The norms of Bloch vectors and classification of four qudits quantum states

TL;DR

This paper derives tight bounds on the norms of Bloch vectors for four-qudit quantum states, providing new criteria for entanglement and separability, and offers a comprehensive classification of such states.

Contribution

It introduces tight bounds on Bloch vector norms for four-qudit states and uses these to classify states and analyze entanglement and separability.

Findings

Derived tight upper bounds for Bloch vector norms.

Established necessary conditions for separability in four-partite systems.

Provided a complete classification scheme for four-qudit quantum states.

Abstract

We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. Theses upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four qudits quantum systems.

Equations

$$\rho =\frac{1}{d}{\mathrm{I}}_d+\frac{1}{2}\sum _{i=1}^{d^2-1}⟨{\lambda }_i⟩{\lambda }_i.$$

$$\rho =\frac{1}{d^2}I\otimes I+\frac{1}{2d}\sum _{k=1}^{d^2-1}{r}_k{\lambda }_k\otimes I+\frac{1}{2d}\sum _{l=1}^{d^2-1}{s}_{l}I\otimes {\lambda }_l+\frac{1}{4}\sum _{k=1}^{d^2-1}\sum _{l=1}^{d^2-1}{t}_{kl}{\lambda }_k\otimes {\lambda }_l,$$

$$\Vert T^{(12)}{\Vert }^2\le \frac{4(d^2-1)}{d^2},$$

$$\rho$$

$$+\frac{1}{4d}(\sum t_{ij}^{12}{\lambda }_i\otimes {\lambda }_j\otimes I+\sum t_{ik}^{13}{\lambda }_i\otimes I\otimes {\lambda }_k+\sum t_{jk}^{23}I\otimes {\lambda }_j\otimes {\lambda }_k)$$

$$+\frac{1}{8}\sum t_{ijk}^{123}{\lambda }_i\otimes {\lambda }_j\otimes {\lambda }_k,$$

$$\Vert T^{(123)}{\Vert }^2\le \frac{1}{d^3}(8d^3-24d+16)$$

$$\rho =\frac{1}{d^4}I\otimes I\otimes I\otimes I+\frac{1}{2d^3}{M}_1+\frac{1}{4d^2}{M}_2+\frac{1}{8d}{M}_3+\frac{1}{16}{M}_4,$$

$$M_1$$

$$M_2$$

$$M_3$$

$$M_4$$

$$\Vert T^{(1234)}{\Vert }^2\le \frac{16(d^2-1{)}^2}{d^4}.$$

$$E_T(|\psi ⟩)=\frac{d^N}{2^N}||T^{(N)}||-(\frac{d(d-1)}{2}{)}^{\frac{N}{2}},$$

$$E_T(|\psi ⟩)\le \{\begin{array}{r}\sqrt{\frac{d^3(d-1{)}^2}{8}}(\sqrt{d+2}-\sqrt{d-1}),\text{N=3;}\\ \frac{d^2(d-1)}{2},\text{N=4.}\end{array}$$

$$\sum _{1\le x<y<z\le 4}{\Vert {\mathrm{T}}^{(xyz)}\Vert }^2\le \frac{8{(d^2-1)}^3}{d^3(d^2-2)},$$

$$\Vert T^{(1234)}{\Vert }^2\le \{\begin{array}{rlr}\frac{16}{d^4}((d-1)(d^3-3d+2)),& & \text{if ρ is 1-3 separable;}\\ \frac{16}{d^4}(d^2-1{)}^2,& & \text{if ρ is 2-2 separable;}\\ \frac{16}{d^4}(d^2-1)(d-1{)}^2,& & \text{if ρ is 1-1-2 separable;}\\ \frac{16}{d^4}(d-1{)}^4,& & \text{if ρ is 1-1-1-1 separable}.\end{array}$$

$$\rho =x|\psi ⟩⟨\psi |+\frac{1-x}{16}I,$$

$$\mathrm{Tr}{\rho }^2$$

$$\frac{1}{d^3}+$$

$${\rho }_1=\frac{1}{d}I+\frac{1}{2}\sum t_i^1{\lambda }_i^1,{\rho }_{23}=\frac{1}{d^2}I\otimes I+\frac{1}{2d}(\sum t_j^2{\lambda }_j^2\otimes I+\sum t_k^{3}I\otimes {\lambda }_k^3)+\frac{1}{4}\sum t_{jk}^{23}{\lambda }_j\otimes {\lambda }_k.$$

