}^A\otimes V_j{\rho }^{ABC}{I}^A\otimes V_j^{†}.$$

$$|\psi {⟩}^{ABC}=\sum _{k,i,j}{a}_{kij}|k_a⟩|i_b⟩|j_c⟩.$$

$$p_s{\rho }_s^{AB}={\Phi }_s\left({\rho }^{ABC}\right)=|{\psi }_s^{\prime }⟩⟨{\psi }_s^{\prime }|,$$

$${\rho }_s^A={\rho }^A,\forall s,$$

$$|\psi {⟩}^{ABC}=|\varphi {⟩}^{A{B}_1}|\eta {⟩}^{B_{2}C},$$

$$N(\rho )=\frac{{\Vert {\rho }^{T_A}\Vert }_{\mathrm{Tr}}-1}{2},\rho \in {\mathcal{S}}^{AB},$$

$$E_N(\rho )={\log }_{2}N(\rho ).$$

$${\rho }^{T_A}=(1+a){\rho }^{+}-a{\rho }^{-},$$

$$p_k{\sigma }_k^{T_A}$$

$$E_r({\rho }^{AB})\equiv \underset{{\sigma }^{AB}}{\min }S\left({\rho }^{AB}||{\sigma }^{AB}\right),$$

$$U(A\otimes P_{\alpha })U^{†}=\sum _{i,j}{V}_{i}A{V}_j^{†}\otimes |i_c⟩⟨j_c|.$$

$${\mathrm{Tr}}_C\left[I^{AB}\otimes P_{i}U\left({\rho }^{AB}\otimes P_{\alpha }\right)U^{†}{I}^{AB}\otimes P_i\right]$$

$$=V_i{\rho }^{AB}{V}_i^{†}\equiv {\Phi }_i\left({\rho }^{AB}\right)=p_i{\rho }_i^{AB}.$$

$$\sum _i{p}_{i}S\left({\rho }_i^{AB}/p_i||{\sigma }_i^{AB}/q_i\right)=S({\rho }^{AB}||{\sigma }^{AB})$$

$${\Phi }_i(X)\equiv V_{i}X{V}_i^{†}=p_{i}UX{U}^{†}$$

Full text

††preprint: APS/123-QED

Entanglement is one of the most crucial features of quantum theory as compared to classical theory, which is also considered to be a valuable resource for quantum information processing Nielsenbook ; Wildebook . To quantify the amount of entanglement contained in a composite quantum system is a fundamental problem in quantum information science and quantum physics Horodecki2009 ; Guhne ; Plenio2007 ; Donald2002jmp . The first significant milestone in this field came from the discovery that entanglement can be used as a resource for distributed quantum information processing in the frame work of local operations and classical communication (LOCC) Bennett1996 . Consequently, to identify certain a priori axioms for a good measure of entanglement, Vedral et al. Vedral1997 proposed three conditions for a quantity to be such a measure for the first time. Later, Vidal in Ref. Vidal2000 explored a more restrictive requirement on LOCC, and an additional demand of convexity is needed, and there the satisfactory measure is called an entanglement monotone.

It is interesting that these constraints on entanglement measures can be easily checked Vidal2000 : For any convex roof extended entanglement measure, it is an entanglement monotone if it can be defined by both a locally unitary invariant and a concave function on the reduced states of the pure states [see Eqs. (3) and (4) below]. Recently, we found that, for almost all entanglement measures so far, the associated functions are not only concave, but also strictly concave GG2 . More significantly, this strict concavity guarantees the monogamy of entanglement GG2 where the monogamy law is a key feature of entanglement distribution among multiparties (see Refs. GG2 ; GG ; Dhar and references therein for details). This motivates us to investigate entanglement measures deeply. In this paper, we investigate this strict concavity in a more general sense: We show that entanglement is strictly monotonic under LOCC on average for many entanglement monotones. That is, we exploit here a new property of the entanglement monotone.

