1. INTRODUCTION

In quantum information theory, the minimum output von Neumann entropy of a bosonic quantum channel has attracted considerable attention because of its direct implication in determining the channel capacity [1–4]. The exact results are known only for Gaussian channels [5–7], however, and the quantity remains largely unexplored for non-Gaussian channels. This is unsatisfactory because there are important examples of non-Gaussian channels including phase noise channels [8–10], bosonic dephasing channels [11–13], and Gaussian dilatable channels [14–16]. A Gaussian dilatable channel is a natural extension of a Gaussian channel described by a Gaussian unitary interaction with an environment, and it becomes a non-Gaussian channel only if the environment is in a non-Gaussian state. It is often highly challenging to determine the minimum output von Neumann entropy of a bosonic quantum channel, and one needs a reliable method, such as the entropy power inequalities (EPIs), to estimate the minimum output entropy.

In classical information theory, the entropy power inequality was first introduced in Shannon’s seminal paper [17] and was rigorously proven later [18–21]. The EPI provides useful bounds on the capacities of classical channels [17] including Gaussian broadcast channels [22] and Gaussian wiretrap channels [23]. First defined in terms of the Shannon entropy, the classical EPI has been extended to a generalized measure of information, the Rényi entropy [24–26]. Furthermore, inspired by the classical EPIs, quantum EPIs for bosonic Gaussian dilatable channels have been derived using the von Neumann entropy [27,28], providing reliable lower bounds on the output entropy. In addition, the qudit analogs of the quantum EPIs have been established for partial swap channels [29]. Despite the progress, a quantum version of the Rényi entropy power inequality has still not been found.

The quantum Rényi entropy is a natural generalization of the von Neumann entropy and is defined as ${S_\alpha}(\rho) = \ln [{\rm tr}({\rho ^\alpha})]/(1 - \alpha)$ [30]. One particularly interesting instance is the quantum Rényi-2 entropy, ${S_2}(\rho) = - \ln ({\mu _\rho})$, where ${\mu _\rho} = {\rm tr}({\rho ^2})$ is the purity of the state. The latter is directly related to a measure of mixedness called the quantum linear entropy $L(\rho) = 1 - {\mu _\rho}$. The quantum Rényi-2 entropy provides a lower bound to the von Neumann entropy and is much easier to obtain, both theoretically and experimentally. To calculate the von Neumann entropy requires the full density matrix, which in turn requires resource-intensive quantum state tomography [31]. Furthermore, it can be challenging to calculate the von Neumann entropy of a high-dimensional quantum state even when we have full access to its density matrix [32]. By contrast, the purity can be obtained without performing the quantum state tomography and subsequently solving eigenvalue problems [33–37]. For this reason, the purity-based measures have been used in various fields of quantum physics, including area laws in quantum many-body systems [38], quantum thermalization [39], and non-Hermitian phase transitions [40]. In addition, purity-based measures have proved to be a valuable tool in investigating quantum resources. For instance, they play a critical role in Gaussian quantum information [41,42], detecting quantum correlations, i.e., quantum entanglement [43,44], and quantum steering [45,46], and estimating the properties of quantum states, i.e., concurrence [47], non-Gaussianity [48], and quantum mutual information [49].

Here we ask whether we can find a quantum Rényi-2 entropy version of the EPIs derived in [28]. We first prove that the naive substitution of the quantum Rényi-2 entropy into the EPI of [28] works for Gaussian states. We then observe that the inequality is violated for a simple class of non-Gaussian states. To cope with the deficiency, we derive more complete quantum Rényi-2 EPIs that take different forms to the original EPIs: while either of the latter places a lower bound on the output entropy in the form of a weighted sum of two entropy powers, our new bound takes the maximum of the two weighted entropy powers. Our results provide a helpful basis for estimating the output Rényi-2 entropy of a Gaussian dilatable channel and hence place a lower bound to the von Neumann entropy. Lastly, we explore how one of our inequalities finds its use in entanglement generation and detection.