$$\mathrm{Tr}{\rho }_1^2=\frac{1}{d}+\frac{1}{2}{\Vert T^{(1)}\Vert }^2$$

$$\mathrm{Tr}{\rho }_{23}^2=\frac{1}{d^2}+\frac{1}{2d}({\Vert T^{(2)}\Vert }^2+{\Vert T^{(3)}\Vert }^2)+\frac{1}{4}{\Vert T^{(23)}\Vert }^2.$$

$$\mathrm{Tr}{\rho }_2^2=\frac{1}{d}+\frac{1}{2}{\Vert T^{(2)}\Vert }^2,$$

$$\mathrm{Tr}{\rho }_{13}^2=\frac{1}{d^2}+\frac{1}{2d}({\Vert T^{(1)}\Vert }^2+{\Vert T^{(3)}\Vert }^2)+\frac{1}{4}{\Vert T^{(13)}\Vert }^2;$$

$$\mathrm{Tr}{\rho }_3^2=\frac{1}{d}+\frac{1}{2}{\Vert T^{(3)}\Vert }^2,$$

$$\mathrm{Tr}{\rho }_{12}^2=\frac{1}{d^2}+\frac{1}{2d}({\Vert T^{(1)}\Vert }^2+{\Vert T^{(2)}\Vert }^2)+\frac{1}{4}{\Vert T^{(12)}\Vert }^2.$$

$$\mathrm{Tr}{\rho }_i^2=\mathrm{Tr}{\rho }_{jk}^2$$

$$\frac{3}{d}+\frac{1}{2}({\Vert T^{(1)}\Vert }^2+{\Vert T^{(2)}\Vert }^2+{\Vert T^{(3)}\Vert }^2)$$

$$=\frac{3}{d^2}+\frac{1}{d}({\Vert T^{(1)}\Vert }^2+{\Vert T^{(2)}\Vert }^2+{\Vert T^{(3)}\Vert }^2)+\frac{1}{4}({\Vert T^{(12)}\Vert }^2+{\Vert T^{(13)}\Vert }^2+{\Vert T^{(23)}\Vert }^2).$$

$$A={\Vert T^{(1)}\Vert }^2+{\Vert T^{(2)}\Vert }^2+{\Vert T^{(3)}\Vert }^2$$

$$B={\Vert T^{(12)}\Vert }^2+{\Vert T^{(13)}\Vert }^2+{\Vert T^{(23)}\Vert }^2.$$

$$\frac{1}{4}B=(\frac{1}{2}-\frac{1}{d})A+\frac{3}{d}-\frac{2}{d^2}.$$

Full text

The norms of Bloch vectors and classification

of four qudits quantum states

Ming Li1,2, Zong Wang1, Jing Wang1, Shuqian Shen1 and Shao-ming Fei2,3

$~{}^{1}$ College of the Science, China University of Petroleum, 266580 Qingdao

$~{}^{2}$ Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig

$~{}^{3}$ School of Mathematical Sciences, Capital Normal University, 100048 Beijing

Abstract

We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. Theses upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four qudits quantum systems.

PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w

I Introduction

Quantum entanglement, as the remarkable nonlocal feature of quantum mechanics, is recognized as a valuable resource in the rapidly expanding field of quantum information science, with various applications nielsen ; di such as quantum computation , quantum teleportation, dense coding, quantum cryptographic schemes, quantum radar, entanglement swapping and remote states preparation. ${\rm{Tr}}$

It is known that the Bloch vectors give one of the possible descriptions of qudit states. The Bloch vectors are then generalized to composite quantum systems with many subsystems. From the norms of the Bloch vectors in the generalized Bloch representation of a quantum state, separable conditions for both bi- and multi-partite quantum states have been presented in vicente1 ; vicente2 ; hassan ; ming . Two multipartite entanglement measures for N-qubit and N-qudit pure states are given in hassan1 ; hassan2 . A general framework for detecting genuine multipartite entanglement and non full separability in multipartite quantum systems of arbitrary dimensions has been introduced in vicente3 . In horo1995 ; mingbell it has been shown that the norms of the Bloch vectors have a close relationship to the maximal violation of a kind of multi Bell inequalities and to the concurrence GE ; newge . However, with the increasing of the dimensions of the subsystems, the norms of Bloch vectors for density matrices become hard to describeMahler1995 ; Siennicki2001 ; Kimura2003 ; G.K2003 .