Let $\mathcal{H}^{A}\otimes\mathcal{H}^{B}\equiv\mathcal{H}^{AB}$ be a bipartite Hilbert space with finite dimension, where $A,B$ are subsystems of the composite quantum system, and let $\mathcal{S}(\mathcal{H}^{AB})\equiv\mathcal{S}^{AB}$ be the set of density operators acting on $\mathcal{H}^{AB}$. Recall that, a function $E:\mathcal{S}^{AB}\to\mathbb{R}_{+}$ is called a measure of entanglement if it satisfies Vedral1997 : (E1) $E(\rho)=0$ iff $\rho$ is separable [this condition can be replaced by $E(\rho)=0$ if $\rho$ is separable]; (E2) $E$ is invariant under local unitary operations, i.e., $E(\rho)=E(U_{A}\otimes U_{B}\rho U_{A}^{\dagger}\otimes U_{B}^{\dagger})$ for any local unitaries $U_{A}$ and $U_{B}$; (E3) $E$ cannot increase under LOCC, i.e., $E[\Phi(\rho)]\leq E(\rho)$ for any LOCC $\Phi$. Note that (E3) implies (E2). The map $\Phi$ is completely positive and trace preserving (CPTP). In general, LOCC can be stochastic in the sense that $\rho$ can be converted to $\sigma_{j}$ with some probability $p_{j}$. In this case, the map from $\rho$ to $\sigma_{j}$ can not be described in general by a CPTP map. More than (E2), $E$ is said to be an entanglement monotone Vidal2000 if it is nonincreased on average under stochastic LOCC, i.e.,

[TABLE]

Note that Eq. (1) is more restrictive than $E(\rho)\geq E(\sum_{i}p_{j}\sigma_{j})$ since in such a case we cannot select subensembles according to a measurement outcome Plenio2005 . It is possible that $E(\sigma_{j_{0}})>E(\rho)$ for some $j_{0}$. Almost all measures of entanglement studied in literature satisfy (1). The measure is said to be faithful if it is zero only on separable states.

Let $E$ be a measure of entanglement on bipartite states. The entanglement of formation $E_{F}$ associated with $E$ is defined by

[TABLE]

where the minimum is taken over all pure state decompositions of $\rho=\sum_{j=1}^{n}p_{j}|\psi_{j}\rangle\langle\psi_{j}|$. That is, $E_{F}$ is the convex roof extension of $E$. Vidal (Vidal2000, , Theorem 2) showed that $E_{F}$ above is an entanglement monotone on mixed bipartite states if the following concavity condition holds. For a pure state $|\psi\rangle\in\mathcal{H}^{AB}$, $\rho^{A}={\rm Tr}_{B}|\psi\rangle\langle\psi|$, define the function $h:\mathcal{S}^{A}\rightarrow\mathbb{R}_{+}$ by

[TABLE]

Note that since $E$ is invariant under local unitaries we must have $h\left(U\rho^{A}U^{\dagger}\right)=h\left(\rho^{A}\right)$ for any unitary operator $U$ acting on $\mathcal{H}^{A}$. If $h$ is also concave, i.e.

[TABLE]

for any states $\rho_{1}$, $\rho_{2}$, and any $0\leq\lambda\leq 1$, then $E_{F}$ as defined in (2) is an entanglement monotone.

It was shown in Ref. GG2 that for almost all the well-known entanglement measures, the associated function defined as in (3) is strictly concave (from which we proved that $E$ is monogamous on pure tripartite states and $E_{F}$ is monogamous on both pure and mixed tripartite states, according to our definition in Ref. GG ). Then, in the sense of Vidal Vidal2000 , what is the corresponding property of LOCC if $h$ is strict concave? We introduce here the concept of strict entanglement monotone in terms of more restriction on the LOCC in (E3). We then show that $E_{F}$ is a strict entanglement monotone if the associated function $h$ is strict concave. Going further, we will prove that many entanglement measures, such as the negativity VidalWerner , the relative entropy of entanglement Vedral1997 ; Vedral1998pra , and the squashed entanglement Christandl2004jmp (if it can be obtained by some optimal extension) are strict entanglement monotones. Our results would demonstrate that entanglement measures are strict in our sense.

