Bounds on Negativity of Superpositions

TL;DR

This paper investigates the negativity of superposed pure bipartite states, establishing bounds and continuity properties of entanglement measures like concurrence, with implications for estimating entanglement in quantum systems.

Contribution

It provides new bounds on negativity for superposed states and demonstrates the continuity of concurrence even in infinite-dimensional systems.

Findings

High fidelity states have nearly the same entanglement.

Bounds on negativity are derived in terms of component states.

Concurrence is shown to be a continuous function in infinite dimensions.

Abstract

The entanglement quantified by negativity of pure bipartite superposed states is studied. We show that if the entanglement is quantified by the concurrence two pure states of high fidelity to one another still have nearly the same entanglement. Furthermore this conclusion can be guaranteed by our obtained inequality, and the concurrence is shown to be a continuous function even in infinite dimensions. The bounds on the negativity of superposed states in terms of those of the states being superposed are obtained. These bounds can find useful applications in estimating the amount of the entanglement of a given pure state.

Equations

$$E({\Phi }_{AB})\equiv S({\mathrm{Tr}}_B|\Phi {⟩}_{AB}⟨\Phi |)=S({\mathrm{Tr}}_A|\Phi {⟩}_{AB}⟨\Phi |),$$

$$\mathcal{C}({\Phi }_{AB})\equiv \sqrt{2\left(1-\mathrm{Tr}{\rho }_{\mathrm{A}}^2\right)}=\sqrt{2\left(1-{\sum }_i{\mu }_i^2\right)},$$

$$|\varphi {⟩}_{AB}=|00⟩,$$

$$|\psi {⟩}_{AB}=\sqrt{1-ϵ}|\varphi {⟩}_{AB}+\sqrt{\frac{ϵ}{d}}[|11⟩+|22⟩+\cdots +|dd⟩].$$

$$E({\psi }_{AB})\approx ϵ{\log }_{2}d\to \infty ,$$

$${\mathcal{C}}^2({\psi }_{AB})=2\left(2ϵ-{ϵ}^2-\frac{{ϵ}^2}{d}\right)\to 0,$$

$$|{\mathcal{C}}^2({\rho }_{AB})-{\mathcal{C}}^2({\sigma }_{AB})|\le 4T({\rho }_A,{\sigma }_A).$$

$$\sum _i|r_i-s_i|\le T({\rho }_A,{\sigma }_A).$$

$$\left|{\mathcal{C}}^2({\rho }_{AB})-{\mathcal{C}}^2({\sigma }_{AB})\right|$$

$$\mathcal{N}=\frac{1}{2}(\Vert {\rho }^{T_A}\Vert -1).$$

$$\mathcal{N}=\frac{1}{2}\left[{\left(\sum _i\sqrt{{\mu }_i}\right)}^2-1\right].$$

$${\left(\sum _i\sqrt{{\mu }_i}\right)}^2=2\mathcal{N}+1.$$

$${\mu }_i(H)+{\mu }_1(K)\le {\mu }_i(H+K)\le {\mu }_i(H)+{\mu }_n(K),$$

$$\sqrt{{\mu }_i(H)}\le \sqrt{{\mu }_i(H+K)}\le \sqrt{{\mu }_i(H)}+\sqrt{{\mu }_n(K)},$$

$${\rho }_A=\Phi {\Phi }^{†},$$

$$\frac{2|\alpha {|}^2\mathcal{N}({\Phi }_1)+2|\beta {|}^2\mathcal{N}({\Psi }_1)-1}{4}\le \mathcal{N}(\alpha {\Phi }_1+\beta {\Psi }_1)\le \frac{2|\alpha {|}^2\tilde{\mathcal{N}}({\Phi }_1)+2|\beta {|}^2\tilde{\mathcal{N}}({\Psi }_1)-1}{4},$$

$$\tilde{\mathcal{N}}({\Phi }_1)=\mathcal{N}({\Phi }_1)+\frac{n|\beta |\sqrt{{\mu }_n({\Psi }_1)[2\mathcal{N}({\Phi }_1)+1]}}{|\alpha |}+\frac{n^2|\beta {|}^2{\mu }_n({\Psi }_1)}{2|\alpha {|}^2},$$

$$\tilde{\mathcal{N}}({\Psi }_1)=\mathcal{N}({\Psi }_1)+\frac{n|\alpha |\sqrt{{\mu }_n({\Phi }_1)[2\mathcal{N}({\Psi }_1)+1]}}{|\beta |}+\frac{n^2|\alpha {|}^2{\mu }_n({\Phi }_1)}{2|\beta {|}^2}.$$

