Tight Sample Complexity Bounds for Parameter Estimation
Under Quantum Differential Privacy for Qubits

Farhad Farokhi F. Farokhi is with the Department of Electrical and Electronic Engineering at the University of Melbourne.

Abstract

This short note provides tight upper and lower bounds for minimal number of samples (copies of quantum states) required to attain a prescribed accuracy (measured by error variance) for scalar parameters using unbiased estimators under quantum local differential privacy for qubits. In the small privacy budget $\epsilon$ regime, i.e., $\epsilon\ll 1$ , the sample complexity scales as $\Theta(\epsilon^{-2})$ . This bound matches that of classical parameter estimation under local differential privacy. The lower bound loosens (converges to zero) in the large privacy budget regime, i.e., $\epsilon\gg 1$ , but that case is not particularly interesting as tight bounds for parameter estimation in the noiseless case are widely known. That being said, extensions to systems with higher dimensions and tightening the bounds for the large privacy budget regime are interesting avenues for future research.

I Introduction

Differential privacy [1, 2, 3] has taken over the computer science literature as the gold standard definition for private data analysis. Recently these classical definitions have been extended to the quantum domain [4, 5, 6]. Further extensions in the forms of pufferfish privacy [7] and information-theoretic privacy [8] have been also presented.

The definition and analysis of quantum differential privacy has fueled a line of research on understanding fundamental limits of quantum data processing under privacy. Hypothesis testing under quantum differential privacy was studied in [9, 10, 11]. Limits of quantum machine learning differential privacy have been also studied in [12]. This brief note focuses on deterministic (non-Bayesian) parameter estimation under quantum differential privacy. We use quantum Cramér-Rao bound [13, 14, 15] to establish bounds on the number of quantum state copies or samples required to attain a prescribed estimation error variance. We particularly use the Bloch sphere representation for qubit representation and explicit Fisher information formulas in this regime [16].

The rest of this note is organized as follows. We first review some definitions and present some preliminary results in Section II. The main results are then presented in Section III.

II Preliminary Material

II-A Density Operators

The following definitions and preliminary results are adopted from [17].

The set of linear operators from (finite-dimensional) Hilbert space $\mathcal{H}$ to $\mathcal{H}$ is denoted by $\mathcal{L}(\mathcal{H})$ . The set of positive semi-definite linear operators is denoted by $\mathcal{P}(\mathcal{H})\subset\mathcal{L}(\mathcal{H})$ . The set of density operators (i.e., positive semi-definite linear operators with unit trace) is denoted by $\mathcal{S}(\mathcal{H})\subset\mathcal{P}(\mathcal{H})$ . Qubits, which stand for quantum bits, are the basic units of quantum information correspond to 2-dimensional Hilbert spaces. In the so-called Bloch sphere representation [18, p. 105], the density operator $\rho$ for any qubit can be represented as

\displaystyle\rho=\frac{1}{2}\left(I+\omega.\hat{\sigma}\right),

(1)

where $\omega=(\omega_{x},\omega_{y},\omega_{z})\in\mathbb{R}^{3}$ is such that $\|\omega\|_{2}\leq 1$ (with $\|\omega\|_{2}^{2}=\omega\top\omega$ ) and $\hat{\sigma}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$ is the tuple of Pauli matrices

\displaystyle\hat{\sigma}_{x}:=\begin{bmatrix}0&1\\ 1&0\end{bmatrix},\quad\hat{\sigma}_{y}:=\begin{bmatrix}0&-i\\ i&0\end{bmatrix},\quad\hat{\sigma}_{z}:=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}.

Here, these matrices are represented in the so-called computational basis. Note that, in the Bloch sphere representation, the definition of the inner product is expanded to allow for

\displaystyle\omega.\hat{\sigma}:=\omega_{x}\hat{\sigma}_{x}+\omega_{y}\hat{\sigma}_{y}+\omega_{z}\hat{\sigma}_{z}.

A quantum channel, in its most general form, is a mapping on the space of density operators that is both completely positive and trace preserving. In the case of qubits, for each quantum channel $\mathcal{E}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H})$ , there exist $A\in\mathbb{R}^{3\times 3}$ and $c\in\mathbb{R}^{3}$ such that

\displaystyle\mathcal{E}(\rho)=\frac{1}{2}\left(I+(A\omega+c).\hat{\sigma}\right).