For convenience, we fix some terminologies. An entanglement measure $E$ is said to be strictly decreasing on average under LOCC if for any stochastic LOCC,

[TABLE]

there exists $\rho\in\mathcal{S}^{AB}$ such that

[TABLE]

where $p_{j}\sigma_{j}=\Phi_{j}(\rho)$, $U_{j}^{X}$ are unitary operators on $\mathcal{H}^{X}$. Equivalently, an entanglement measure $E$ decreases strictly on average under LOCC if and only if

[TABLE]

holds for all states $\rho\in\mathcal{S}^{AB}$ implies that the LOCC is either a local unitary operation (if the LOCC is a map from system $A+B$ to $A^{\prime}+B^{\prime}$, then it is a local isometric operation; hereafter, we always assume with no loss of generality that the LOCCs are acting from $A+B$ to itself) or a convex mixture of local unitary operations. If an entanglement monotone $E$ is strictly decreasing on average under LOCC, we call it is a strict entanglement monotone (SEM). If an entanglement monotone $E$ is strictly decreasing under LOCC for pure states, we call it is a SEM on pure states.

Applying $\Phi_{k}$ to $\rho$, the state becomes

[TABLE]

with probability $p_{k}={\rm Tr}\Phi_{k}(\rho)$. We assume that $\rho=|\psi\rangle\langle\psi|\in\mathcal{S}^{AB}$ is an entangled pure state. It yields

[TABLE]

If $E$ is a SEM on pure states and the equality holds in (8) for any pure state $|\psi\rangle\in\mathcal{H}^{AB}$, then either $\Phi_{k}\equiv\Phi^{B}$ for some local unitary operation $\Phi^{B}$ or $\Phi_{k}(\cdot)=p_{k}I^{A}\otimes U_{k}^{B}(\cdot)I^{A}\otimes(U_{k}^{B})^{\dagger}$.

Now we assume that $\rho$ is mixed. Perform $\Phi_{k}$ on $\rho$ and denote $\sigma_{k}=\Phi_{k}(\rho)/p_{k}$ with probability $p_{k}={\rm Tr}\Phi_{k}(\rho)$. Observe that there exists an ensemble $\{t_{j},|\eta_{j}\rangle\}$ of $\rho$ such that

[TABLE]

For each $j$, let $\sigma_{jk}=\frac{1}{t_{jk}}\Phi_{k}\left(|\eta_{j}\rangle\langle\eta_{j}|\right)$, where $t_{jk}={\rm Tr}\Phi_{k}(|\eta_{j}\rangle\langle\eta_{j}|)$. Then $\sigma_{k}=\frac{1}{p_{k}}\sum_{j}t_{j}t_{jk}\sigma_{jk}$ and $E(|\eta_{j}\rangle)\geq\sum_{k}t_{jk}E\left(\sigma_{jk}\right)$ by what is proved for pure states above. It follows that

[TABLE]

If $E_{F}(\rho)=\sum_{k}p_{k}E(\sigma_{k})$ for any $\rho\in\mathcal{S}^{AB}$, then $E(|\eta_{j}\rangle)=\sum_{k}t_{jk}E_{F}(\sigma_{jk})$, which completes the proof by the result of the case for pure states. ∎

If $h$ is strictly concave, we assume that the equality holds in (8), which leads to