$${\Gamma }_1{\Gamma }_1^{†}=|\alpha {|}^2{\Phi }_1{\Phi }_1^{†}+|\beta {|}^2{\Psi }_1{\Psi }_1^{†}+\alpha {\beta }^{\ast }{\Phi }_1{\Psi }_1^{†}+{\alpha }^{\ast }\beta {\Psi }_1{\Phi }_1^{†}.$$

$${\Gamma }_1{\Gamma }_1^{†}=|\alpha {|}^2{\Phi }_1{\Phi }_1^{†}+|\beta {|}^2{\Psi }_1{\Psi }_1^{†}.$$

$$|\alpha {|}^2{\mu }_i({\Phi }_1{\Phi }_1^{†})+|\beta {|}^2{\mu }_1({\Psi }_1{\Psi }_1^{†})\le {\mu }_i({\Gamma }_1{\Gamma }_1^{†}).$$

$$|\alpha {|}^2{\mu }_i({\Phi }_1{\Phi }_1^{†})\le {\mu }_i({\Gamma }_1{\Gamma }_1^{†}).$$

$$|\alpha |\sum _i\sqrt{{\mu }_i({\Phi }_1{\Phi }_1^{†})}\le \sum _i\sqrt{{\mu }_i({\Gamma }_1{\Gamma }_1^{†})}.$$

$$\sum _i\sqrt{{\mu }_i({\Gamma }_1{\Gamma }_1^{†}})\le |\alpha |\sum _i\sqrt{{\mu }_i({\Phi }_1{\Phi }_1^{†})}+n|\beta |\sqrt{{\mu }_n({\Psi }_1{\Psi }_1^{†})}.$$

$$|\alpha {|}^2\mathcal{N}({\Phi }_1)+\frac{|\alpha {|}^2-1}{2}$$

$$|\beta {|}^2\mathcal{N}({\Psi }_1)+\frac{|\beta {|}^2-1}{2}$$

$$|\varphi {⟩}_{AB}=\alpha |\phi {⟩}_{AB}+\beta |\psi {⟩}_{AB},$$

$$|\phi {⟩}_{AB}=\frac{1}{\sqrt{2}}|00⟩+\frac{1}{\sqrt{2}}|11⟩,$$

$$|\psi {⟩}_{AB}=\frac{1}{\sqrt{2}}|22⟩+\frac{1}{\sqrt{2}}|33⟩,$$

$$0<\mathcal{N}(|\varphi {⟩}_{AB})=\frac{3}{2}<4,$$

Full text

Quantum entanglement plays an important role both in many aspects of quantum information theorynielsen and in describing quantum phase transition in quantum many-body systemsos ; oo . As such characterization quantification of quantum entanglement is a fundamental issue. Consequently the legitimate measures of entanglement are desirable as a first step. The existing well-known bipartite measure of entanglement with an elegant formula is the concurrence derived analytically by Wootterswootters and the entanglement of formationbennett ; hill is a monotonically increasing function of the concurrence. In general for a multipartite or higher-dimensional system it is a formidable task of quantifying its entanglement since it needs complicate convex-roof extension. In the last 10 years some important properties of quantum entanglement were found, one of which is the monogamy property described by Coffman-Kundu-Wootters inequality in terms of concurrence coffman . In our previous work we have shown that the monogamy inequality can not generalize to higher-dimensional systemsou1 and established a monogamy inequality in terms of negativity giving a different residual entanglementou2 .

On the other hand, quantum entanglement is a direct consequence of the superposition principle. It is an interesting physical phenomenon that the superposition of two separable states may give birth to an entangled state, on the contrary, the superposition of two entangled states may give birth to a separable state. The relation between the entanglement of the state and the entanglement of the individual terms that by superposition yield the state has been studied, where the entanglement is quantified by the von Neumann entropylinden and the concurrenceyu . Recently it was generalized to the superposition of more than two componentsyang . If the entanglement is quantified by negativity, it would be interesting to establish the analogous relation and obtain the bound of entanglement for the superposition state. In this paper, we first show that, by contrast to the von Neumann entropy, the concurrence is a continuous function even in infinite dimensions. We deduce an inequality to guarantee this property. Next we give the bounds of the negativity of the superposition state. The discussion and conclusion are presented in the end.