In this paper, we study the Bloch representations of quantum states with the number of subsystems less than or equal to four. We present tight upper bounds for the norms of Bloch vectors. These upper bounds are then used to derive tight upper bounds for entanglement measure in hassan1 ; hassan2 . A trade-off relation of the norms of Bloch vectors is also discussed by these bounds. Then we investigate different subclasses of bi-separable states in four-partite systems. Necessary conditions are presented for these kinds of separable states. By these analyses we present a complete classification of four qudits quantum states.

II Upper bounds of the norms of Bloch vectors

Let ${\lambda_{i}}$s $({i=1,...,{d^{2}}-1})$ be orthogonal generators of $SU(d)$ which satisfy $\lambda_{i}^{+}={\lambda_{i}},{\rm{Tr}}(\lambda_{i})=0,{\rm{Tr}}(\lambda_{i}\lambda_{j})=2{\delta_{ij}}.$ Denote the identity operator by ${{\rm I}_{d}}$. One finds that ${{\rm I}_{d}}$ and ${\lambda_{i}}$s compose an orthogonal basis of the linear space consisting of all $d\times d$ Hermitian matrices with respect to the Hilbert-Schmidt inner product. By using ${\rm{Tr}}\rho=1$ and $\langle{{\lambda_{i}}}\rangle={\rm{Tr}}({\rho{\lambda_{i}}})$ , we get that any density operator $\rho$ can be written in the form:

[TABLE]

The Bloch vectorBloch1946 ; Hioe1981 ; Pottinger1985 ; Lendi1986 ; Alicki1987 ; Mahler1995 ; G.K2003 ; Siennicki2001 ; Kimura2003 is defined by ${\bf{b}}=({{b_{1}},...,{b_{{d^{2}}-1}}})\equiv({\langle{{\lambda_{1}}}\rangle,...,\langle{{\lambda_{{d^{2}}-1}}}\rangle}).$ The state can be determined by measuring values of ${\lambda_{i}}$s, the state $\rho$ can also be given by the map ${\bf{b}}\to\rho=\frac{1}{d}{{\rm I}_{d}}{\rm{+}}\frac{{\rm{1}}}{{\rm{2}}}\sum\limits_{i=1}^{{d^{2}}-1}{{b_{i}}{\lambda_{i}}}.$ The set of all the Bloch vectors that constitute a density operator is known as the Bloch vector space $B({\mathbb{R}}^{d^{2}-1})$.

A matrix of the form (1) is of unit trace and Hermitian, but it might not be positive. To guarantee the positivity restrictions must be imposed on the Bloch vector. It is shown that $B({\mathbb{R}}^{d^{2}-1})$ is a subset of the ball $D_{R}({\mathbb{R}}^{d^{2}-1})$ of radius $R=\sqrt{2(1-\frac{1}{d})}$, which is the minimum ball containing it, and that the ball $D_{r}({\mathbb{R}}^{d^{2}-1})$ of radius $r=\sqrt{\frac{2}{d(d-1)}}$ is included in $B({\mathbb{R}}^{d^{2}-1})$ Harriman , that is, $D_{r}({\mathbb{R}}^{d^{2}-1})\subseteq B({\mathbb{R}}^{d^{2}-1})\subseteq D_{R}({\mathbb{R}}^{d^{2}-1}).$

Using the generators of $SU(d)$, any quantum state $\rho\in H_{1}^{d}\otimes H_{2}^{d}$ can be writing as:

[TABLE]

where $r_{k}={\rm{Tr}}(\rho\lambda_{k}\otimes I),s_{l}={\rm{Tr}}(\rho I\otimes\lambda_{l})$ and $t_{kl}={\rm{Tr}}(\rho\lambda_{k}\otimes\lambda_{l})$. We denote by $T^{(12)}$ a vector with entries $t_{kl}$. By using ${\rm{Tr}}(\rho^{2})\leq 1$, one obtains that

[TABLE]

where $||\cdot||$ stands for the Hilbert-Schmidt norm or Frobenius norm.

