Quantum coherence in mutually unbiased bases

TL;DR

This paper explores the $l_{1}$ norm of quantum coherence in mutually unbiased bases, deriving new results for specific quantum states and introducing the concept of autotensor of mutually unbiased bases to analyze coherence.

Contribution

It introduces the autotensor of mutually unbiased bases (AMUB) and provides analytical results for the $l_{1}$ norm of coherence in various quantum states within this framework.

Findings

Sum of squared $l_{1}$ coherence for single qubit is less than two.

The $l_{1}$ norm of coherence for three classes of $X$ states in 4D is equal.

Coherence levels of Bell-diagonal, Werner, and isotropic states are characterized in AMUB.

Abstract

We investigate the $l_{1}$ norm of coherence of quantum states in mutually unbiased bases. We find that the sum of squared $l_{1}$ norm of coherence of the mixed state single qubit is less than two. We derive the $l_{1}$ norm of coherence of three classes of $X$ states in nontrivial mutually unbiased bases for $4$-dimensional Hilbert space is equal. We proposed "autotensor of mutually unbiased basis(AMUB)" by the tensor of mutually unbiased bases, and depict the level surface of constant the sum of the $l_{1}$ norm of coherence of Bell-diagonal states in AMUB. We find the $l_{1}$ norm of coherence of Werner states and isotropic states in AMUB is equal respectively.

Equations

$$C_{l_1}(\rho )=\sum _{i\ne j}|{\rho }_{i,j}|,$$

$$C_r(\rho )=S({\rho }_{diag})-S(\rho ),$$

$$|⟨i_k|j_l⟩{|}^2=\frac{1}{d},\forall i,j\in \{0,\cdot \cdot \cdot ,d-1\},$$

$${\alpha }_1$$

$${\alpha }_2$$

$${\alpha }_3$$

$${\beta }_1$$

$${\beta }_2$$

$${\beta }_3$$

$${\beta }_4$$

$${\rho }_s=\frac{I+\overrightarrow{r}\cdot \overrightarrow{\sigma }}{2}$$

$${\rho }_s$$

$$=$$

$$C_{l_1}({\rho }_s{)}_{{\alpha }_1}=|\frac{1}{2}(x-iy)|+|\frac{1}{2}(x+iy)|=\sqrt{x^2+y^2}.$$

$${\rho }_s$$

$$\{\begin{array}{l}\frac{1}{2}(a_{11}+a_{12}+a_{21}+a_{22})=\frac{1}{2}(1+z)\\ \frac{1}{2}(a_{11}-a_{12}+a_{21}-a_{22})=\frac{1}{2}(x-iy)\\ \frac{1}{2}(a_{11}+a_{12}-a_{21}-a_{22})=\frac{1}{2}(x+iy)\\ \frac{1}{2}(a_{11}-a_{12}-a_{21}+a_{22})=\frac{1}{2}(1-z)\end{array}.$$

$$\{\begin{array}{l}{a}_{11}=\frac{1+x}{2}\\ a_{12}=\frac{z+iy}{2}\\ a_{21}=\frac{z-iy}{2}\\ a_{22}=\frac{1-x}{2}\end{array}.$$

$$C_{l_1}({\rho }_s{)}_{{\alpha }_2}=|\frac{1}{2}(z+iy)|+|\frac{1}{2}(z-iy)|=\sqrt{z^2+y^2}.$$

$${\rho }_s$$

$$C_{l_1}({\rho }_s{)}_{{\alpha }_3}=|\frac{1}{2}(z-ix)|+|\frac{1}{2}(z+ix)|=\sqrt{z^2+x^2}.$$

$$\displaystyle\rho_{X}=\left(\begin{array}[]{ccc}x&0&z\\ 0&1-x-y&0\\ z&o&y\\ \end{array}\right),$$

$$\displaystyle\rho_{X}=\left(\begin{array}[]{ccc}b_{11}&b_{12}&b_{13}\\ b_{21}&b_{22}&b_{23}\\ b_{31}&b_{32}&b_{33}\\ \end{array}\right),$$

