Sharp Continuity Bounds for Entropy and Conditional Entropy

TL;DR

This paper derives tight continuity bounds for the Renyi entropy in quantum information theory, generalizing the Fannes inequality for von Neumann entropy and relating entropy differences to trace norm distances.

Contribution

It introduces a sharp inequality connecting Renyi entropy differences to trace norm distance, which is tight and encompasses the Fannes inequality as a special case.

Findings

Derived a tight inequality for Renyi entropy differences

Included the Fannes inequality as a special case

Established the inequality's attainability for all trace norm distances

Abstract

The Renyi entropy plays an essential role in quantum information theory. We study the continuity estimation of the Renyi entropy. An inequality relating the Renyi entropy difference of two quantum states to their trace norm distance is derived. This inequality is shown to be tight in the sense that equality can be attained for every prescribed value of the trace norm distance. It includes the sharp Fannes inequality for von Neumann entropy as a special case.

Equations

$$S(\rho ):=-\mathrm{Tr}[\rho {\log }_2\rho ].$$

$$H(p):=\sum _i^{d}H(p_i)=-\sum _i^d{p}_i{\log }_2{p}_i,$$

$$|S(\rho )-S(\sigma )|⩽2T{\log }_2(d)-2T{\log }_2(2T),$$

$$|S(\rho )-S(\sigma )|⩽T{\log }_2(d-1)-H((T,1-T)).$$

$$H_{\alpha }(\rho ):=\frac{1}{1-\alpha }\log [\mathrm{Tr}({\rho }^{\alpha })],\alpha >0,$$

$$|H_{\alpha }(\rho )-H_{\alpha }(\sigma )|⩽\frac{d^{\alpha -1}}{1-\alpha }[1-(1-T{)}^{\alpha }-(d-1{)}^{1-\alpha }{T}^{\alpha }],\alpha >1,$$

$$|H_{\alpha }(\rho )-H_{\alpha }(\sigma )|⩽\frac{1}{1-\alpha }[1-(1-T{)}^{\alpha }-(d-1{)}^{1-\alpha }{T}^{\alpha }],\alpha <1,$$

$$|S_2({\rho }_{AB}|{\rho }_B)-S_2({\sigma }_{AB}|{\sigma }_B)|⩽4ϵ+2h_2(ϵ,1-ϵ).$$

$$|D_2({\rho }_{AB})-D_2({\sigma }_{AB})|⩽8ϵ+4h_2(ϵ,1-ϵ).$$

$$|H_{\alpha }(\rho )-H_{\alpha }(\sigma )|⩽d^{\alpha -1}|S_{\alpha }(\rho )-S_{\alpha }(\sigma )|\alpha >1,$$

$$|H_{\alpha }(\rho )-H_{\alpha }(\sigma )|⩽|S_{\alpha }(\rho )-S_{\alpha }(\sigma )|\alpha <1.$$

$$|S_{\alpha }(\rho )-S_{\alpha }(\sigma )|⩽1-(1-T{)}^{\alpha }-(d-1{)}^{1-\alpha }{T}^{\alpha }.$$

$$|S_{\alpha }(p)-S_{\alpha }(q)|⩽1-(1-T{)}^{\alpha }-(d-1{)}^{1-\alpha }{T}^{\alpha },$$

$$S_{\alpha }(q)-S_{\alpha }(p)=S_{\alpha }(p_1+T)-S_{\alpha }(p_1)$$

$$+S_{\alpha }((1-p_1)r-Ts)-S_{\alpha }((1-p_1)r).$$

$$S_{\alpha }((1-p_1)r-Ts)-S_{\alpha }((1-p_1)r)$$

$$=S_{\alpha }((1-p_1-T)\eta )-S_{\alpha }((1-p_1-T)\eta +Ts).$$

$$S_{\alpha }((1-p_1-T)\eta )-S_{\alpha }((1-p_1-T)\eta +Ts)$$

$$=S_{\alpha }(1-p_1-T)-S_{\alpha }(1-p_1-T(1-s_1))$$

$$-S_{\alpha }(T(1-s_1)\varphi )$$

$$\triangleq \Delta .$$

$$\Delta =S_{\alpha }(1-p_1-T)-S_{\alpha }(1-p_1-T(1-s_1))$$

$$-(d-2{)}^{1-\alpha }{S}_{\alpha }(T(1-s_1)).$$

$$\frac{\partial \Delta }{\partial s_1}=-T\alpha [(1-p_1-T(1-s_1){)}^{\alpha -1}-(d-2{)}^{1-\alpha }(T(1-s_1){)}^{1-\alpha }]=0,$$