(2)

Note that it must be that $\|A\omega+c\|_{2}\leq 1$ for all $\omega$ such that $\|\omega\|_{2}\leq 1$ . This is to ensure that the output $\mathcal{E}(\rho)$ is still a density operator. A necessary condition for this is that $\|c\|_{2}\leq 1$ (because $1\geq\|A\omega+c\|_{2}$ for $\omega=0$ ) and $\|A\|_{2}=\sigma_{\max}(A)\leq 2$ (because $1\geq\|A\omega+c\|_{2}\geq\|A\omega\|_{2}-\|c\|_{2}$ ). Given the equivalence in (2), we may abuse the notation by referring to quantum channel $\mathcal{E}$ with $(A,c)$ .

II-B Quantum Fisher Information

The following definitions and preliminary results are adopted from [16].

Let density operator $\rho_{\lambda}\in\mathcal{S}(\mathcal{H})$ depend on a scalar parameter $\lambda\in\mathbb{R}$ . Assume that $\rho_{\lambda}$ is continuously differentiable with respect to $\lambda$ . The quantum Fisher information is

\displaystyle\mathcal{F}(\rho_{\lambda}):=\tr(\rho_{\lambda}L_{\lambda}^{2})=\tr\left(\left(\frac{\partial}{\partial\lambda}\rho_{\lambda}\right)L_{\lambda}\right),

(3)

where symmetric logarithmic derivative operator $L_{\lambda}\in\mathcal{L}(\mathcal{H})$ is any Hermitian operator that satisfies

\displaystyle\frac{\partial}{\partial\lambda}\rho_{\lambda}=\frac{1}{2}\left(\rho_{\lambda}L_{\lambda}+L_{\lambda}\rho_{\lambda}\right).

For qubits, this definition can be simplified to

\displaystyle\mathcal{F}(\rho_{\lambda})=\begin{cases}\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}+\displaystyle\frac{|\innerproduct{\omega_{\lambda}}{\partial_{\lambda}\omega_{\lambda}}|^{2}}{1-\|\omega_{\lambda}\|_{2}^{2}},&\|\omega_{\lambda}\|_{2}<1,\\ \|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2},&\|\omega_{\lambda}\|_{2}=1,\end{cases}

(4)

where $\partial_{\lambda}\omega_{\lambda}=\partial\omega_{\lambda}/\partial\lambda$ . Note that the quantum Fisher information is not necessarily continuous everywhere (particularly as $\|\omega_{\lambda}\|_{2}\rightarrow 1$ ) [19]. Assume that we can gather measurements from $N\geq 1$ copies of $\rho_{\lambda}$ , denoted by $\rho_{\lambda}^{\otimes N}$ , by implementing a positive operator-valued measure (POVM). The measurement outcomes can be used to estimate parameter $\lambda$ . Let $\hat{\lambda}$ denote any unbiased estimate of the parameter $\lambda$ . The so-called quantum Cramér-Rao theorem implies that

\displaystyle\mathbb{E}\{(\lambda-\hat{\lambda})^{2}\}\geq\frac{1}{N\mathcal{F}(\rho_{\lambda})}.

(5)

In the scalar parameter case discussed above, the lower bound can be saturated [14, 15]; see [20, 21] for generalized saturability results.

II-C Quantum Differential Privacy

The following definitions and preliminary results are adopted from [6, 22].

The quantum local differential privacy [22] is akin to quantum differential privacy with the exception of removing the so-called “neighboring quantum states”. Local differential privacy is a stronger or more robust approach to privacy removing the need for a trusted curator [23, 22].

Definition 1

For $\epsilon\geq 0$ , quantum channel $\mathcal{E}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H})$ is $\epsilon$ -locally differentially private if

\displaystyle\tr(M\mathcal{E}(\rho))\leq e^{\epsilon}\tr(M\mathcal{E}(\sigma)),

(6)

for all operators $0\preceq M\preceq I$ , where $A\preceq B$ means $B-A\in\mathcal{P}(\mathcal{H})$ , and all density operators $\rho,\sigma\in\mathcal{S}(\mathcal{H})$ . The set of all quantum channel that are $\epsilon$ -locally differentially private is denoted by ${\rm LDP}_{\epsilon}$ .