$$\{\begin{array}{l}\frac{1}{3}(b_{11}+b_{12}+b_{13}+b_{21}+b_{22}+b_{23}+b_{31}+b_{32}+b_{33})=x\\ \frac{1}{3}(b_{11}+{\omega }^2{b}_{12}+\omega b_{13}+b_{21}+{\omega }^2{b}_{22}+\omega b_{23}+b_{31}+{\omega }^2{b}_{32}+\omega b_{33})=0\\ \frac{1}{3}(b_{11}+\omega b_{12}+{\omega }^2{b}_{13}+b_{21}+\omega b_{22}+{\omega }^2{b}_{23}+b_{31}+\omega b_{32}+{\omega }^2{b}_{33})=z\\ \frac{1}{3}(b_{11}+b_{12}+b_{13}+\omega b_{21}+\omega b_{22}+\omega b_{23}+{\omega }^2{b}_{31}+{\omega }^2{b}_{32}+{\omega }^2{b}_{33})=0\\ \frac{1}{3}(b_{11}+{\omega }^2{b}_{12}+\omega b_{13}+\omega b_{21}+b_{22}+{\omega }^2{b}_{23}+{\omega }^2{b}_{31}+\omega b_{32}+b_{33})=1-x-y\\ \frac{1}{3}(b_{11}+\omega b_{12}+{\omega }^2{b}_{13}+\omega b_{21}+{\omega }^2{b}_{22}+b_{23}+{\omega }^2{b}_{31}+b_{32}+\omega b_{33})=0\\ \frac{1}{3}(b_{11}+b_{12}+b_{13}+{\omega }^2{b}_{21}+{\omega }^2{b}_{22}+{\omega }^2{b}_{23}+\omega b_{31}+\omega b_{32}+\omega b_{33})=z\\ \frac{1}{3}(b_{11}+{\omega }^2{b}_{12}+\omega b_{13}+{\omega }^2{b}_{21}+\omega b_{22}+b_{23}+\omega b_{31}+b_{32}+{\omega }^2{b}_{33})=0\\ \frac{1}{3}(b_{11}+\omega b_{12}+{\omega }^2{b}_{13}+{\omega }^2{b}_{21}+b_{22}+\omega b_{23}+\omega b_{31}+{\omega }^2{b}_{32}+b_{33})=y\end{array}.$$

$$\{\begin{array}{l}{b}_{11}=\frac{1+2z}{3},b_{12}=\frac{(3x+z-1)-\sqrt{3}(x+2y+z-1)i}{6},b_{13}=\frac{(3x+z-1)+\sqrt{3}(x+2y+z-1)i}{6},\\ b_{21}=\overline{b_{12}},b_{22}=\frac{1-z}{3},b_{23}=\frac{(3x-2z-1)-\sqrt{3}(x+2y-2z-1)i}{6},\\ b_{31}=\overline{b_{13}},b_{32}=\overline{b_{23}},b_{33}=\frac{1-3z}{3}.\end{array}$$

$$C_{l_1}({\rho }_X{)}_{{\beta }_2}=2(|b_{12}|+|b_{13}|+|b_{23}|).$$

$$\displaystyle\rho_{X}=\left(\begin{array}[]{ccc}b_{22}&\overline{b_{12}}&b_{23}\\ b_{12}&b_{11}&b_{13}\\ \overline{b_{23}}&\overline{b_{13}}&b_{33}\\ \end{array}\right),$$

$$C_{l_1}({\rho }_X{)}_{{\beta }_3}=2(|b_{12}|+|b_{13}|+|b_{23}|).$$

$$\displaystyle\rho_{X}=\left(\begin{array}[]{ccc}b_{22}&b_{12}&\overline{b_{23}}\\ \overline{b_{12}}&b_{11}&\overline{b_{13}}\\ b_{23}&b_{13}&b_{33}\\ \end{array}\right),$$

$$C_{l_1}({\rho }_X{)}_{{\beta }_4}=2(|b_{12}|+|b_{13}|+|b_{23}|).$$

$$C_{l_1}({\rho }_X{)}_{{\beta }_2}=C_{l_1}({\rho }_X{)}_{{\beta }_3}=C_{l_1}({\rho }_X{)}_{{\beta }_4}.$$