$$T(1-s_1)=\frac{(1-p_1)(d-2)}{(d-1)}\equiv \omega .$$

$$-S_{\alpha }(p_1)+S_{\alpha }(p_1+T)+(1-p_1-T{)}^{\alpha }-\frac{(1-p_1{)}^{\alpha }}{(d-1{)}^{\alpha -1}}.$$

$$1-(1-T{)}^{\alpha }⩽(1-\frac{T}{d-2}{)}^{\alpha }-(1-\frac{(d-1)T}{d-2}{)}^{\alpha }.$$

$$S_{\alpha }(p)-S_{\alpha }(q)=S_{\alpha }(p_1)-S_{\alpha }(p_1+T)$$

$$+S_{\alpha }((1-p_1)r)-S_{\alpha }((1-p_1)r-Ts),$$

$$S_{\alpha }((1-p_1)r)-S_{\alpha }((1-p_1)r-Ts)$$

$$=S_{\alpha }((1-p_1-T)\eta +Ts)-S_{\alpha }((1-p_1-T)\eta ).$$

$$S_{\alpha }((1-p_1-T)\eta +Ts)-S_{\alpha }((1-p_1-T)\eta )$$

$$=S_{\alpha }(1-p_1-T(1-s_1))+S_{\alpha }(T(1-s_1)\varphi )$$

$$-S_{\alpha }(1-p_1-T)\triangleq \Delta .$$

$$(1-T{)}^{\alpha }+(d-1{)}^{1-\alpha }{T}^{\alpha }-1$$

Full text

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111Corresponding author (email: †mazhihaoquantum@126.com, *feishm@cnu.edu.cn)

Sharp Continuity Bounds for Entropy and Conditional Entropy

Zhihua Chen

  

Zhihao Ma

  

Ismail Nikoufar

  

Shao-Ming Fei

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310014, China;

School of Mathematical Sciences, Shanghai Jiaotong University, Shanghai, 200240, China;

Department of Mathematics, Payame Noor University, 19395-3697 Tehran, Iran;

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

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

Dear Editor,

The von Neumann entropy captures many operational quantities in the quantum information theory such as quantum capacity of the communication channel. Von Neumann entropy is continuous and is represented by Fannes inequality, which was originally given in[1]. Quantum correlations such as entanglement and quantum discord , et al., are important resources in quantum information processing. In the last year enormous progress on the generation, concentration, detection and quantification of entanglement has been achieved [2]. Fannes inequality has many applications in the quantum information theory, such as the investigation of continuity of entanglement measures, including entanglement of formation, relative entropy of entanglement, squashed entanglement and conditional entanglement of mutual information, and the continuity of quantum channel capacities [3]. Recently Fannes inequality was improved to get a sharp one and it was also generalized to Tsallis entropy [4, 5].

However, in non-asymptotic settings, the natural quantities that arise are R$\acute{e}$nyi entropies and the properties of R$\acute{e}$nyi entropies were also investigated in many papers, such as [7]. R$\acute{e}$nyi entropies have many applications, as in the case of one-shot problems, typically arising in cryptographic settings, the min- and max-entropies are widely used . In [8], the authors found that R$\acute{e}$nyi-2 entropy was a proper measure of information for any multimode Gaussian state, and they defined and analyzed the measures of Gaussian entanglement and quantum correlation by using R$\acute{e}$nyi-2 entropy, and found its properties such as monogamy. In our work, we study the continuity property of R$\acute{e}$nyi-$\alpha$ entropy, which includes R$\acute{e}$nyi-2 entropy as a special case. Our result is also useful in studying the continuity of the entanglement measure of the Gaussian state in quantum harmonic systems.

On the other hand, the authors found that Tsalli-2 entropy (i.e., linear entropy) was natural to define the measure of quantum correlation for the the discrete system [9]. They called this measure as linear discord and used conditional linear entropy to define the linear discord. They found that the linear discord has deep connection with the original discord defined by von Neumann entropy. Moreover, they gave the analytical formula for arbitrary $2\otimes d$ state of the linear discord. However, a question still remains open: if two states are close, is their linear discord also close to each other? In other words, is the linear discord continuous? For the original discord, the answer is affirmative, see [10]. For the linear discord, there is no answer yet. Hence it is worthwhile to study the continuity of conditional linear entropy.