For density operators $\rho,\sigma\in\mathcal{S}(\mathcal{H})$ , the quantum hockey-stick divergence is

\displaystyle E_{\gamma}(\rho\|\sigma)=\frac{1}{2}\|\rho-\gamma\sigma\|_{1}+\frac{1}{2}(1-\gamma),

(7)

where $\|M\|_{1}:=\tr(|M|)$ is the trace norm of operator $M\in\mathcal{L}(\mathcal{H})$ and $|M|=\sqrt{M^{\dagger}M}$ .

Lemma 1

Quantum channel $\mathcal{E}\in{\rm LDP}_{\epsilon}$ if and only if $E_{e^{\epsilon}}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))=0$ for all $\rho,\sigma\in\mathcal{S}(\mathcal{H})$ .

Proof:

The proof follows from [6, Lemma III.2] by setting $\delta=0$ . ∎

We can prove the following lemma for differentially private quantum channels acting on qubits. This results, particularly the “only if” part, plays a pivotal role in establishing the sample complexity bounds in the next section.

Lemma 2

$(A,c)\in{\rm LDP}_{\epsilon}$ if and only if

\displaystyle\|A(\omega-\nu)+(1-e^{\epsilon})(A\nu+c)\|_{2}\leq e^{\epsilon}-1,

(8)

for all $\omega,\nu\in\mathbb{R}^{3}$ such that $\|\omega\|\leq 1$ and $\|\nu\|\leq 1$ .

Proof:

Let $\rho=(I+\omega.\hat{\sigma})/2$ and $\sigma=(I+\nu.\hat{\sigma})/2$ . Therefore, $\mathcal{E}(\rho)=(I+\bar{\omega}.\hat{\sigma})/2$ and $\mathcal{E}(\sigma)=(I+\bar{\nu}.\hat{\sigma})/2$ , where $\bar{\omega}=(A\omega+c)$ and $\bar{\nu}=(A\nu+c)$ . Note that

$\displaystyle\\|\mathcal{E}(\rho)-e^{\epsilon}\mathcal{E}(\sigma)\\|_{1}=$	$\displaystyle\frac{1}{2}\left\\|(1-e^{\epsilon})+(\bar{\omega}-e^{\epsilon}\bar{\nu}).\hat{\sigma}\right\\|_{1}$
$\displaystyle=$	$\displaystyle\frac{1}{2}\left\|(1-e^{\epsilon})+\\|\bar{\omega}-e^{\epsilon}\bar{\nu}\\|_{2}\right\|$
	$\displaystyle+\frac{1}{2}\left\|(1-e^{\epsilon})-\\|\bar{\omega}-e^{\epsilon}\bar{\nu}\\|_{2}\right\|,$	(9)

where the second equality follows from Lemma A in the appendix. Because $e^{\epsilon}\geq 1$ (or equivalently $1-e^{\epsilon}\leq 0$ ) for all $\epsilon\geq 0$ , we have

\displaystyle|(1-e^{\epsilon})-

\displaystyle\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}|=\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}-(1-e^{\epsilon}).

(10)

We analyze the other term for the following two cases.

•

Case I: Assume that $\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}\geq-(1-e^{\epsilon})$ . In this case, we have

\displaystyle|(1\!-\!e^{\epsilon})\!+\!\|\bar{\omega}\!-\!e^{\epsilon}\bar{\nu}\|_{2}|=\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}\!+\!(1\!-\!e^{\epsilon}).

(11)

Combining (10) and (11) with (9), we get $\|\bar{\rho}-e^{\epsilon}\bar{\sigma}\|_{1}=\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2},$ which results in

\displaystyle E_{e^{\epsilon}}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))=

\displaystyle\frac{1}{2}\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}+\frac{1}{2}(1-e^{\epsilon}).

(12)

•

Case II: Assume that $\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}<-(1-e^{\epsilon})$ . In this case, we have

\displaystyle|(1\!-\!e^{\epsilon})\!+\!\|\bar{\omega}\!-\!e^{\epsilon}\bar{\nu}\|_{2}|=-\|\bar{\omega}\!-\!e^{\epsilon}\bar{\nu}\|_{2}\!-\!(1\!-\!e^{\epsilon}).

(13)

Combining (10) and (13) with (9), we get $\|\bar{\rho}-e^{\epsilon}\bar{\sigma}\|_{1}=-(1-e^{\epsilon}),$ which results in

\displaystyle E_{e^{\epsilon}}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))=0.