$$\displaystyle\rho_{\vartriangle}=\left(\begin{array}[]{ccc}1-x-y&0&0\\ 0&x&z\\ 0&z&y\\ \end{array}\right),$$

$$\displaystyle\rho_{\triangledown}=\left(\begin{array}[]{ccc}x&z&0\\ z&y&0\\ 0&0&1-x-y\\ \end{array}\right),$$

$$|(⟨i{|}_k\otimes ⟨j{|}_k)(|m{⟩}_l\otimes |n{⟩}_l)|=\frac{1}{d},$$

Full text

I Introduction

Quantum coherence is a special feature of quantum mechanic like entanglement and other quantum correlations. Quantum coherence is an essential factor in quantum information processing Jha ; Bagan ; Kammerlander , quantum optics Glauber ; Sudarshan ; Mandel , quantum metrology Giovannetti ; Demkowicz ; Giovannetti1 , low-temperature thermodynamics Lostaglio ; Lostaglio1 ; Vazquez ; Wacker ; pati ; aberg ; Narasimhachar ; Oppenheim and quantum biology Lloyd ; Li ; Huelga ; levi ; Plenio ; Rebentrost . Recently, a structure to quantify coherence has been proposed Baumgratz , and various quantum coherence measures, such as the $l_{1}$ norm of coherence Baumgratz , trace norm of coherence shao , relative entropy of coherence Baumgratz , Tsallis relative $\alpha$ entropies Rastegin and Relative Rényi $\alpha$ monotones Chitambar , have been defined. With the help of of the coherence measures, a variety of properties of quantum coherence, such as the relations between quantum correlations and quantum coherence Ma ; Streltsov ; Radhakrishnan ; Yao ; Xi , the freezing phenomenon of coherence Bromley ; Yu , have been studied.

Mutually unbiased bases are used in detection of quantum entanglement Spengler , quantum state reconstruction Wootters , quantum error correction Gottesman ; Calderbank , and the mean king s problem Vaidman ; Englert . Many features of mutually unbiased bases are reviewed in reference Durt . When d is power of a prime number, maximal sets of $d+1$ mutually unbiased bases have been built for the case. Maximal sets of MUBs are an open problem Durt , when the dimensionality is another composite number. Entropic uncertainty relations for $d+1$ mutually unbiased bases in d-dimensional Hilbert space were obtained in references Ivanovic ; sanchez . The fine-grained uncertainty relation for mutually unbiased bases is derived in ren . The relation between mutually unbiased bases and unextendible maximally entangled is investigated in chen .

In this article, we investigate the $l_{1}$ norm of coherence of quantum states in mutually unbiased bases. We evaluate analytically the sum of squared $l_{1}$ norm of coherence of the mixed state single qubit. We derive the relation of the $l_{1}$ norm of coherence of three classes of $X$ states in nontrivial mutually unbiased bases for $4$-dimensional Hilbert space. We propose “autotensor of mutually unbiased basis(AMUB)” by the tensor of mutually unbiased bases, and depict the level surface lang of constant the sum of the $l_{1}$ norm of coherence of Bell-diagonal states in AMUB. We obtain the relations of the $l_{1}$ norm of coherence of Werner states and isotropic states in AMUB respectively.

II The $l_{1}$ norm of coherence of quantum states in $2$ dimension mutually unbiased bases

Under fixed reference basis, the $l_{1}$ norm of coherence of state $\rho$ is defined by

[TABLE]

and the relative entropy of coherence is given by

[TABLE]

where $S(\rho)=-Tr{\rho\log\rho}$ is von Neumann entropy.

A set of orthonormal bases $\{B_{k}\}$ for a Hilbert space $H=C^{d}$ where $\{B_{k}\}=\{|0_{k}\rangle,\cdot\cdot\cdot,|d-1_{k}\rangle$ is called mutually unbiased (MU) iff

[TABLE]

holds for all basis vectors $|i_{k}\rangle$ and $|j_{l}\rangle$ that belong to different bases, i.e. $\forall k\neq l$.