2. QUANTUM RÉNYI-2 ENTROPY POWER INEQUALITY FOR GAUSSIAN STATES

Consider the action of a bosonic channel acting on an input state ${\rho _{\rm A}}$ to produce an output state ${{\tilde \rho} _{\rm A}}$ as depicted in Fig. 1. The channel can be either the beam-splitting or the squeezing type and require an auxiliary mode in state ${\rho _{\rm B}}$. In such scenarios, it was found that the EPIs for bosonic quantum systems can be written as [28]

Naively, one may ask whether the following Rényi-2 EPIs hold:

A. Gaussian States

The purity of a single-mode Gaussian state $\sigma$ is determined by [50]

$\hat Q = \{\hat q,\hat p{\} ^T}$ is the set of quadrature operators defined as $\hat q = ({{{\hat a}^\dagger} + \hat a})/\sqrt 2$ and $\hat p = i({{{\hat a}^\dagger} - \hat a})/\sqrt 2$ and ${\langle {\hat O} \rangle _\sigma} = {\rm tr}({\sigma \hat O})$ is the expectation value of the operator $\hat O$. We note that the quantum Rényi-$\alpha$ entropy of a single-mode Gaussian state is fully determined by a function of $\det \,{\Gamma _\sigma}$ [51], i.e., ${S_\alpha}(\sigma) = {{\ln [{{{({\sqrt {\det {\Gamma _\sigma}} + 1/2})}^\alpha} - {{({\sqrt {\det {\Gamma _\sigma}} - 1/2})}^\alpha}}]} / {(\alpha - 1)}}$.

In the Heisenberg picture, the beam-splitting operation is described by $\hat U_\lambda ^\dagger \hat a{\hat U_\lambda} = \sqrt \lambda \hat a + \sqrt {1 - \lambda} \hat b$ [52], which yields $\hat U_\lambda ^\dagger {\hat q_{\rm A}}{\hat U_\lambda} = \sqrt \lambda {\hat q_{\rm A}} + \sqrt {1 - \lambda} {\hat q_{\rm B}}$ and $\hat U_\lambda ^\dagger {\hat p_{\rm A}}{\hat U_\lambda} = \def\LDeqbreak{}\sqrt \lambda {\hat p_{\rm A}} + \sqrt {1 - \lambda} {\hat p_{\rm B}}$. From these, it is straightforward to show that the action of a beam-splitter on ${\rho _{\rm A}} \otimes {\rho _{\rm B}}$ yields the following expression for the covariance matrix of the output state ${\tilde \rho _{\rm A}}$ [27]:

 

Fig. 1. (a) Quantum channel yields an output state ${\tilde \rho _{\rm A}}$ from an input state ${\rho _{\rm A}}$. The action of a quantum channel is modeled by a partial trace in conjunction with a unitary interaction $\hat U$ between the input and ancillary modes, i.e., ${\tilde \rho _{\rm A}} = {{\rm tr}_{\rm B}}[\hat U({\rho _{\rm A}} \otimes {\rho _{\rm B}}){\hat U^\dagger}]$. (b) A Gaussian dilatable channel using a beam-splitting interaction ${\hat U_\lambda} = \exp [\theta ({\hat a^\dagger}\hat b - \hat a{\hat b^\dagger})]$ with a transmittance $\lambda = \mathop {\cos}\nolimits^2 \theta$. (c) A Gaussian dilatable channel using a two-mode squeezing interaction ${\hat U_\kappa} = \exp [\nu ({\hat a^\dagger}{\hat b^\dagger} - \hat a\hat b)]$ with a gain factor $\kappa = \mathop {\cosh}\nolimits^2 \nu$.

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Invoking the Minkowski determinant inequality [53], we obtain

For the two-mode squeezing operation, we similarly obtain

Given that the naive substitution of the quantum Rényi-$\alpha$ entropy into the EPIs for bosonic quantum systems works for Gaussian states when $\alpha = 2$, one may ask whether our EPIs for Gaussian states could be extended to all $\alpha$. One can, however, easily find counterexamples, e.g., thermal states, for $\alpha \gt 2$.