(14)

Combining Case I, i.e., (12), and Case II, i.e., (14), shows that

\displaystyle E_{e^{\epsilon}}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))=\max\left\{0,\frac{1}{2}\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}+\frac{1}{2}(1-e^{\epsilon})\right\}.

and, as a result,

	$\displaystyle\sup_{\rho,\sigma\in\mathcal{S}(\mathcal{H})}E_{e^{\epsilon}}$	$\displaystyle(\mathcal{E}(\rho)\\|\mathcal{E}(\sigma))$
		$\displaystyle=\max\left\{0,\max_{\bar{\omega},\bar{\nu}}\frac{1}{2}\\|\bar{\omega}-e^{\epsilon}\bar{\nu}\\|_{2}+\frac{1}{2}(1-e^{\epsilon})\right\}.$

Therefore, Lemma 1 implies that $\mathcal{E}\in{\rm LDP}_{\epsilon}$ if and only if $\|\bar{\omega}-e^{\epsilon}\bar{\nu}\|_{2}\leq e^{\epsilon}-1$ for all $\bar{\omega},\bar{\nu}$ . ∎

III Parameter Estimation Sample Complexity

We first need to define the notion of sample complexity for parameter estimation based on multiple copies of quantum states.

Definition 2 (Sample Complexity)

For $\alpha,\epsilon>0$ , the minimum number of samples required for obtaining estimation accuracy of $\alpha$ is

\displaystyle N_{\alpha,\epsilon}=\!\!\inf_{\hat{\gamma}:\mathbb{E}\{\hat{\gamma}\}=\gamma}\!\!\inf\{N:\mathbb{E}\{(\lambda-\hat{\lambda})^{2}\}\leq\alpha\mbox{ based on }\rho_{\lambda}^{\otimes N}\}.

Now, we can present the main result of this note regarding the sample complexity of parameter estimation under quantum differential privacy for qubits.

Theorem 1

Assume that $\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}\neq 0$ . Then,

\displaystyle\frac{C_{1}}{\alpha(e^{\epsilon}-1)^{2}}\leq N_{\alpha,\epsilon}\leq\frac{C_{2}(e^{\epsilon}+1)^{2}}{\alpha(e^{\epsilon}-1)^{2}},

where

	$\displaystyle C_{1}$	$\displaystyle=\frac{1}{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}}\left(4+\frac{1}{4}\frac{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}}{\|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}\|^{2}}\right)^{-1},$
	$\displaystyle C_{2}$	$\displaystyle=\frac{1}{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}}.$

Particularly, $N_{\alpha,\epsilon}=\Theta\left(\alpha^{-1}\epsilon^{-2}\right)$ for $\epsilon\ll 1$ .

Proof:

Proving the Lower Bound on $N_{\alpha,\epsilon}$ : We prove three important inequalities that enable us to bound the quantum Fisher information. For the first inequality, let $\omega=\omega_{\lambda}$ and

\nu=\omega_{\lambda}-\underbrace{\frac{\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}}{\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}}}_{:=\beta}\partial_{\lambda}\omega_{\lambda}.

We have $\|\nu\|_{2}^{2}=\|\omega_{\lambda}\|_{2}^{2}-|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}|^{2}/\|\omega_{\lambda}\|_{2}^{2}\leq\|\omega_{\lambda}\|_{2}^{2}\leq 1.$ Substituting $\omega$ and $\nu$ in Lemma 2 results in

	$\displaystyle(e^{\epsilon}-1)^{2}\geq$	$\displaystyle\\|\beta A\partial_{\lambda}\omega_{\lambda}+(1-e^{\epsilon})(A\omega_{\lambda}-\beta A\partial_{\lambda}\omega_{\lambda}+c)\\|_{2}^{2}$
	$\displaystyle=$	$\displaystyle\\|(1-e^{\epsilon})(A\omega_{\lambda}+c)+\beta e^{\epsilon}A\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}$
	$\displaystyle=$	$\displaystyle(1-e^{\epsilon})^{2}\\|A\omega_{\lambda}+c\\|_{2}^{2}+\beta^{2}e^{2\epsilon}\\|A\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}$
		$\displaystyle-2\beta(e^{\epsilon}-1)e^{\epsilon}\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}$
	$\displaystyle\geq$	$\displaystyle(1-e^{\epsilon})^{2}\\|A\omega_{\lambda}+c\\|_{2}^{2}$
		$\displaystyle-2\beta(e^{\epsilon}-1)e^{\epsilon}\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}.$

Noting that $e^{\epsilon}\geq 1$ , we get

	$\displaystyle-\beta\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}$	$\displaystyle\leq\frac{1}{e^{\epsilon}}\frac{e^{\epsilon}-1}{2}(1-\\|A\omega_{\lambda}+c\\|_{2}^{2})$
		$\displaystyle\leq\frac{e^{\epsilon}-1}{2}(1-\\|A\omega_{\lambda}+c\\|_{2}^{2}).$		(15)

For the second inequality, let $\nu=\omega_{\lambda}$ and

\omega=\omega_{\lambda}-\underbrace{\frac{\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}}{\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}}}_{:=\beta}\partial_{\lambda}\omega_{\lambda}.