We then consider the upper bounds of the Hilbert-Schmidt norm of the Bloch vectors for tripartite quantum systems. Let $\rho\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}$ be a quantum state, which can be represented by Bloch vectors as follow:

[TABLE]

where $t_{i}^{1}={\rm{Tr}}({\rho{\lambda_{i}}\otimes I\otimes I}),t_{j}^{2}={\rm{Tr}}({\rho I\otimes{\lambda_{j}}\otimes I}),t_{k}^{3}={\rm{Tr}}({\rho I\otimes I\otimes{\lambda_{k}}}),t_{ij}^{{\rm{12}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes{\lambda_{j}}\otimes I}),$ $t_{ik}^{{\rm{13}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes I\otimes{\lambda_{k}}}),t_{jk}^{{\rm{23}}}={\rm{Tr}}({\rho I\otimes{\lambda_{j}}\otimes{\lambda_{k}}}),t_{ijk}^{{\rm{123}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes{\lambda_{j}}\otimes{\lambda_{k}}})$ in the above representation. Define further ${T^{(x)}},{T^{({xy})}},{T^{({123})}}$ be the vectors with entries $t_{i}^{x},t_{ij}^{xy},t_{ijk}^{123}$, and $1\leq x<y<z\leq 3$.

Theorem 1: For $\rho\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}$ with Bloch representation $(\ref{s3}),$ we have:

[TABLE]

See supplemental material for the proof of the theorem.

We further consider four-partite quantum states. Let $\rho\in{{\rm H}_{1234}}{\rm{=H}}_{1}^{d}\otimes{\rm{H}}_{2}^{d}\otimes{\rm{H}}_{3}^{d}\otimes{\rm{H}}_{4}^{d}$ be a mixed quantum state with the Bloch representation

[TABLE]

where

[TABLE]

We have defined $t_{i}^{1}={\rm{Tr}}({\rho{\lambda_{i}}\otimes I\otimes I\otimes I}),t_{j}^{2}={\rm{Tr}}({\rho I\otimes{\lambda_{j}}\otimes I\otimes I}),t_{k}^{3}={\rm{Tr}}({\rho I\otimes I\otimes{\lambda_{k}}\otimes I})$, $t_{l}^{4}={\rm{Tr}}({\rho I\otimes I\otimes I\otimes{\lambda_{l}}})$, $t_{ij}^{{\rm{12}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes{\lambda_{j}}\otimes I\otimes I}),t_{ik}^{{\rm{13}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes I\otimes{\lambda_{k}}\otimes I}),t_{il}^{{\rm{14}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes I\otimes I\otimes{\lambda_{l}}})$,$t_{jk}^{{\rm{23}}}={\rm{Tr}}({\rho I\otimes{\lambda_{j}}\otimes{\lambda_{k}}\otimes I}),t_{jl}^{{\rm{24}}}={\rm{Tr}}({\rho I\otimes{\lambda_{j}}\otimes I\otimes{\lambda_{l}}}),t_{kl}^{34}={\rm{Tr}}({\rho I\otimes I\otimes{\lambda_{k}}\otimes{\lambda_{l}}})$, $t_{ijk}^{{\rm{123}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes{\lambda_{j}}\otimes{\lambda_{k}}\otimes I}),t_{ijl}^{{\rm{124}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes{\lambda_{j}}\otimes I\otimes{\lambda_{l}}}),t_{ikl}^{{\rm{134}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes I\otimes{\lambda_{k}}\otimes{\lambda_{l}}})$,$t_{jkl}^{{\rm{234}}}={\rm{Tr}}({\rho I\otimes{\lambda_{j}}\otimes{\lambda_{k}}\otimes{\lambda_{l}}})$, $t_{ijkl}^{{\rm{1234}}}={\rm{Tr}}({\rho{\lambda_{i}}\otimes{\lambda_{j}}\otimes{\lambda_{k}}\otimes{\lambda_{l}}})$ in the above representation. Define further ${T^{(x)}},{T^{({xy})}},{T^{({xyz})}},{T^{({1234})}}$ be the vectors with entries $t_{i}^{x},t_{ij}^{xy},t_{ijk}^{xyz},t_{ijkl}^{1234}$, and $1\leq x<y<z\leq 4$.