B. Noisy Single-Photon States

An obvious question is whether Eqs. (3) and (4) are valid for arbitrary ${\rho _{\rm A}}$ and ${\rho _{\rm B}}$. We here show that the simple case of noisy single-photon states in the form of ${\rho _f} = (1 - f)|0\rangle\def\LDeqbreak{} \langle 0| + f|1\rangle \langle 1|$ violates the inequalities.

In Fig. 2, we compare the left- and right-hand sides of Eqs. (3) and (4) when noisy single-photon states ${\rho _{\rm A}} = {\rho _{\rm B}} = {\rho _f}$ undergo the corresponding Gaussian interactions. We observe that the output Rényi-2 entropy power, i.e., $\exp [{{S_2}({{\tilde \rho}_{\rm A}})}]$, becomes smaller than the lower bound provided by Eqs. (3) and (4) in the shaded regions. The inequalities are thus invalid for non-Gaussian states in general.

 

Fig. 2. Shaded regions indicate that noisy single-photon states $(1 - f)|0\rangle \langle 0| + f|1\rangle \langle 1|$ can violate the quantum Rényi-2 EPIs for (a) beam-splitting and (b) squeezing interactions in Eqs. (3) and (4), respectively.

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The violation of Eq. (3) under the constraint of ${\rho _{\rm A}} = {\rho _{\rm B}}$ signifies that the output purity ${\mu _{{{\tilde \rho}_{\rm A}}}}$ is greater than the input purity ${\mu _{{\rho _{\rm A}}}}$. Therefore, our results show that a two-copy deterministic purification, using a beam-splitting operation, is possible for some non-Gaussian states. Such a phenomenon occurs because the beam-splitting interaction acts as a Gaussification protocol [54] and there are some non-Gaussian states that are less pure than their reference Gaussian states [48]. Here, the reference Gaussian state of a quantum state $\rho$ refers to a Gaussian state having the same first- and second-order quadrature moments as $\rho$. We remark that comparing measures between a quantum state and its reference Gaussian state, e.g., in proving or disproving Gaussian extremality, has been a topic of considerable interest [55–59] in the field of continuous variable quantum information.

3. GENERALIZED QUANTUM RÉNYI-2 ENTROPY POWER INEQUALITY

The above counter-example begs for generalizations of Eqs. (3) and (4). In this section, we prove that a generalization is possible, and it takes a simple form: instead of the weighted summation of the input entropy powers, one takes the maximum of the two weighted entropy powers. Our generalization of the quantum Rényi-2 EPI for a beam-splitting interaction can thus be written as

To prove this, we note that the action of a beam-splitter with transmittance $\lambda$ is described by [14]

To derive Eq. (14), we use the fact that the absolute value of the characteristic function of a quantum state $\tau$ is bounded by unity, i.e., $|{C_\tau}(\lambda)| \le 1$ [60]. Therefore, upon using Hölder’s inequality [61] and introducing ${\xi ^\prime} = \sqrt \lambda \xi$ and ${\xi ^{{\prime \prime}}} = \sqrt {1 - \lambda} \xi$, we have

The last line is a monotonic function with respect to $\chi \in [0,1]$. Therefore, we find its minimum at $\chi = 0$ or $\chi = 1$, which yields Eq. (14).

The proof for the squeezing interaction follows analogously and yields

The main difference to the beam-splitting case is that the action of a two-mode squeezing operation is described by [14]

It is interesting to note that one of our results, i.e., ${\mu _{{{\tilde \rho}_{\rm A}}}} \le {\mu _{{\rho _{\rm A}}}}/\kappa$, shows that the output purity ${\mu _{{{\tilde \rho}_{\rm A}}}}$ is always smaller than the input purity ${\mu _{{\rho _{\rm A}}}}$ for all $\kappa \gt 1$, indicating that a two-copy deterministic purification using a two-mode squeezing interaction is impossible.

