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Uncertainty relations with quantum memory for the Wehrl entropy

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Abstract

We prove two new fundamental uncertainty relations with quantum memory for the Wehrl entropy. The first relation applies to the bipartite memory scenario. It determines the minimum conditional Wehrl entropy among all the quantum states with a given conditional von Neumann entropy and proves that this minimum is asymptotically achieved by a suitable sequence of quantum Gaussian states. The second relation applies to the tripartite memory scenario. It determines the minimum of the sum of the Wehrl entropy of a quantum state conditioned on the first memory quantum system with the Wehrl entropy of the same state conditioned on the second memory quantum system and proves that also this minimum is asymptotically achieved by a suitable sequence of quantum Gaussian states. The Wehrl entropy of a quantum state is the Shannon differential entropy of the outcome of a heterodyne measurement performed on the state. The heterodyne measurement is one of the main measurements in quantum optics and lies at the basis of one of the most promising protocols for quantum key distribution. These fundamental entropic uncertainty relations will be a valuable tool in quantum information and will, for example, find application in security proofs of quantum key distribution protocols in the asymptotic regime and in entanglement witnessing in quantum optics.

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Correspondence to Giacomo De Palma.

Additional information

I acknowledge financial support from the European Research Council (ERC Grant Agreements Nos. 337603 and 321029), the Danish Council for Independent Research (Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059), and the Marie Skłodowska-Curie Action GENIUS (Grant No. 792557).

Appendix A

Appendix A

Lemma 1

(Jensen’s trace inequality) Let us consider the operator

$$\begin{aligned} \hat{A} = \sum _{k=0}^\infty a_k\,|\phi _k\rangle \langle \phi _k|\;,\qquad 0\le a_k\le 1\qquad \forall \;k\in \mathbb {N}\;,\qquad \sum _{k=0}^\infty a_k<\infty \;, \end{aligned}$$
(66)

where the vectors \(|\phi _k\rangle \) form a resolution of the identity:

$$\begin{aligned} \sum _{k=0}^\infty |\phi _k\rangle \langle \phi _k| = \hat{\mathbb {I}}\;. \end{aligned}$$
(67)

Then, for any concave function \(f:[0,1]\rightarrow [0,\infty )\),

$$\begin{aligned} \mathrm {Tr}\,f\left( \hat{A}\right) \ge \sum _{k=0}^\infty f(a_k)\,\langle \phi _k|\phi _k\rangle \;. \end{aligned}$$
(68)

Proof

From (67) we get

$$\begin{aligned} \langle \phi _k|\phi _k\rangle \le 1\qquad \forall \;k\in \mathbb {N}\;. \end{aligned}$$
(69)

We then have

$$\begin{aligned} \mathrm {Tr}\,\hat{A} = \sum _{k=0}^\infty a_k\,\langle \phi _k|\phi _k\rangle \le \sum _{k=0}^\infty a_k < \infty \;. \end{aligned}$$
(70)

\(\hat{A}\) has then discrete spectrum, and we can diagonalize it:

$$\begin{aligned} \hat{A} = \sum _{l=0}^\infty \lambda _l\,|v_l\rangle \langle v_l|\;,\qquad \langle v_k|v_l\rangle = \delta _{kl}\quad \forall \;k,\,l\in \mathbb {N}\;,\qquad \sum _{l=0}^\infty |v_l\rangle \langle v_l| = \hat{\mathbb {I}}\;. \end{aligned}$$
(71)

From the completeness of the \(|\phi _k\rangle \), for any \(l\in \mathbb {N}\)

$$\begin{aligned} \sum _{k=0}^\infty |\langle v_l|\phi _k\rangle |^2 = \langle v_l|v_l\rangle =1\;; \end{aligned}$$
(72)

hence, \(|\langle v_l|\phi _k\rangle |^2\) is a probability distribution on \(\mathbb {N}\). We then have from Jensen’s inequality

$$\begin{aligned} \mathrm {Tr}\,f\left( \hat{A}\right)&= \sum _{l=0}^\infty f(\lambda _l) = \sum _{l=0}^\infty f\left( \sum _{k=0}^\infty |\langle v_l|\phi _k\rangle |^2 a_k\right) \ge \sum _{k,\,l=0}^\infty |\langle v_l|\phi _k\rangle |^2\,f(a_k)\nonumber \\&= \sum _{k=0}^\infty \langle \phi _k|\phi _k\rangle \,f(a_k)\;, \end{aligned}$$
(73)

where we have used that for the completeness of the \(|v_l\rangle \), for any \(k\in \mathbb {N}\)

$$\begin{aligned} \sum _{l=0}^\infty |\langle v_l|\phi _k\rangle |^2 = \langle \phi _k|\phi _k\rangle \;. \end{aligned}$$
(74)

\(\square \)

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De Palma, G. Uncertainty relations with quantum memory for the Wehrl entropy. Lett Math Phys 108, 2139–2152 (2018). https://doi.org/10.1007/s11005-018-1067-y

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