Theorem 2: For $\rho\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}\otimes{\rm H}_{4}^{d}$ with Bloch representation $(\ref{rho4}),$ we have:

[TABLE]

See supplemental material for the proof of the theorem.

The two upper bounds for norms of Bloch vectors are tight and useful as that will be shown in the following remarks.

Remark 1: The Bloch vectors are used to define a valid entanglement measure in hassan1 ; hassan2 as follows. For a N-qudit pure state, the entanglement measure is defined as

[TABLE]

where $T^{(N)}$ is defined as a tensor with elements $t_{i_{1}i_{2}\cdots i_{N}}^{{\rm{12\cdots N}}}={\rm{Tr}}({\rho\lambda_{i_{1}}\otimes\lambda_{i_{2}}\otimes\cdots\otimes\lambda_{i_{N}}})$.

By Theorem 1 and 2, one obtains the upper bounds of $E_{T}(|\psi\rangle)$ for $N=3$ and $N=4$ as follows.

[TABLE]

By considering the the tripartite-qutrit state $|\psi\rangle=\frac{1}{\sqrt{3}}(|000\rangle+|111\rangle+|222\rangle)$ and the four-qubit state $|\phi\rangle=\frac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle)$, one computes the upper bounds of $E_{T}(|\psi\rangle)$ are 3.01969 and 2 respectively(coincide with that in hassan2 ). Thus the upper bounds of $E_{T}(|\psi\rangle)$ are tight.

Remark 2: We consider four-partite quantum systems. In zong we have shown that for the state $\rho$ with representation (6), we have:

[TABLE]

where $||\cdot||$ stands for the $l_{2}$ norm of a vector.

Set $d=2$, we get $\sum\limits_{1\leq x<y<z\leq 4}\|\rm{T}^{(xyz)}\|^{2}\leq 13.5$. By theorem 1 one has $\|\rm{T}^{(xyz)}\|^{2}\leq 4$. Thus we obtain that it is impossible for $\|\rm{T}^{(123)}\|,\|\rm{T}^{(124)}\|,\|\rm{T}^{(134)}\|,$ and $\|\rm{T}^{(234)}\|$ attaining 4 simultaneously.

III Necessary conditions for bi-separable states.

In this section, we investigate subclasses of the bi-separable states in four partite quantum systems by the upper bounds of norms for Bloch vectors. Let’s start with the following definition.

Definition: Let $\rho\in{\rm H}_{\rm{1}}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}\otimes{\rm H}_{4}^{d}$ be a quantum state with $d$ being the dimension of the subsystems ${\rm H}_{i},i=1,2,3,4.$ If $\rho$ can be written as $\rho=\sum_{k}p_{k}|x_{k}\rangle\langle x_{k}|,$ where $\sum_{k}p_{k}=1,|x_{k}\rangle$ is in one of the following sets: $\{|\phi_{1}\rangle\otimes|\phi_{234}\rangle,|\phi_{2}\rangle\otimes|\phi_{134}\rangle,|\phi_{3}\rangle\otimes|\phi_{124}\rangle,|\phi_{4}\rangle\otimes|\phi_{123}\rangle\}$, $\{|\psi_{12}\rangle\otimes|\psi_{34}\rangle,|\psi_{13}\rangle\otimes|\psi_{24}\rangle,|\psi_{14}\rangle\otimes|\psi_{23}\rangle\}$, $\{|\xi_{1}\rangle\otimes|\xi_{2}\rangle\otimes|\xi_{34}\rangle,|\xi_{1}\rangle\otimes|\xi_{3}\rangle\otimes|\xi_{24}\rangle,|\xi_{1}\rangle\otimes|\xi_{4}\rangle\otimes|\xi_{23}\rangle,|\xi_{14}\rangle\otimes|\xi_{2}\rangle\otimes|\xi_{3}\rangle,|\xi_{13}\rangle\otimes|\xi_{2}\rangle\otimes|\xi_{4}\rangle,|\xi_{12}\rangle\otimes|\xi_{3}\rangle\otimes|\xi_{4}\rangle\}$ and $\{|\chi_{1}\rangle\otimes|\chi_{2}\rangle\otimes|\chi_{3}\rangle\otimes|\chi_{4}\rangle\}$, then $\rho$ is called $1-3$ separable, $2-2$ separable, $1-1-2$ separable, and $1-1-1-1$ separable respectively.