In dimension $d=2$, a set of three mutually unbiased bases is readily obtained from the eigenvectors of the three Pauli matrices $\sigma_{z}$, $\sigma_{x}$ and $\sigma_{y}$:

[TABLE]

In dimension $d=3$, there are four mutually unbiased bases as fowllow:

[TABLE]

where $\omega=e^{i\frac{2\pi}{3}}$.

An arbitrary density matrix for a mixed state single qubit may be written as

[TABLE]

where $\overrightarrow{r}=(x,y,z)$ is a real three-dimensional vector such that $x^{2}+y^{2}+z^{2}\leq 1$, and $\overrightarrow{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. In particular, $\rho$ is pure if and only if $x^{2}+y^{2}+z^{2}=1$.

Next, we will consider the relation of the $l_{1}$ norm of coherence among $\rho_{s}$ in three mutually unbiased bases $\alpha_{1},\alpha_{2},\alpha_{3}$.

The density matrix of mixed state single qubit $\rho_{s}$ in base $\alpha_{1}=\{\alpha_{11},\alpha_{12}\}=\{|0\rangle,|1\rangle\}$ is

[TABLE]

Using Eq. (1) directly, the $l_{1}$ norm of coherence of state $\rho_{s}$ in base $\alpha_{1}$ is

[TABLE]

The density matrix of $\rho_{s}$ in base $\alpha_{2}=\{\alpha_{21},\alpha_{22}\}$ is

[TABLE]

As $\rho_{s}$ in Eq. (6) and Eq. (12) is the same, using the method of undeterminated coefficients, we obtain

[TABLE]

The solution of the equation is

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{s}$ in base $\alpha_{2}$ is

[TABLE]

The density matrix of $\rho_{s}$ in base $\alpha_{3}=\{\alpha_{31},\alpha_{32}\}$ by the above method is

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{s}$ in base $\alpha_{3}$ is

[TABLE]

As $x^{2}+y^{2}+z^{2}\leq 1$, $[C_{l_{1}}(\rho_{s})_{\alpha_{1}}]^{2}+[C_{l_{1}}(\rho_{s})_{\alpha_{2}}]^{2}+[C_{l_{1}}(\rho_{s})_{\alpha_{3}}]^{2}\leq 2.$

III The $l_{1}$ norm of coherence of $X$ states in the tensor of $3$ dimension mutually unbiased bases

For the three classes of $X$ states in base $\beta_{1}=\{\beta_{11},\beta_{12},\beta_{13}\}=\{|0\rangle,|1\rangle,|2\rangle\}$

[TABLE]

where $x,y,z$ are all real number, we will consider the $l_{1}$ norm of coherence of $\rho_{X}$ in the $3$ dimension mutually unbiased bases $\beta_{2},\beta_{3},\beta_{4}$.

Let the density matrix of $\rho_{X}$ in base $\beta_{2}=\{\beta_{21},\beta_{22},\beta_{23}\}$ be

[TABLE]

and $\rho_{X}=b_{11}\beta_{21}\beta_{21}^{{\dagger}}+b_{12}\beta_{21}\beta_{22}^{{\dagger}}+b_{13}\beta_{21}\beta_{23}^{{\dagger}}+b_{21}\beta_{22}\beta_{21}^{{\dagger}}+b_{22}\beta_{22}\beta_{22}^{{\dagger}}+b_{23}\beta_{22}\beta_{23}^{{\dagger}}+b_{31}\beta_{23}\beta_{21}^{{\dagger}}+b_{32}\beta_{23}\beta_{22}^{{\dagger}}+b_{33}\beta_{23}\beta_{23}^{{\dagger}}$. As $\rho_{X}$ in Eq. (22) and Eq. (26) is the same, using the method of undeterminated coefficients, we obtain

[TABLE]

The solution of the equation is

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{X}$ in base $\beta_{2}$ is

[TABLE]

Similarly, the density matrix of $\rho_{X}$ in base $\beta_{3}$ is

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{X}$ in base $\beta_{3}$ is

[TABLE]