	$\displaystyle(e^{\epsilon}-1)^{2}\geq$	$\displaystyle\\|-\beta A\partial_{\lambda}\omega_{\lambda}+(1-e^{\epsilon})(A\omega_{\lambda}+c)\\|_{2}^{2}$
	$\displaystyle=$	$\displaystyle(1-e^{\epsilon})^{2}\\|A\omega_{\lambda}+c\\|_{2}^{2}+\beta^{2}\\|A\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}$
		$\displaystyle+2\beta(e^{\epsilon}-1)\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}$
	$\displaystyle\geq$	$\displaystyle(1-e^{\epsilon})^{2}\\|A\omega_{\lambda}+c\\|_{2}^{2}$
		$\displaystyle+2\beta(e^{\epsilon}-1)\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}.$

Thus

\displaystyle\beta\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}

\displaystyle\leq\frac{e^{\epsilon}-1}{2}(1-\|A\omega_{\lambda}+c\|_{2}^{2}).

(16)

Combining (15) and (16) while recalling definition of $\beta$ , we get

\displaystyle\Bigg{|}\frac{\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}}{\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}}

\displaystyle\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}\Bigg{|}\!\leq\!\frac{e^{\epsilon}\!-\!1}{2}(1\!-\!\|A\omega_{\lambda}+c\|_{2}^{2}),

and hence

\displaystyle\frac{|\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}|}{(1-\|A\omega_{\lambda}+c\|_{2}^{2})}\leq\frac{e^{\epsilon}-1}{2}\frac{\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}}{|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}|}.

(17)

For the third inequality, let $\nu=0$ . We get $e^{\epsilon}-1\geq\|A\omega+(1-e^{\epsilon})c\|_{2}\geq\|A\omega\|_{2}-(e^{\epsilon}-1)\|c\|_{2}$ and thus, it must be that

\displaystyle\|A\omega\|_{2}\leq(e^{\epsilon}-1)(1+\|c\|)\leq 2(e^{\epsilon}-1).

(18)

Now, we are ready to bound quantum Fisher information. Note that

	$\displaystyle\mathcal{F}(\mathcal{E}(\rho_{\lambda}))\leq$	$\displaystyle\\|A\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}+\displaystyle\frac{\|\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}\|^{2}}{1-\\|A\omega_{\lambda}+c\\|_{2}^{2}}$
	$\displaystyle\leq$	$\displaystyle 4(e^{\epsilon}-1)^{2}\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}$
		$\displaystyle+\frac{(e^{\epsilon}-1)^{2}}{4}\frac{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{4}}{\|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}\|^{2}}(1-\\|A\omega_{\lambda}+c\\|_{2}^{2})$
	$\displaystyle\leq$	$\displaystyle 4(e^{\epsilon}-1)^{2}\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}\left(1+\frac{1}{16}\frac{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}}{\|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}\|^{2}}\right).$

Finally, from the quantum Cramér-Rao bound, we get

	$\displaystyle\alpha\geq$	$\displaystyle\mathbb{E}\{(\lambda-\hat{\lambda})^{2}\}$
	$\displaystyle\geq$	$\displaystyle\frac{1}{N}\frac{1}{4(e^{\epsilon}-1)^{2}\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}}\left(1+\frac{1}{16}\frac{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}}{\|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}\|^{2}}\right)^{-1}.$

Proving the Upper Bound on $N_{\alpha,\epsilon}$ : Select

\displaystyle\mathcal{E}(\rho)=\frac{p}{2}I+(1-p)\rho.

This is the so-called global depolarizing channel. From Lemma IV.2 in [6], we know that $\mathcal{E}\in{\rm LDP}_{\epsilon}$ if $p=2/(1+e^{\epsilon})$ . We have

\displaystyle\mathcal{F}(\mathcal{E}(\rho_{\lambda}))\geq

\displaystyle(1-p)^{2}\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}=\left(\frac{e^{\epsilon}-1}{e^{\epsilon}+1}\right)^{2}\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}.