The following theorem gives necessary conditions of these kinds of separable states.

Theorem 3: Let $\rho\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}\otimes{\rm H}_{4}^{d}$ be a four-qudit quantum state. We have

[TABLE]

See supplemental material for the proof of the theorem.

The following two examples show that the upper bounds in theorem 3 are nontrivial and are tight.

Example 1: Consider the quantum state $\rho\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}\otimes{\rm H}_{4}^{d}$,

[TABLE]

where $|\psi\rangle=\frac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle)$ and $I$ stands for the identity operator. By theorem 3, we compute that $\|T^{(1234)}\|^{2}=9x^{2}$. Thus for $\frac{2}{3}<x\leq 1$ and $\frac{1}{\sqrt{3}}<x\leq\frac{2}{3}$, $\rho$ will be not 1-3 separable and not 1-1-2 separable respectively. While for $\frac{1}{3}<x\leq\frac{1}{\sqrt{3}}$, $\rho$ is not 1-1-1-1 separable.

Example 2: Consider bi-separable state $\rho^{d}=|\psi_{+}^{d}\rangle\langle\psi_{+}^{d}|$ with $|\psi_{+}^{d}\rangle=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}|ii\rangle\otimes\frac{1}{\sqrt{d}}\sum_{i=1}^{d}|ii\rangle$. One computes that $\|T^{(1234)}\|^{2}=\frac{16}{d^{4}}(d^{2}-1)^{2}$ which means that the upper bound for 2-2 separable states in theorem 3 is saturated. Actually, the upper bound can be also attained by considering the maximal entangled states as shown in remark 1.

Remark 3: With above theorems and examples, we are ready to classify the four-partite quantum states by using the norms of the Bloch vector $\|T^{(1234)}\|$, as shown in Fig.1. It is worth mentioning that the 1-3 separable quantum states are always in the interior of the bi-separable set, while for some 2-2 separable quantum states the boundary of the bi-separable set is attainable. Since the upper bound for 2-2 separable states is just the upper bound for any four qudits states, we conclude that it is possible that the 2-2 separable state is on the boundary of the set of states(see Fig. 1).

IV Conclusions and Remarks

It is a basic and fundamental question in quantum entanglement theory to classify and detect entanglement states. In this paper, we have investigated the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms have been derived, which are used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. A trade-off relation of the norms of Bloch vectors is also discussed by these bounds. Then these upper bounds have been applied on the separability. Necessary conditions have been presented for 1-3, 2-2, 1-1-2 and 1-1-1-1 separable quantum states in four-partite quantum systems. With these bounds a complete classification of four qudits quantum states is presented.

Acknowledgments    This work is supported by the NSFC No.11775306, and 11701568; the Fundamental Research Funds for the Central Universities Grants No.16CX02049A, 17CX02033A and 18CX02023A; the Shandong Provincial Natural Science Foundation No.ZR2016AQ06, and ZR2017BA019.