A. Input-Independent Inequality for the Squeezing Interaction

Another type of inequality which is independent of the input quantum Rényi-2 entropies can be derived for the squeezing interaction as follows:

The second line follows from the change of variables ${\xi ^\prime} = \sqrt {2\kappa - 1} \xi$ and ${\eta ^\prime} = \kappa /(2\kappa - 1)$, the third line follows from the definition ${\bar \rho _{\rm A}} = {{\rm tr}_{\rm B}}[{{{\hat U}_{{\eta ^\prime}}}({\rho _{\rm A}} \otimes {\cal T}[{\rho _{\rm B}}])\hat U_{{\eta ^\prime}}^\dagger}]$, where ${\hat U_{{\eta ^\prime}}}$ is the beam-splitter unitary with a transmittance ${\eta ^\prime}$, and the final line follows trivially because ${\mu _\tau} \le 1$. Rewriting in terms of the quantum Rényi-2 entropy gives

Notably, the inequality is saturated when both ${\rho _{\rm A}}$ and ${\rho _{\rm B}}$ are squeezed coherent states $\hat S({\zeta _i})\hat D({\alpha _i})|0\rangle$ having the same squeezing strength and symmetric squeezing angles, i.e., ${\zeta _1} = \zeta _2^*$, where $\hat S(\zeta) = \exp \{{[{- {\zeta ^2}{{({{{\hat a}^\dagger}})}^2} + {{({{\zeta ^*}})}^2}{{\hat a}^2}}]/2} \}$ denotes a single mode squeezing operation with a squeezing strength $r = |\zeta |$ and a squeezing angle $\theta = {\rm arg} \zeta$. This is a direct consequence of the saturation of inequality (12) for pure Gaussian states.

4. APPLICATIONS

The quantum von Neumann EPIs provide useful lower bounds to the minimum output entropy, which in turn lead to upper bounds on the channel capacities of a selection of bosonic quantum channels. Due to the ordering relation between the von Neumann entropy and the quantum Rényi-2 entropy, viz, $S(\rho) \ge {S_2}(\rho)$, our Rényi-2 EPIs place lower bounds to the minimum output (von Neumann) entropy. Although the corresponding von Neumann EPIs provide tighter bounds, the Rényi-2 versions have the advantage of being easier to determine both experimentally and theoretically. The Rényi-2 EPIs derived in the previous section, therefore, provide practical lower bounds to the minimum output entropy, which in turn can be used to determine upper bound to the channel capacities of some Gaussian dilatable channels.

In this section, we propose two further applications of the new EPIs, i.e., Eqs. (18) and (22) for the squeezing interaction. We first derive an inequality that places a lower bound on the amount of squeezing required to generate entanglement starting from a product state input ${\rho _{\rm A}} \otimes {\rho _{\rm B}}$. Then, we show that Eq. (22) can be used as an entanglement detection condition.

A. Entanglement Generation from a Product State

With the squeezing interaction, a sufficient condition to generate quantum entanglement from a product state ${\rho _{{\rm AB}}} = {\rho _{\rm A}} \otimes {\rho _{\rm B}}$ is

Its proof is as follows. We first introduce ${\Delta _j}({\rho _{{\rm AB}}}) \equiv {S_2}({\rho _j}) - {S_2}({\rho _{{\rm AB}}})$ with $j \in \{{\rm A},{\rm B}\}$, which reveals the entanglement of the quantum state ${\rho _{{\rm AB}}}$ when at least one of ${\Delta _{\rm A}}({\rho _{{\rm AB}}})$ and ${\Delta _{\rm B}}({\rho _{{\rm AB}}})$ is positive. This follows from the fact that all separable states satisfy ${S_2}({\sigma _j}) \le {S_2}({\sigma _{{\rm AB}}})$ for $j \in \{{\rm A},{\rm B}\}$ [43]. For the case of ${\tilde \rho _{{\rm AB}}} = {\hat U_\kappa}({\rho _{\rm A}} \otimes {\rho _{\rm B}})\hat U_\kappa ^\dagger$, we obtain