From[14, 15], we know that, in the case of scalar parameters, there exists an unbiased estimator for which the quantum Cramér-Rao bound is saturated. That is, $\alpha=\mathbb{E}\{(\lambda-\hat{\lambda})^{2}\}=1/(N\mathcal{F}(\mathcal{E}(\rho_{\lambda}))).$ Therefore, for this specific unbiased estimator, we get

\displaystyle N\leq\frac{1}{\alpha}\left(\frac{e^{\epsilon}+1}{e^{\epsilon}-1}\right)^{2}\frac{1}{\|\partial_{\lambda}\omega_{\lambda}\|_{2}^{2}}.

This concludes the proof. ∎

Remark 1

A similar bound for the classical case is shown to hold [24].

Acknowledgment

The author would like to thank Theshani Nuradha for pointing out a mathematical typo in the proof of Theorem 1.

References

[1] C. Dwork, F. McSherry, K. Nissim, and A. Smith, “Calibrating noise to sensitivity in private data analysis,” in Theory of Cryptography: Third Theory of Cryptography Conference, TCC 2006, New York, NY, USA, March 4-7, 2006. Proceedings 3, pp. 265–284, Springer, 2006.
[2] C. Dwork, “Differential privacy: A survey of results,” in Theory and Applications of Models of Computation: 5th International Conference, TAMC 2008, Xi’an, China, April 25-29, 2008. Proceedings 5, pp. 1–19, Springer, 2008.
[3] C. Dwork and A. Roth, “The algorithmic foundations of differential privacy,” Foundations and Trends in Theoretical Computer Science, vol. 9, no. 3–4, pp. 211–407, 2014.
[4] L. Zhou and M. Ying, “Differential privacy in quantum computation,” in 2017 IEEE 30th Computer Security Foundations Symposium (CSF), pp. 249–262, IEEE, 2017.
[5] S. Aaronson and G. N. Rothblum, “Gentle measurement of quantum states and differential privacy,” in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pp. 322–333, 2019.
[6] C. Hirche, C. Rouzé, and D. S. França, “Quantum differential privacy: An information theory perspective,” IEEE Transactions on Information Theory, vol. 69, no. 9, pp. 5771–5787, 2023.
[7] T. Nuradha, Z. Goldfeld, and M. M. Wilde, “Quantum pufferfish privacy: A flexible privacy framework for quantum systems,” IEEE Transactions on Information Theory, pp. 5731–5762, 2024.
[8] F. Farokhi, “Barycentric and pairwise Rényi quantum leakage with application to privacy-utility trade-off,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 480, no. 2303, p. 20240319, 2024.
[9] F. Farokhi, “Quantum privacy and hypothesis-testing,” in 2023 62nd IEEE Conference on Decision and Control (CDC), pp. 2841–2846, IEEE, 2023.
[10] T. Nuradha and M. M. Wilde, “Contraction of private quantum channels and private quantum hypothesis testing,” IEEE Transactions on Information Theory, 2025. In Press.
[11] H.-C. Cheng, C. Hirche, and C. Rouzé, “Sample complexity of locally differentially private quantum hypothesis testing,” in 2024 IEEE International Symposium on Information Theory (ISIT), pp. 2921–2926, IEEE, 2024.
[12] W. M. Watkins, S. Y.-C. Chen, and S. Yoo, “Quantum machine learning with differential privacy,” Scientific Reports, vol. 13, no. 1, p. 2453, 2023.
[13] C. W. Helstrom, “Minimum mean-squared error of estimates in quantum statistics,” Physics letters A, vol. 25, no. 2, pp. 101–102, 1967.
[14] S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Physical Review Letters, vol. 72, no. 22, p. 3439, 1994.
[15] S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and lorentz invariance,” Annals of Physics, vol. 247, no. 1, pp. 135–173, 1996.
[16] W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Physical Review A, vol. 87, no. 2, p. 022337, 2013.
[17] M. Wilde, Quantum Information Theory. Quantum Information Theory, Cambridge University Press, 2013.
[18] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 10th anniversary ed., 2010.
[19] D. Šafránek, “Discontinuities of the quantum fisher information and the bures metric,” Physical Review A, vol. 95, no. 5, p. 052320, 2017.
[20] H. I. Nurdin, “Saturability of the quantum Cramér–Rao bound in multiparameter quantum estimation at the single-copy level,” IEEE Control Systems Letters, vol. 8, pp. 376–381, 2024.
[21] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, “Compatibility in multiparameter quantum metrology,” Physical Review A, vol. 94, no. 5, p. 052108, 2016.
[22] A. Angrisani and E. Kashefi, “Quantum local differential privacy and quantum statistical query model,” arXiv preprint arXiv:2203.03591, 2022.
[23] J. C. Duchi, M. I. Jordan, and M. J. Wainwright, “Local privacy and statistical minimax rates,” in 2013 IEEE 54th annual symposium on foundations of computer science, pp. 429–438, IEEE, 2013.
[24] L. P. Barnes, W.-N. Chen, and A. Özgür, “Fisher information under local differential privacy,” IEEE Journal on Selected Areas in Information Theory, vol. 1, no. 3, pp. 645–659, 2020.