[TABLE]

since $E(|\psi\rangle\langle\psi|)=h(\rho^{A})=h(\sigma^{A})$ and $\sum_{k}p_{k}h\left(\sigma^{A}_{k}\right)=\sum_{k}p_{k}E\left(\sigma_{k}\right)=\sum_{k}p_{k}E_{F}\left(\sigma_{k}\right)$, where $\rho^{A}={\rm Tr}_{B}|\psi\rangle\langle\psi|$, $\sigma^{A}_{k}={\rm Tr}_{B}\sigma_{k}$ and $\sigma^{A}=\sum_{k}p_{k}\sigma^{A}_{k}$. Then $\sigma^{A}_{k}=\sigma^{A}_{l}$ for any $k$ and $l$, which implies that either $\Phi_{k}\equiv\Phi^{B}$ for some local unitary operation $\Phi^{B}$ or $\Phi_{k}(\cdot)=p_{k}I^{A}\otimes U_{k}^{B}(\cdot)I^{A}\otimes(U_{k}^{B})^{\dagger}$, where $U_{k}^{B}$’s are unitary operators on $\mathcal{H}^{B}$, $\sum_{k}p_{k}=1$. The proof is completed. ∎

Note that, many entanglement measures, such as entanglement of distillation $E_{d}$, entanglement cost $E_{c}$, the squashed entanglement $E_{\rm sq}$ Christandl2004jmp , and the relative entropy of entanglement $E_{r}$ coincide with the entanglement of formation $E_{f}$ (hereafter, we denote by $E_{f}$ the original entanglement formation HillWotters ) for pure states Donald2002jmp ; Vedral1998pra ; Christandl2004jmp . In addition, $E_{d}\leq E_{\rm sq}\leq E_{f}$ Christandl2004jmp , $E_{r}\leq E_{f}$ Vedral1998pra , and $E_{c}\leq E_{f}$ Christandl2003 ; Hayden2001jpa . Thus $E_{f}$, $E_{c}$, $E_{\rm sq}$, and $E_{r}$ are SEMs on pure states, and $E_{d}$ decreases strictly under LOCC on average for pure states ($E_{d}$ is not an entanglement monotone since it is not convex, see Table 1).

[TABLE]

for some $j_{0}$ we have (6) holds, where $p_{j}\sigma_{j}=\Phi_{j}(\rho)$, $X^{A}={\rm Tr}_{B}X$.

It is interesting that we can give another proof of part 1 in Ref. (GG2, , Theorem) from condition (11). We recall part 1 of the Theorem in Ref. GG2 : Let $E$ be an entanglement monotone for which $h$, as defined in Eq. (3), is strictly concave. If $\rho^{ABC}=|\psi\rangle\langle\psi|^{ABC}$ is pure and $E(\rho^{A|BC})=E(\rho^{AB})$, then $\mathcal{H}^{B}$ has a subspace isomorphic to $\mathcal{H}^{B_{1}}\otimes\mathcal{H}^{B_{2}}$ and up to local unitary on system $B_{1}B_{2}$,

[TABLE]

where $|\phi\rangle^{AB_{1}}\in\mathcal{H}^{AB_{1}}$ and $|\eta\rangle^{B_{2}C}\in\mathcal{H}^{B_{2}C}$ are pure states. In particular, $\rho^{AC}$ is a product state [and, consequently, $E(\rho^{AC})=0$], so that $E$ is monogamous on pure tripartite states. In order to see this, we let $\{|i_{b}\rangle\}$ and $\{|j_{c}\rangle\}$ be orthonormal bases of $\mathcal{H}^{B}$ and $\mathcal{H}^{C}$, respectively. Define

[TABLE]

It follows that

[TABLE]

Let $\rho^{ABC}=|\psi\rangle\langle\psi|^{ABC}$ and assume that it satisfies $E(\rho^{A|BC})=E(\rho^{AB})$, $\rho^{AB}={\rm Tr}_{C}\rho^{ABC}$. Let $\{|k_{a}\rangle\}$ be an orthonormal basis of $\mathcal{H}^{A}$, then

[TABLE]

The action of $\Phi_{s}(\cdot)\equiv I^{A}\otimes V_{s}(\cdot)I^{A}\otimes V_{s}^{\dagger}$ on $\rho^{ABC}$ gives