If the right-hand side is positive, ${\Delta _{\rm A}}({\tilde \rho _{{\rm AB}}})$ is guaranteed to be positive. A sufficient condition for entanglement generation is therefore

An identical consideration for ${\Delta _{\rm B}}({\tilde \rho _{{\rm AB}}}) = {S_2}({\tilde \rho _{\rm B}}) - {S_2}({\tilde \rho _{{\rm AB}}})$ yields

Finally, a quick comparison of Eqs. (26) and (27) gives Eq. (23).

We note that one may construct another sufficient condition for the entanglement generation as

An interesting consequence of Eq. (23) is that quantum entanglement can be generated from a product state ${\rho _{\rm A}} \otimes {\rho _{\rm B}}$, even when one of the input modes has an extremely low purity. As long as the gain factor $\kappa$ is greater than the Rényi-2 entropy power for an input mode, e.g., $\kappa \gt \exp [{S_2}({\rho _{\rm A}})]$, the output state ${\tilde \rho _{{\rm AB}}} = \hat U_\kappa ^\dagger ({\rho _{\rm A}} \otimes {\rho _{\rm B}})\hat U_\kappa ^\dagger$ is guaranteed to be entangled regardless of the purity of the other mode.

B. Entanglement Detection via Purity Measurement

Equation (22) or its equivalent form

Using ${\tilde \rho _{{\rm A},k}} = {{\rm tr}_{\rm B}}[{{{\hat U}_\kappa}({\rho _{{\rm A},k}} \otimes {\rho _{{\rm B},k}})\hat U_\kappa ^\dagger}]$ in conjunction with Eqs. (22) and (31), we readily see that

The violation of the inequality therefore signifies that ${\rho _{{\rm AB}}}$ is entangled.

The quantum EPI provides an analogous entanglement condition, $S({\tilde \rho _{\rm A}}) \lt \ln (2\kappa - 1)$, in terms of the von Neumann entropy. We would like to stress, however, that ${S_2}({\tilde \rho _{\rm A}}) \lt\def\LDeqbreak{} \ln (2\kappa - 1)$ detects more entanglement than its von Neumann counterpart. This is because if the latter is satisfied, ${S_2}({\tilde \rho _{\rm A}}) \lt\def\LDeqbreak{} \ln (2\kappa - 1)$ is necessarily satisfied due to the ordering $S(\tau) \ge {S_2}(\tau)$. Therefore, whenever ${S_2}({\tilde \rho _{\rm A}}) \lt \ln (2\kappa - 1) \lt S({\tilde \rho _{\rm A}})$, Eq. (30) detects entanglement that its von Neumann counterpart cannot do.

In Fig. 3, we investigate the case of the photon number entangled states [62] in the form of $|{\Psi _f}\rangle = \sqrt {1 - f} |0{\rangle _{\rm A}}|0{\rangle _{\rm B}} + \sqrt f |2{\rangle _{\rm A}}|2{\rangle _{\rm B}}$. While $|{\Psi _f}\rangle$ is entangled for $0 \lt f \lt 1$, its quantum entanglement cannot be witnessed by a Gaussian entanglement condition [63] because the covariance matrix of $|{\Psi _f}\rangle$ is the same as that of a two-mode thermal state ${\sigma _{2f}} \otimes {\sigma _{2f}}$, where ${\sigma _{\bar n}} = \sum\nolimits_{k = 0}^\infty [{{{\bar n}^k}/(1 + \bar n{)^{k + 1}}}]|k\rangle \langle k|$ denotes a single-mode thermal state with a mean photon number $\bar n$. Figure 3(a) shows that Eq. (30) enables one to detect entanglement missed by the Gaussian conditions. There is a threshold value of the gain factor, ${\kappa _{{\min}}}$, above which entanglement is detected. This value increases with the fraction $f$, as shown in Fig. 3(b). Furthermore, we find that ${\kappa _{{\min}}}$ approaches infinity near $f \simeq 0.743$, showing that our approach can detect the quantum entanglement of $|{\Psi _f}\rangle$ for $0 \lt f \lt 0.743$.