Appendix A Useful Lemma

Lemma A

For any $m\in\mathbb{R}$ and $n\in\mathbb{R}^{3}$ , $\|mI+n.\hat{\sigma}\|_{1}=|m-\|n\|_{2}|+|m+\|n\|_{2}|$ .

Proof:

Note that

	$\displaystyle\det(mI+n.\hat{\sigma}-sI)=$	$\displaystyle\det\left(\begin{bmatrix}m+n_{z}-s&n_{x}-in_{y}\\ n_{x}+in_{y}&m-n_{z}-s\end{bmatrix}\right)$
	$\displaystyle=$	$\displaystyle(s-m)^{2}-\\|n\\|_{2}^{2}.$

Therefore, the eigenvalues of $mI+n.\hat{\sigma}$ are $s_{\pm}=m\pm\|n\|_{2}$ . The rest follows from that $\|mI+n.\hat{\sigma}\|_{1}=|s_{+}|+|s_{-}|$ . ∎

$\displaystyle\\|\mathcal{E}(\rho)-e^{\epsilon}\mathcal{E}(\sigma)\\|_{1}=$	$\displaystyle\frac{1}{2}\left\\|(1-e^{\epsilon})+(\bar{\omega}-e^{\epsilon}\bar{\nu}).\hat{\sigma}\right\\|_{1}$
$\displaystyle=$	$\displaystyle\frac{1}{2}\left\|(1-e^{\epsilon})+\\|\bar{\omega}-e^{\epsilon}\bar{\nu}\\|_{2}\right\|$
	$\displaystyle+\frac{1}{2}\left\|(1-e^{\epsilon})-\\|\bar{\omega}-e^{\epsilon}\bar{\nu}\\|_{2}\right\|,$	(9)

	$\displaystyle\mathcal{F}(\mathcal{E}(\rho_{\lambda}))\leq$	$\displaystyle\\|A\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}+\displaystyle\frac{\|\innerproduct{A\omega_{\lambda}+c}{A\partial_{\lambda}\omega_{\lambda}}\|^{2}}{1-\\|A\omega_{\lambda}+c\\|_{2}^{2}}$
	$\displaystyle\leq$	$\displaystyle 4(e^{\epsilon}-1)^{2}\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}$
		$\displaystyle+\frac{(e^{\epsilon}-1)^{2}}{4}\frac{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{4}}{\|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}\|^{2}}(1-\\|A\omega_{\lambda}+c\\|_{2}^{2})$
	$\displaystyle\leq$	$\displaystyle 4(e^{\epsilon}-1)^{2}\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}\left(1+\frac{1}{16}\frac{\\|\partial_{\lambda}\omega_{\lambda}\\|_{2}^{2}}{\|\innerproduct{\partial_{\lambda}\omega_{\lambda}}{\omega_{\lambda}}\|^{2}}\right).$

Tight Sample Complexity Bounds for Parameter Estimation Under Quantum Differential Privacy for Qubits

Abstract

I Introduction

II Preliminary Material

II-A Density Operators

II-B Quantum Fisher Information

II-C Quantum Differential Privacy

Definition 1

Lemma 1

Proof:

Lemma 2

Proof:

III Parameter Estimation Sample Complexity

Definition 2 (Sample Complexity)

Theorem 1

Proof:

Remark 1

Acknowledgment

References

Appendix A Useful Lemma

Lemma A

Proof:

Tight Sample Complexity Bounds for Parameter Estimation
Under Quantum Differential Privacy for Qubits