[TABLE]

where $|\psi^{\prime}_{s}\rangle=\sum_{k,i}a_{kis}|k_{a}\rangle|i_{b}\rangle$. That is, $\rho_{s}^{AB}$ is a pure state for any $s$. On the other hand, $E$ obeys (11), which results in

[TABLE]

where $\rho_{s}^{A}={\rm Tr}_{B}\rho_{s}^{AB}$.Note that $\rho^{AB}=\sum_{s}|\psi_{s}\rangle\langle\psi_{s}|^{AB}$, then following the proof of the Theorem in Ref. GG2 , we can conclude that $\mathcal{H}^{B}$ has a subspace isomorphic to $\mathcal{H}^{B_{1}}\otimes\mathcal{H}^{B_{2}}$ and up to local unitary on system $B_{1}B_{2}$,

[TABLE]

where $|\phi\rangle^{AB_{1}}\in\mathcal{H}^{AB_{1}}$ and $|\eta\rangle^{B_{2}C}\in\mathcal{H}^{B_{2}C}$ are pure states.

In what follows, we discuss whether or not the entanglement monotones that are not derived via the convex roof structure are SEMs as well. The well known one is the computable measure of entanglement, negativity, which is defined by VidalWerner

[TABLE]

where $\|X\|_{\rm Tr}={\rm Tr}\sqrt{X^{\dagger}X}$ and $\rho^{T_{A}}$ denotes the partial transposition with respect to part $A$ under some given orthonormal bases of $\mathcal{H}^{A}$ and $\mathcal{H}^{B}$. The logarithmic negativity $E_{N}$ is defined as VidalWerner

[TABLE]

It is known that the negativity $N$ is a SEM on pure states GG2 and thus $N_{F}$ is also a SEM by Proposition 2. In what follows we will show that $N$ is also a SEM on mixed states, and thus it is a SEM.

[TABLE]

where $(1+a)\rho^{+}$ and $a\rho^{-}$ are the positive part and the negative part of ${\rho}^{T_{A}}$, respectively. That is, $N(\rho)=a$, $\rho^{+}\rho^{-}=\rho^{-}\rho^{+}=0$. It follows that

[TABLE]

It is clear that $N(\sigma_{k})\leq q_{k}a/p_{k}$, $q_{k}={\rm Tr}\Phi_{k}(\rho^{-})$. Thus, if Eq. (7) holds, then $N(\sigma_{k})=q_{k}a/p_{k}$, and thus $\Phi_{k}(\rho^{+})\Phi_{k}(\rho^{-})=\Phi_{k}(\rho^{-})\Phi_{k}(\rho^{+})=0$. Take $\rho=|\psi\rangle\langle\psi|$ with $|\psi\rangle=\sum_{j}\lambda_{j}|j_{a}\rangle|j_{b}\rangle$ as the Schmidt decomposition of $|\psi\rangle$. Then $|\psi\rangle\langle\psi|^{T_{A}}=\sum_{j}\lambda_{j}^{2}|j_{a}\rangle\langle j_{a}|\otimes|j_{b}\rangle\langle j_{b}|+\sum_{i<j}\lambda_{i}\lambda_{j}|\psi_{ij}^{+}\rangle\langle\psi_{ij}^{+}|-\sum_{i<j}\lambda_{i}\lambda_{j}|\psi_{ij}^{-}\rangle\langle\psi_{ij}^{-}|$, where $|\psi_{ij}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|i_{a}\rangle|j_{b}\rangle\pm|j_{a}\rangle|i_{b}\rangle)$. Denoting by $M_{k}|j_{b}\rangle\equiv|j^{\prime}_{b}\rangle$, it follows that $\langle i^{\prime}_{b}|i^{\prime}_{b}\rangle=\langle j^{\prime}_{b}|j^{\prime}_{b}\rangle$ holds for any $i$ and $j$, from which we can conclude that $M_{k}$ is a scalar multiple of some unitary operator. This guarantees that $\Phi_{k}$ is either a local unitary operation or $\Phi_{k}(\cdot)=q_{k}I^{A}\otimes U_{k}^{B}(\cdot)I^{A}\otimes(U_{k}^{B})^{\dagger}$ with $\sum_{k}q_{k}=1$ provided that $N(\rho)=\sum_{k}p_{k}N({\sigma_{k}})$. Therefore, $N$ decreases strictly on average under LOCC. ∎