We have two aims in this work: first, we study the continuity estimation of the R$\acute{e}$nyi entropy and present a tight inequality relating the R$\acute{e}$nyi entropy difference of two quantum states to their trace norm distance, which includes the sharp Fannes inequality for von Neumann entropy as a special case. Second, we study the continuity of conditional linear entropy, and prove a useful property for a measure of the quantum correlation: linear discord.

The von Neumann entropy of a quantum state $\rho$ is defined by

[TABLE]

For the classical probability distributions, the von Neumann entropy reduces to the Shannon entropy,

[TABLE]

where $p:=(p_{i})=(p_{1},p_{2},...,p_{d})$ is a $d$-dimensional probability vector, $p_{i}\geqslant 0$, $\sum_{i}^{d}p_{i}=1$ and $H(p_{i}):=-p_{i}\log_{2}p_{i}$.

In [1] Fannes proved his famous inequality for the continuity of the von Neumann entropy,

[TABLE]

where $T:=\frac{||\rho-\sigma||_{1}}{2}$ is half of the trace norm distance between the states $\rho$ and $\sigma$, $||\rho-\sigma||_{1}={\rm Tr}[|\rho-\sigma|]$, and $|X|:=\sqrt{(X)^{\dagger}(X)}$ denotes the absolute value of an operator $X$. Obviously $T\in[0,1]$. The inequality (3) is valid for $0\leqslant T\leqslant 1/2e$, where $e$ is Euler’s number. The inequality (3) is further improved to be a sharp one by Audenaert [4]:

[TABLE]

The R$\acute{e}$nyi entropy is a more general form of the von Neumann entropy,

[TABLE]

when $\alpha$ goes to one, R$\acute{e}$nyi entropy becomes the von Neumann entropy. In the following we show that for the R$\acute{e}$nyi entropy, an improved sharp Fannes-type inequality exists.

Theorem 1. For all $d$-dimensional quantum states $\rho$ and $\sigma$,

[TABLE]

[TABLE]

where $T$ is the trace norm distance of $\rho$ and $\sigma$. See proof in Appendix.

We investigated the continuity estimation of the R$\acute{e}$nyi entropy, by presenting an inequality which relates the R$\acute{e}$nyi entropy difference of two quantum states to their trace norm distance. In our inequality, equality can be attained for every prescribed value of the trace norm distance. It is direct to verify that for $\alpha\to 1$, our inequality (6) and (7) give rise to the sharp Fannes inequality for von Neumann entropy. It has potential applications in investigating the continuity of entanglement measure and more general correlations for multimode Gaussian states, since R$\acute{e}$nyi-2 entropy is a proper information measure for this kind of state [8].

Besides R$\acute{e}$nyi entropy, linear entropy is also used to measure quantum correlations, such as linear discord [9], which is defined as the minimal difference of the two conditional linear entropy, before and after the local projective measurement, $D_{2}(\rho_{AB}):=\min\limits_{P_{i}}(S_{2}(A|B)-S_{2}(P_{i}|B))$, where $S_{2}(A|B)$ is the conditional linear entropy of the original state $\rho_{AB}$, while $S_{2}(P_{i}|B)$ is the conditional linear entropy of the post measurement state after local measurement $P_{i}$, and the minimum runs over all local projection measurements $P_{i}$. Therefore it is also important to study the continuity of the linear entropy, especially conditional linear entropy. It can help us get the continuity of the linear discord.

We can prove the following conclusion: conditional linear entropy is continuous, see proof in Appendix.

Theorem 2. For bipartite quantum states $\rho_{AB}$ and $\sigma_{AB}$, if $\epsilon:=||\rho_{AB}-\sigma_{AB}||_{1}<1$, then the following inequality holds,

[TABLE]

By using the method of [10], it is straightforward to show that the linear discord $D_{2}(\rho_{AB})$ is also continuous:

Theorem 3. For bipartite quantum states $\rho_{AB}$ and $\sigma_{AB}$, if $\epsilon:=||\rho_{AB}-\sigma_{AB}||_{1}<1$, then

[TABLE]

In summary, we have investigated the continuity of R$\acute{e}$nyi entropy and conditional linear entropy. Through the continuity of conditional linear entropy, the continuity of linear discord has also been obtained, which means that the linear discord varies as the quantum state changes continuously. This fact can guarantee that the errors in state tomograph would not significantly affect the result of the quantum correlations in the state. As the sharp Fannes inequality is the special case of our theorem about the continuity of R$\acute{e}$nyi entropy, our results can also be used to verify the continuity of entanglement measures for continuous variable quantum states.

Acknowledgments    This work is supported by the NSFC under number 11371247, 11275131, 11675113 and 11571313.