 

Fig. 3. (a) Entropic quantities ${S_2}({\tilde \rho _{\rm A}})$ (blue solid curve) and $\ln (2\kappa - 1)$ (black dashed curve) for ${\rho _{{\rm AB}}} = |{\Psi _{1/2}}\rangle \langle {\Psi _{1/2}}|$ are plotted against the gain factor $\kappa$. Quantum entanglement is witnessed in the gray shaded region in which ${S_2}({\tilde \rho _{\rm A}}) \lt \ln (2\kappa - 1)$. The amount of information is quantified by “nats” instead of “bits.” (b) The minimum required gain factor ${\kappa _{{\min}}}$ to reveal the quantum entanglement of $|{\Psi _f}\rangle$ using our method is plotted against the fraction $f$.

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5. CONCLUSION

We have derived quantum Rényi-2 EPIs for bosonic quantum systems. We first showed that a naive approach works for Gaussian states: The quantum Rényi-2 EPIs for Gaussian states have the same forms as the quantum EPIs, i.e., $\exp [{S_2}({\tilde \rho _{\rm A}})] \ge \lambda \exp [{S_2}({\rho _{\rm A}})] + (1 - \lambda)\exp [{S_2}({\rho _{\rm B}})]$ and $\exp [{S_2}({\tilde \rho _{\rm A}})] \ge \kappa \exp [{S_2}({\rho _{\rm A}})] + (\kappa - 1)\exp [{S_2}({\rho _{\rm B}})]$ for a beam-splitting interaction and a two-mode squeezing interaction, respectively. We then showed that these inequalities are violated by noisy single-photon states, suggesting that the naive expectation, i.e., $\exp [{S_\alpha}({\tilde \rho _{\rm A}})] \ge \lambda \exp [{S_\alpha}({\rho _{\rm A}})] + (1 - \lambda)\exp [{S_\alpha}({\rho _{\rm B}})]$ or $\exp [{S_\alpha}({\tilde \rho _{\rm A}})] \ge \kappa \exp [{S_\alpha}({\rho _{\rm A}})] + (\kappa - 1)\exp [{S_\alpha}({\rho _{\rm B}})]$, fails in general for $\alpha \ne 1$.

To cope with the deficiency, we have derived simple and general quantum Rényi-2 EPIs for beam-splitting and two-mode squeezing interactions. The new EPIs have $\max \{\lambda \exp [{S_2}({\rho _{\rm A}})],(1 - \lambda)\exp [{S_2}({\rho _{\rm B}})] \}$ or $\max \{\kappa \exp [{S_2}({\rho _{\rm A}})],(\kappa - 1)\exp [{S_2}({\rho _{\rm B}})] \}$ on the right-hand side, instead of the weighted summations of the input entropy powers. As potential applications of the new EPIs, we have addressed how the inequality for the squeezing interaction yields sufficient conditions for entanglement generation and detection. These topics have received much less attention than sufficient conditions for entanglement generation and detection using the beam-splitting interaction [64–74].

In the future, it will be interesting to find more elaborate quantum Rényi-2 EPIs that directly enhance the performance in estimating the output quantum entropies and help establish tighter bounds for entanglement generation and detection using two-mode squeezing interaction. While our inequalities are useful for estimating the output quantum Rényi-2 entropies, the gap between the estimated and the actual output quantum entropies can be large for non-Gaussian states in general. Another worthwhile direction for investigation is to examine how the non-Gaussianity measures [48,56–58,75–78] can be involved in estimating the output quantum entropies beyond our inequalities.