The authors inlinden have shown that two states of high fidelity to one another may not have the same entanglement, i.e., $|\langle\psi|\phi\rangle|^{2}\rightarrow 1$ may not generally result in $E(\psi)\rightarrow E(\phi)$, where $E$ is the von Neumann entropy. For a bipartite pure state $|\Phi\rangle_{AB}$ the von Neumann entropy is defined as

[TABLE]

where $S(\rho)=-\mathrm{Tr}(\rho\log\rho)$, and the concurrence is defined as

[TABLE]

where $\rho_{A}=\mathrm{Tr}_{B}|\Phi\rangle_{AB}\langle\Phi|$ with the eigenvalues $\mu_{i}$. However, if we employ the concurrence to quantify the entanglement, $|\langle\psi|\phi\rangle|^{2}\rightarrow 1$ must result in $\mathcal{C}(\psi)\rightarrow\mathcal{C}(\phi)$. Let us see their example letting

[TABLE]

and

[TABLE]

It is obviously true that $E(\phi_{AB})=\mathcal{C}(\phi_{AB})=0$, while according to linden the von Neumann entropy of the state $|\psi\rangle_{AB}$ is

[TABLE]

specially when $d$ is as large as we expect. It follows from Eq.(2) that the concurrence of the state $|\psi\rangle_{AB}$ give us the result

[TABLE]

when $\epsilon$ is adequately small. By contrast to $E(\psi_{AB})$ in Eq.(5), $\mathcal{C}^{2}(\psi_{AB})$ in Eq.(6) is independent of $d$. Note that when $\epsilon$ is small the two states have high fidelity $|\langle\psi|\phi\rangle|^{2}=1-\epsilon\rightarrow 1$. Comparing Eq.(5) to Eq.(6), we can draw a conclusion that if the entanglement is quantified by the concurrence two states of high fidelity to one another still have nearly the same entanglement.

It is indeed that the difference of the von Neumann entropy between two pure states of fixed dimension can be bounded using Fannes’ inequalityfannes , while the von Neumann entropy is not a continuous function and no such bound applies in infinite dimensions. However, as we will show here, a similar bound still works if the entanglement is quantified by the concurrence and the concurrence is a continuous function even in infinite dimensions. In order to explain our above viewpoint we present the following Theorem which is similar to the original Fannes’ inequality except that the entanglement is quantified by the concurrence.

Theorem 1. Suppose $\rho_{AB}$ and $\sigma_{AB}$ are density matrices of two bipartite pure states in arbitrary dimensions. For the trace distance $T(\rho_{A},\sigma_{A})\equiv\mathrm{Tr}|\rho_{A}-\sigma_{B}|$ between $\rho_{A}=\mathrm{Tr}_{B}\rho_{AB}$ and $\sigma_{A}=\mathrm{Tr}_{B}\sigma_{AB}$ we have

[TABLE]

Proof. Let $r_{1}\geq r_{2}\geq\dots\geq r_{d}$ be the eigenvalues of $\rho_{A}$, in decreasing order, and $s_{1}\geq s_{2}\geq\dots\geq s_{d}$ be the eigenvalues of $\sigma_{A}$, also in decreasing order. According tonielsen , it follows that

[TABLE]

From the observation of the definition of the concurrence in Eq.(2), we can rewrite the left-hand-side of Eq.(7) as

[TABLE]

The second formula is obtained from the observation that $\left|a+b+\cdot\cdot\cdot+k\right|\leq\left|a\right|+\left|b\right|+\cdot\cdot\cdot+\left|k\right|$ for any complex quantities $a,b,\cdot\cdot\cdot,k.$ In the derivation of the last formula we have taken into account the fact that $r_{i}+s_{i}\leq 2$ since each eigenvalue of $r_{i}$ and $s_{i}$ is not greater than one. Combining Eqs.(8) and (9) can give Eq.(7). Thus the proof is completed.

From the Theorem 1 it can be seen that the difference of the concurrences of two pure states is a function of fidelity and can be bounded by Eq.(7). What’s more, by contrast to the von Neumann entropylinden the concurrence is a continuous function and such a bound still works in infinite dimensions. Note that whether a similar bound in Eq.(7) holds for the negativity is still open. In the next paragraphs we are devoted to deducing the bounds on the negativity of any bipartite pure state as a superposition of two terms $|\Gamma\rangle_{AB}=\alpha|\Psi\rangle+\beta|\Phi\rangle$.

