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abstract: 'In a recent paper (Phys. Rev. Lett. 109, 160501 (2012). arXiv:1201.0849), it is claimed that any quantum protocol for classical two-sided computation between Alice and Bob can be proven completely insecure for Alice if it is secure against Bob. Here we show that the proof is not sufficiently general, because the security definition it based on is only a sufficient condition but not a necessary condition.'
author:
- Guang Ping He
title: 'Comment on Complete insecurity of quantum protocols for classical two-party computation'
---

Let us first look at the security definition in [@qbc61]. As stated in the paragraph below its FIG. 1, let $\varepsilon \geq 0$ and write $\rho \simeq _{\varepsilon }\sigma $ (i.e., $\rho $ is $\varepsilon $-close to $\sigma $) if the purified distance $\sqrt{1-(tr\sqrt{\sqrt{\rho }%
\sigma \sqrt{\rho }})^{2}}$ between the density matrices $\rho $ and $%
\sigma $ is not greater than $\varepsilon $. Then a two-party quantum protocol corresponding to a completely positive trace-preserving (CPTP) map $\pi $ is defined as $\varepsilon
$-secure against dishonest Bob if for any real adversary $B^{\prime }$ there exists an ideal adversary $\hat{B}%
^{\prime }$ such that $[id_{R}\otimes \pi _{A,B^{\prime }}](\rho
_{UVR})\simeq _{\varepsilon }[id_{R}\otimes \mathcal{F}_{\hat{A},\hat{B}%
^{\prime }}](\rho _{UVR})$. Here $A$ denotes the real honest Alice, $%
B^{\prime }$ the dishonest Bob, and $\hat{A}$, $\hat{B}^{\prime }$ the ideal versions. Both parties obtain an input (Alice’s $u$ in register $U$ and Bob’s $v$ in register $V$) drawn from the distribution $p(u,v)$. $%
[id_{R}\otimes \pi _{A,B^{\prime }}](\rho _{UVR})$ is the output state of the protocol augmented by the reference $R$, where $\rho _{UVR}$ is a purification of $\sum\nolimits_{u,v}p(u,v)\left\vert u\right\rangle
\left\langle u\right\vert _{U}\left\vert v\right\rangle \left\langle
v\right\vert _{V}$. And $\mathcal{F}$ is an ideal functionality which measures the inputs and outputs orthogonal states that correspond to the function values of the classical two-sided computation. Please see [qbc61]{} for more detailed explanations of the notations.

In simple words, as can be seen from Sec. 1.6 of [@qi1087] (i.e., Ref. \[12\] of [@qbc61]), the meaning of this definition can be understood as follows. Let $\alpha $ and $\beta $ be the physical systems accessible to Alice and Bob, respectively. Denote the density matrices of $\alpha $, $%
\beta $ as $\rho _{\alpha }$, $\rho _{\beta }$ when Bob plays honestly, or as $\rho _{\alpha }^{\prime }$, $\rho _{\beta }^{\prime }$ when he applies a certain cheating strategy. If there is $\rho _{\alpha }^{\prime }\simeq
_{\varepsilon }\rho _{\alpha }$, the cheating strategy will be nearly undetectable to Alice so that Bob can pass the security checks in the protocol successfully, while if there is $\rho _{\beta }^{\prime
}\simeq _{\varepsilon }\rho _{\beta }$, a dishonest Bob can hardly gain any extra information other than what is accessible to an honest Bob. Then the above security definition means that a protocol is secure against Bob if for any cheating strategy, there is always $\rho _{\beta }^{\prime }\simeq
_{\varepsilon }\rho _{\beta }$. For simplicity, we call such a cheating strategy as a type I strategy.