The density matrix of $\rho_{X}$ in base $\beta_{4}$ is

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{X}$ in base $\beta_{4}$ is

[TABLE]

At last, we find that the $l_{1}$ norm of coherence of state $\rho_{X}$ in base $\beta_{2},\beta_{3},\beta_{4}$ is equal, i.e

[TABLE]

Furthermore, let

[TABLE]

and

[TABLE]

where $x,y,z$ are all real number, using above method, we can find that the $l_{1}$ norm of coherence of state $\rho_{\vartriangle}$ and $\rho_{\triangledown}$ in base $\beta_{2},\beta_{3},\beta_{4}$ is also equal respectively.

IV The $l_{1}$ norm of coherence of Bell-diagonal states in the tensor of $2$ dimension mutually unbiased bases

In this section, we extend the concept of mutually unbiased basis by the tensor.

Definition. For the set of mutually unbiased bases $\{B_{k}\}$ for a Hilbert space $H=C^{d}$ where $\{B_{k}\}=\{|0_{k}\rangle,\cdot\cdot\cdot,|d-1_{k}\rangle\}$, we call the set $\{\gamma_{k}\}=\{|i\rangle_{k}\otimes|i\rangle_{k}|\forall i,j\in\{0,\cdot\cdot\cdot,d-1\}\}$ autotensor of mutually unbiased basis(AMUB) if

[TABLE]

where $k\neq l$. Furthermore, we can construct a set of AMUB by $d=2$ dimension mutually unbiased bases. For example, let

[TABLE]

Next, we will consider the relation of the coherence of quantum states in above AMUB.

A two-qubit Bell-diagonal states can be written as

[TABLE]

where $\{\sigma_{i}\}_{i=1}^{3}$ are the Pauli matrices, and $c_{1},c_{2},c_{3}\in[-1,1]$. The density matrix of $\rho_{B}$ in base $\gamma_{1}=\{\gamma_{11},\gamma_{12},\gamma_{13},\gamma_{14}\}=\{|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle\}$ is:

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{B}$ in base $\gamma_{1}$ is

[TABLE]

Let the density matrix of $\rho_{B}$ in base $\gamma_{2}=\{\gamma_{21},\gamma_{22},\gamma_{23},\gamma_{24}\}$ is

[TABLE]

and $\rho_{B}=d_{11}\gamma_{21}\gamma_{21}^{{\dagger}}+d_{12}\gamma_{21}\gamma_{22}^{{\dagger}}+d_{13}\gamma_{21}\gamma_{23}^{{\dagger}}+d_{14}\gamma_{21}\gamma_{24}^{{\dagger}}+d_{21}\gamma_{22}\gamma_{21}^{{\dagger}}+d_{22}\gamma_{22}\gamma_{22}^{{\dagger}}+d_{23}\gamma_{22}\gamma_{23}^{{\dagger}}+d_{24}\gamma_{22}\gamma_{24}^{{\dagger}}+d_{31}\gamma_{23}\gamma_{21}^{{\dagger}}+d_{32}\gamma_{23}\gamma_{22}^{{\dagger}}+d_{33}\gamma_{23}\gamma_{23}^{{\dagger}}+d_{34}\gamma_{23}\gamma_{24}^{{\dagger}}+d_{41}\gamma_{24}\gamma_{21}^{{\dagger}}+d_{42}\gamma_{24}\gamma_{22}^{{\dagger}}+d_{43}\gamma_{24}\gamma_{23}^{{\dagger}}+d_{44}\gamma_{24}\gamma_{24}^{{\dagger}}.$ As $\rho_{B}$ in Eq. (55) and Eq. (61) is the same, using the method of undeterminated coefficients, we obtain

[TABLE]

The solution of the equation is

[TABLE]

So, the density matrix of $\rho_{B}$ in base $\gamma_{2}=\{\gamma_{21},\gamma_{22},\gamma_{23},\gamma_{24}\}$ is

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{B}$ in base $\gamma_{2}$ is

[TABLE]

Similarly, the density matrix of $\rho_{B}$ in base $\gamma_{3}=\{\gamma_{31},\gamma_{32},\gamma_{33},\gamma_{34}\}$ is