Before embarking on this study, we first recall some basic definitions of the negativity. As for detecting entangled state in higher-dimensional Hilbert space, Peres-Horodecki criterion based on partial transposepe ; ho is a convenient method. Given a density matrix $\rho$ in a bipartite pure system of $A$ and $B$, the partial transpose with respect to $A$ subsystem is described by $(\rho^{T_{A}})_{ij,kl}=(\rho)_{kj,il}$ and the negativity is defined as

[TABLE]

The trace norm $\|R\|$ is given by $\|R\|=\mathtt{Tr}\sqrt{RR^{\dagger}}$. Note that $\mathcal{N}>0$ is the necessary and sufficient condition for entangled bipartite pure states.

There are two key ingredients to obtain the bounds of the negativity for bipartite superposition pure states. One is that the negativity can be expressed by means of Schmidt coefficients of a pure state. Suppose that a pure $m\otimes n(m\leq n)$ quantum state has the standard Schmidt form $|\psi\rangle_{AB}=\sum_{i}\sqrt{\mu_{i}}|a_{i}b_{i}\rangle$, where $\sqrt{\mu_{i}}(i=1,\cdots,m)$ are the Schmidt coefficients, $a_{i}$ and $b_{i}$ are the orthogonal basis in $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, respectively. For the pure bipartite state we can derive $\|\rho^{T_{A}}\|=\left(\sum_{i}\sqrt{\mu_{i}}\right)^{2}$fei , and therefore Eq.(10) can be reexpressed as

[TABLE]

In order for the later use we can transform Eq.(11) into

[TABLE]

The other is the Theoremhorn , which states that for any two Hermitian matrix $H$ and $K$ defined in $\mathcal{C}^{n\times n}$,

[TABLE]

holds, where $\mu_{i}(\cdot)$ are the eigenvalues in increasing order. If $\mu_{1}(K)\geq 0$, from Eq.(13) it is easy to check that

[TABLE]

holds also. Then Eq.(14) will be used repeatedly in what follows.

For the negativity of the arbitrary superposition state let us first see the simplest case in which two bipartite states we are superposing, $\Phi_{1}$ and $\Psi_{1}$, are biorthogonallinden , i.e., $\Phi_{1}\Psi_{1}^{{\dagger}}=\Psi_{1}\Phi_{1}^{{\dagger}}=0$yu . Since the matrix representation of a reduced density matrix will be used, we explain the corresponding notations in the following. For the pure state $|\Phi\rangle_{AB}$ defined in $m\otimes n$ dimensions, generally it can be considered as a vector: $|\Phi\rangle_{AB}=[a_{00},a_{01},\cdots,a_{0m},a_{10},a_{11},\cdots,a_{mn}]^{T}$ with the superscript $T$ denoting transpose operation. With the matrix notation, the reduced density matrix reads

[TABLE]

whose eigenvalues are $\mu_{i}$ appearing in Eq.(11).

Theorem 2. Suppose that two biorthogonal pure states $\Phi_{1}$ and $\Psi_{1}$, which are defined in $m\otimes n(n\leq m)$ dimensions. The negativity of their superposed states $\Gamma_{1}=\alpha\Phi_{1}+\beta\Psi_{1}$ with $|\alpha^{2}|+|\beta|^{2}=1$ satisfies

[TABLE]

where

[TABLE]

and

[TABLE]

Proof. From Eq.(15) the reduced density matrix of the state $\Gamma_{1}$ can read

[TABLE]

The biorthogonal condition with $\Phi_{1}\Psi_{1}^{{\dagger}}=0$ and $\Psi_{1}\Phi_{1}^{{\dagger}}=0$ makes Eq.(17) reduce to

[TABLE]

Substituting Eq.(18) into the left inequality of Eq.(13) we have

[TABLE]

Since $\Psi_{1}\Psi_{1}^{{\dagger}}$ is positive semidefinite, $\mu_{1}(\Psi_{1}\Psi_{1}^{{\dagger}})\geq 0$. Thus Eq.(19) becomes

[TABLE]

Taking the square root of both sides in Eq.(20) and the sum of $\sqrt{\mu_{i}(\cdot)}$ over all index $i$, we have

[TABLE]

In a similar way, substituting Eq.(18) into the right inequality of Eq.(14) and taking the sum of $\sqrt{\mu_{i}(\cdot)}$ over all index $i$, we have

[TABLE]

Substituting Eqs.(21) and (22) into Eq.(12), respectively, we can obtain

[TABLE]

If we replace the matrix $|\alpha|^{2}\Phi_{1}\Phi_{1}^{{\dagger}}$ with $|\beta|^{2}\Psi_{1}\Psi_{1}^{{\dagger}}$ in Eqs.(20) and (21), i.e., equivalently exchange the matrixes $H$ and $K$ in Eq.(14), finally we can also obtain

[TABLE]

Then combining Eqs.(23) and (24) gives Eq.(16). Thus the proof is completed.