Supplemental material for “The norms of Bloch vectors and classification of four qudits quantum states”

Ming Li

College of the Science, China University of Petroleum, 266580 Qingdao, China

Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

  

Zong Wang

College of the Science, China University of Petroleum, 266580 Qingdao, China

  

Jing Wang

College of the Science, China University of Petroleum, 266580 Qingdao, China

  

Shuqian Shen

College of the Science, China University of Petroleum, 266580 Qingdao, China

  

Shao-ming Fei

Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

School of Mathematical Sciences, Capital Normal University, 100048 Beijing, China

IV.1 Proof of Theorem 1

Proof: We start with the pure state. For an arbitrary pure state $\rho=\left|\psi\right\rangle\left\langle\psi\right|$ one has ${\rm{Tr}}{\rho^{2}}=1,$ which means

[TABLE]

Set ${\left\|{{T^{(1)}}}\right\|^{2}}={\sum{(t_{i}^{1})}^{2}},{\left\|{{T^{(2)}}}\right\|^{2}}={\sum{(t_{j}^{2})}^{2}},{\left\|{{T^{(3)}}}\right\|^{2}}={\sum{(t_{k}^{3})}^{2}},{\left\|{{T^{(12)}}}\right\|^{2}}={\sum{(t_{ij}^{12})}^{2}},{\left\|{{T^{(13)}}}\right\|^{2}}={\sum{(t_{ij}^{13})}^{2}},{\left\|{{T^{(23)}}}\right\|^{2}}={\sum{(t_{jk}^{23})}^{2}}$ and ${\left\|{{T^{(123)}}}\right\|^{2}}={\sum{(t_{ijk}^{123})}^{2}}.$

Then we have:

[TABLE]

One computes that:

[TABLE]

Thus we have:

[TABLE]

and

[TABLE]

Similarly we get:

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

By noticing that we are now considering pure state $\rho=\left|\psi\right\rangle\left\langle\psi\right|,$ one has

[TABLE]

for $ijk\in\{123,213,312\}$. Then we get:

[TABLE]

Set

[TABLE]

and

[TABLE]

We have:

[TABLE]

Substitute (S4) into (S2), one has

[TABLE]

Furthermore, we have

[TABLE]

Thus one has:

[TABLE]

Let $\rho\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}$ be an arbitrary mixed state with ensemble decomposition $\rho=\sum{{p_{\alpha}}}\left|{{\psi_{\alpha}}}\right\rangle\left\langle{{\psi_{\alpha}}}\right|.$ We have:

[TABLE]

IV.2 Proof of Theorem 2

Proof: We start the proof with pure state situation. Let $\rho=|\psi\rangle\langle\psi|$ be a pure quantum state in ${\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}\otimes{\rm H}_{4}^{d}$ with Bloch representation (6) in the main tex. By setting $A=\sum_{i=1}^{4}||T^{(i)}||^{2},B=\sum_{1\leq i<j\leq 4}||T^{(ij)}||^{2},C=\sum_{1\leq i<j<k\leq 4}||T^{ijk}||^{2}$, and $D=||T^{(1234)}||^{2}$, one gets

[TABLE]

One can further computes for any $1\leq i\leq 4,1\leq s<t\leq 4$ and $1\leq x<y<z\leq 4$ that

[TABLE]

Since we are considering the pure state $\rho=|\psi\rangle\langle\psi|$, we have

[TABLE]

holds for any $ijkl\in\{1234,2134,3124,4123\}$. Then by summing the equations in (S7), we obtain

[TABLE]

Substituting (S8) into (S6), we get

[TABLE]

which is just $D\leq\frac{16(d^{2}-1)^{2}}{d^{4}}$.

Then we consider a mixed state $\rho$ with ensemble representation $\rho=\sum_{\alpha}p_{\alpha}|\psi_{\alpha}\rangle\langle\psi_{\alpha}|$, where $\sum_{\alpha}p_{\alpha}=1$. By the convexity of the Frobenius norm, one derives $D\leq\sum_{\alpha}p_{\alpha}D_{\alpha}\leq\frac{16(d^{2}-1)^{2}}{d^{4}}$, which ends the proof.

IV.3 Proof of Theorem 3

Proof: Let $|\psi\rangle\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}\otimes{\rm H}_{4}^{d}$ be pure state. Without lose of generality, one sets

[TABLE]

We have

[TABLE]

Thus

[TABLE]

Then for any mixed state $\rho\in{\rm H}_{1}^{d}\otimes{\rm H}_{2}^{d}\otimes{\rm H}_{3}^{d}\otimes{\rm H}_{4}^{d}$, one has

[TABLE]

which ends the proof.