[TABLE]

The $l_{1}$ norm of coherence of state $\rho_{B}$ in base $\gamma_{3}$ is

[TABLE]

In Fig. 1, the $l_{1}$ norm of coherence of Bell-diagonal states $\rho_{B}$ in base $\gamma_{1}$ as a function of $c_{1}$ and $c_{2}$ is depicted. When $c_{1}=c_{2}=0$, the coherence reach minimal value [math]. As $|c_{1}|$ and $|c_{2}|$ increase, the coherence increase. When $|c_{1}|=1$ or $|c_{2}|=1$, the coherence obtain maximum value. Similar situation also appear in the coherence in bases $\gamma_{2}$ and $\gamma_{3}$.

Next, we denote the sum of the $l_{1}$ norm of coherence of Bell-diagonal states $\rho_{B}$ in bases $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ by $C_{l_{1}}(\rho_{B})_{\gamma}$, i. e

[TABLE]

In Fig. 2, we plot the surfaces lang of the sum of the $l_{1}$ norm of coherence $C_{l_{1}}(\rho_{B})_{\gamma}$ of Bell-diagonal states $\rho_{B}$ in bases $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ in (a), (b), and (c). It show that the surface of the sum of the coherence is tetrahexahedron. As the sum increase, its volume expand, i. e. $|c_{1}|$, $|c_{2}|$, $|c_{3}|$ increase simultaneously.

In Eq. (55), let $c_{1}=c_{2}=c_{3}=\frac{4p}{3}-1$, where $0\leq p\leq 1$ ,Bell-diagonal states $\rho_{B}$ turn into Werner state

[TABLE]

We denoted the $l_{1}$ norm of coherence of Werner states $\rho_{W}$ in bases $\gamma_{1},\gamma_{2},\gamma_{3}$ by $C_{l_{1}}(\rho_{W})_{\gamma_{1}},C_{l_{1}}(\rho_{W})_{\gamma_{2}},C_{l_{1}}(\rho_{W})_{\gamma_{3}}$ respectively. By Eqs. (56), (69), (75), we find that $C_{l_{1}}(\rho_{W})_{\gamma_{1}}=C_{l_{1}}(\rho_{W})_{\gamma_{2}}=C_{l_{1}}(\rho_{W})_{\gamma_{3}}=|\frac{4p}{3}-1|$.

In Eq. (55), let $c_{1}=\frac{4F-1}{3},c_{2}=-\frac{4F-1}{3},c_{3}=\frac{4F-1}{3}$, where $0\leq F\leq 1$ ,Bell-diagonal states $\rho_{B}$ turn into isotropic state

[TABLE]

We denoted the $l_{1}$ norm of coherence of isotropic states $\rho_{iso}$ in bases $\gamma_{1},\gamma_{2},\gamma_{3}$ by $C_{l_{1}}(\rho_{iso})_{\gamma_{1}},C_{l_{1}}(\rho_{iso})_{\gamma_{2}},C_{l_{1}}(\rho_{iso})_{\gamma_{3}}$ respectively. By Eqs. (56), (69), (75), we find that $C_{l_{1}}(\rho_{iso})_{\gamma_{1}}=C_{l_{1}}(\rho_{iso})_{\gamma_{2}}=C_{l_{1}}(\rho_{iso})_{\gamma_{3}}=|\frac{4F-1}{3}|$.

V summary

In this work, we studied the $l_{1}$ norm of coherence of quantum states in mutually unbiased bases. We have found the sum of squared $l_{1}$ norm of coherence of the mixed state single qubit is less than two. We have obtained the $l_{1}$ norm of coherence of three classes of $X$ states in nontrivial mutually unbiased bases for $4$-dimensional Hilbert space is equal. We have proposed “autotensor of mutually unbiased basis(AMUB)” by the tensor of mutually unbiased bases, and given the level surfacelang of constant the sum of the $l_{1}$ norm of coherence of Bell-diagonal states in AMUB. We have found the $l_{1}$ norm of coherence of Werner states and isotropic states in AMUB is equal respectively.