Obviously, if *any* cheating strategy currently known or potentially exists in the world belongs to type I, then the corresponding protocol is surely secure. Thus it is a sufficient condition for guaranteeing the security of a protocol. But it is important to question whether the reversed statement is also true. That is, if a protocol is secure, does it necessarily guarantee that *all* cheating strategies have to be type I strategies? In fact, if there is a cheating strategy which does not satisfy $\rho _{\alpha }^{\prime }\simeq _{\varepsilon }\rho _{\alpha }$, then it will be detectable to Alice, so that the protocol can remain secure against Bob no matter $\rho _{\beta }^{\prime }\simeq
_{\varepsilon }\rho _{\beta }$ is satisfied or not. We call strategies satisfying neither $\rho _{\alpha }^{\prime }\simeq
_{\varepsilon }\rho _{\alpha }$ nor $\rho _{\beta }^{\prime }\simeq
_{\varepsilon }\rho _{\beta }$ as type II strategies. Actually, they are no strangers to quantum cryptography. In many existing protocols, there are security checks in which the parties agree to continue with the protocols only when some conditions are met. Otherwise they can choose to abort in the middle of the process, and the protocols output failinstead of the output obtained by honest players. This implies that the protocols are designed against type II strategies. Thus it is clear that the existence of type II strategies does not necessarily hurt the security of protocols. If a protocol is secure, then both types I and II strategies are possible. That is, all cheating strategies belong to type I is not the necessary condition for a protocol to be secure. Therefore, while the security definition in [@qbc61] is a true statement, it cannot be used as a two-party quantum protocol is $\varepsilon $-secure against Bob *if and only if* for any real adversary $B^{\prime }$ there exists an ideal adversary $\hat{B}^{\prime }$ such that $[id_{R}\otimes \pi
_{A,B^{\prime }}](\rho _{UVR})\simeq _{\varepsilon }[id_{R}\otimes \mathcal{F%
}_{\hat{A},\hat{B}^{\prime }}](\rho _{UVR})$, since the reversed statement for any real adversary $B^{\prime }$, there exists an ideal adversary $\hat{B}^{\prime }$ such that $[id_{R}\otimes \pi
_{A,B^{\prime }}](\rho _{UVR})\simeq _{\varepsilon }[id_{R}\otimes \mathcal{F%
}_{\hat{A},\hat{B}^{\prime }}](\rho _{UVR})$ if the protocol is $\varepsilon $-secure against Bob is not true. There can be type II strategies which are not $\varepsilon $-close to any ideal adversary.

Now back to the no-go proof for two-sided computation in [@qbc61]. In brief, the key starting points of the proof are as follows. Suppose that there is a quantum protocol for classical two-sided computation which is already assumed to be secure against a dishonest Bob. To prove that it must be insecure against Alice, in the paragraph before Eq. (1) of [@qbc61], the following cheating strategy of Bob is considered. He plays the honest but purified strategy and outputs the purification of the protocol (register $Y_{1}^{\prime }$) and the output values $f(u,v)$ (register $Y$). We call it strategy $B_{0}^{\prime }$ hereafter. Since the protocol is $\varepsilon $-secure against Bob, in the opinion of [@qbc61] there exists a secure state $\sigma _{RX\tilde{V}Y^{\prime }}$ satisfying $\sigma
_{RXY^{\prime }}\simeq _{\varepsilon }\rho _{RXY^{\prime }}$, where $Y^{\prime }=Y_{1}^{\prime }Y$. Applying Uhlmann’s theorem on $\sigma _{RXY^{\prime }}\simeq _{\varepsilon }\rho _{RXY^{\prime
}}$, Eq. (1) of [@qbc61] can be obtained, which further leads to the rest part of the no-go proof.

However, according to our above discussion on the security definition, the protocol is $\varepsilon $-secure against Bob does not necessarily guarantees that all cheating strategies (including strategy $B_{0}^{\prime }$) must be type I strategies, because the latter statement is not the necessary condition of the former. If $B_{0}^{\prime }$ belongs to type II, then the protocol can still be secure against Bob, while the equation $\sigma
_{RXY^{\prime }}\simeq _{\varepsilon }\rho _{RXY^{\prime }}$ no longer holds. Consequently, Eq. (1) does not necessarily remain valid so that the no-go proof will lose its base. Thus we can see that the proof in [@qbc61] may apply to a protocol for which $B_{0}^{\prime }$ can be proven to be a type I strategy (given that all other features of the protocols studied in [@qbc61] are also met). But it is not sufficient general to cover all protocols, since there is no evidence (at least not provided in [@qbc61]) showing that $B_{0}^{\prime }$ always has to be a type I strategy for any protocol potentially exists. By designing proper security checks which can make $B_{0}^{\prime }$ appear as a type II strategy, it is possible to build protocols not covered by the proof in [@qbc61]. Therefore, the door for finding secure quantum protocols for classical two-party computation is not closed completely.

The work was supported in part by the NSF of China under grant No. 10975198, the NSF of Guangdong province, and the Foundation of Zhongshan University Advanced Research Center.

[9]{} H. Buhrman, M. Christandl, and C. Schaffner, *Phys. Rev. Lett.* **109**, 160501 (2012). *arXiv:1201.0849*. Complete insecurity of quantum protocols for classical two-party computation

D. Unruh, *quant-ph/0409125*. Simulatable security for quantum protocols