Note that the lower bound in Eq.(16) can provide a nonzero value only when $2|\alpha|^{2}\mathcal{N}(\Phi_{1})+2|\beta|^{2}\mathcal{N}(\Psi_{1})>1$. Next we show an example to illustrate the validity of our bound. Consider the state

[TABLE]

with

[TABLE]

[TABLE]

where $\alpha=\beta=1/\sqrt{2}$. It is easy to check that $|\varphi\rangle_{AB}$ and $|\psi\rangle_{AB}$ are biorthogonal, $\mathcal{N}(|\phi\rangle_{AB})=3/2$, $\mathcal{N}(|\varphi\rangle_{AB})=\mathcal{N}(|\psi\rangle_{AB})=1/2$, and $\mu_{4}(|\varphi\rangle_{AB})=\mu_{4}(|\psi\rangle_{AB})=1/2$. Accordingly from Eq.(16) we obtain the lower and upper bounds

[TABLE]

which work well.

Finally we directly present the main Theorem of this paper, in which the two states being superposed can be biorthoganal, orthogonal, or nonorthogonal.

Theorem 3. Suppose that two arbitrary normalized pure states $\Phi_{2}$ with rank $r_{1}$ and $\Psi_{2}$ with rank $r_{2}$, which are defined in any dimensions. The negativity of their superposed states $\Gamma_{2}=\alpha\Phi_{2}+\beta\Psi_{2}$ with rank $r_{3}$ and $|\alpha^{2}|+|\beta|^{2}=1$ satisfies

[TABLE]

where

[TABLE]

[TABLE]

where $r=\max\{r_{1},r_{2},r_{3}\}$.

Proof. Consider the matrix

[TABLE]

which can be rewritten as

[TABLE]

where $\Gamma_{2}^{-}=\alpha\Phi_{2}-\beta\Psi_{2}$, $\widehat{\Gamma}_{2}=\Gamma_{2}/\|\Gamma_{2}\|$, and $\widehat{\Gamma}_{2}^{-}=\Gamma_{2}^{-}/\|\Gamma_{2}^{-}\|$. Thus Eqs.(13) shows that

[TABLE]

and

[TABLE]

Since $\mu_{1}(\Psi_{2}\Psi_{2}^{{\dagger}})\geq 0$ and $\mu_{1}\left(\widehat{\Gamma}_{2}^{-}(\widehat{\Gamma}_{2}^{-})^{{\dagger}}\right)\geq 0$, observing the left inequality of Eq.(33) and the right inequality in Eq.(32) we have

[TABLE]

Substituting Eqs.(34) into Eq.(12) we have

[TABLE]

Likewise, if we replace the two matrixes $|\alpha|^{2}\Phi_{2}\Phi_{2}^{{\dagger}}$ with $|\beta|^{2}\Psi_{2}\Psi_{2}^{{\dagger}}$ in Eq.(32), we can obtain

[TABLE]

Combining Eqs.(35) and (36) gives Eq.(29). Thus the proof is completed.

Since there exists a extra term of the maximal eigenvalue in the second inequality in Eq.(33), generally it is difficult to achieve the universal formula for the lower bound of the negativity in this case. But it is our interest in the future work.

In conclusion, we have shown that if the entanglement is quantified by the concurrence two pure states of high fidelity to one another still have nearly the same entanglement and obtained an inequality that can guarantee that the concurrence is a continuous function even in infinite dimensions. However, whether the similar property can apply to the negativity is still open. The bounds on the negativity of superposed states in terms of those of the states being superposed were obtained. So far some bounds of the wildly-studied measures of entanglement like the von Neumann entropylinden , the concurrenceyu and the negativity in this paper for the superposition states have been provided. In view of that the concurrence can be directly accessible in laboratory experimentwa , these bounds can find useful applications in estimating the amount of the entanglement of a given pure state.

The author Y.C.O. was supported from China Postdoctoral Science Foundation and the author H.F. was supported by ’Bairen’ program NSFC grant and ’973’ program (2006CB921107).