Another important entanglement monotone that is not derived from the convex roof extension is the relative entropy of entanglement Vedral1997 ; Vedral1998pra :

[TABLE]

where $S(\rho^{AB}||\sigma^{AB})\equiv{\rm Tr}[\rho^{AB}(\ln\rho^{AB}-\ln\sigma^{AB})]$ is the quantum relative entropy and the minimum is taken over all separable states $\sigma^{AB}$ in $\mathcal{S}^{AB}$. This measure, as one might expect, is a SEM.

[TABLE]

It is clear that

[TABLE]

According to the proof of Theorem 2 in Ref. Vedral1998pra , we only need to verify that if

[TABLE]

holds for any $\rho^{AB}$ and $\sigma^{AB}$, then

[TABLE]

for some unitary operator $U$, where $q_{i}\sigma_{i}^{AB}=\Phi_{i}(\sigma^{AB})$. Note that

[TABLE]

thus (24) holds and leads to $\sum_{i}p_{i}\ln\dfrac{p_{i}}{q_{i}}=0$, which is equivalent to $p_{i}=q_{i}$ for any $i$. Therefore $\Phi_{i}$ has the form as in (25). Taking $V_{j}=V_{j}^{A}\otimes V_{j}^{B}$, the proof is completed. ∎

The squashed entanglement $E_{\rm sq}$ Christandl2004jmp is an additive entanglement monotone and has a nice operational meaning. For any state $\rho^{AB}\in\mathcal{S}^{AB}$, $E_{\rm sq}$ is defined by Christandl2004jmp

[TABLE]

where $I(A;B|E)=S(\rho^{AE})+S(\rho^{BE})-S(\rho^{ABE})-S(\rho^{E})$, $S(\cdot)$ denotes the von Neumann entropy and the infimum is taken over all extensions of $\rho^{ABE}$ of $\rho^{AB}$. We show below that $E_{\rm sq}$ is also a SEM with the assumption that it can be attained by some optimal extension [i.e., $E_{\rm sq}(\rho^{AB})=\frac{1}{2}I(A;B|E)$ for some extension $\rho^{ABE}$]. Note that, if there does not exist some optimal extension, whether or not $E_{sq}$ is a SEM remains open since it is defined in terms of the infimum process over all states extension which cannot give an accurate equality between the state and its extension state for the conditional mutual information. However, we still do not know such an extension exists or not for any state Christandl2004jmp .

At last, we present a list of the properties of all entanglement measures that are well-known by now for convenience of readers (see Table 1). As one might expect, almost all the entanglement measures are decreasing strictly under LOCC on average for pure states. In addition, one can see from the table that, apart from the strict concavity of the associated function $h$, monogamy is another property that is also closely related with the strict monotonicity of LOCC. We also found that other properties, such as additivity, convexity and faithfulness, seem not to be the nature of the entanglement measures so far.

To summarize, we explored the action of entanglement under LOCC for many entanglement measures so far, and we showed that the axiomatic definition of entanglement monotone can be improved: $E$ is defined to be an entanglement monotone if it is convex, vanishes on separable states, and decreases strictly on average under LOCC in the sense of (6). Together with the result in Ref. GG2 , our results here support the conclusion that entanglement is monogamous. But we still can not prove whether the squashed entanglement (it is defined via the infimum over all extensions), entanglement of distillation, and the entanglement cost are strict entanglement monotones or not.