Achieving Heisenberg limit under General Markov Noise without Ancilla

TL;DR

This paper demonstrates that the Heisenberg limit in quantum metrology can be achieved under general Markovian noise without using ancillas, by applying full and rapid control when the Hamiltonian has a perpendicular component to the noise.

Contribution

It shows that Heisenberg limit can be restored under Markovian noise without ancillas, given specific Hamiltonian conditions, using full and fast control.

Findings

Heisenberg limit achievable without ancillas under Markovian noise

Full and fast control restores quantum precision limits

Perpendicular Hamiltonian component is crucial for HL achievement

Abstract

We consider probing qubits infested by general Makovian noise. It can be shown that one can restore Heisenberg limit (HL) via full and fast control without any ancilla, as long as the Hamiltonian of the system has non-vanishing component perpendicular to the noise

Equations

$$\frac{\mathrm{d}\hat{\rho }}{\mathrm{d}t}=-i\omega [\hat{H},\hat{\rho }]+\sum _{k=1}^R({\hat{L}}_k\hat{\rho }{\hat{L}}_k^{†}-\frac{1}{2}\{{\hat{L}}_k^{†}{\hat{L}}_k,\hat{\rho }\}),$$

$$\hat{H}\notin {\mathrm{span}}_{\mathbb{R}}\left\{\text{openone},{\hat{L}}_j^{\mathrm{H}},{\hat{L}}_j^{\mathrm{AH}},({\hat{L}}_j^{†}{\hat{L}}_k{)}^{\mathrm{H}},({\hat{L}}_j^{†}{\hat{L}}_k{)}^{\mathrm{AH}}\right\}=:\mathbb{S}.$$

$${\hat{H}}_{\mathrm{tot}}:=\sum _{s=1}^S{\hat{H}}^{(n)}.$$

$${\hat{\Pi }}_{\mathrm{C}}{\hat{L}}_j^{(n)}{\hat{\Pi }}_{\mathrm{C}}={\lambda }_k^{(n)}{\hat{\Pi }}_{\mathrm{C}}\text{and}{\hat{\Pi }}_{\mathrm{C}}[{\hat{L}}_j^{(n)}{]}^{†}{\hat{L}}_k^{(m)}{\hat{\Pi }}_{\mathrm{C}}={\mu }_{jk}^{(nm)}{\hat{\Pi }}_{\mathrm{C}},$$

$${\hat{\Pi }}_{\mathrm{C}}{\hat{H}}_{\mathrm{tot}}{\hat{\Pi }}_{\mathrm{C}}/\propto {\hat{\Pi }}_{\mathrm{C}}.$$

$${\mathcal{E}}_{\delta t}^{\otimes N}({\hat{\rho }}_{\mathrm{tot}}):={\hat{K}}_0{\hat{\rho }}_{\mathrm{tot}}{\hat{K}}_0^{†}+\sum _{k=1}^R\sum _{n=1}^N{\hat{K}}_k^{(n)}{\hat{\rho }}_{\mathrm{tot}}[{\hat{K}}_k^{(n)}{]}^{†}$$

$${\hat{K}}_0:=\text{openone}-\left(i\omega {\hat{H}}_{\mathrm{tot}}+\frac{1}{2}\sum _{n=1}^N\sum _{k=1}^R[{\hat{L}}_k^{(n)}{]}^{†}{\hat{L}}_k^{(n)}\right)\delta t,$$

$${\hat{K}}_k^{(n)}:={\hat{L}}_k^{(n)}\sqrt{\delta t}.$$

$$\mathcal{C}\circ {\mathcal{E}}_{\mathrm{d}t}^{\otimes N}\left({\hat{\rho }}_{\mathrm{tot}}\right)={\hat{\rho }}_{\mathrm{tot}}-i\omega [{\hat{H}}_{\mathrm{eff}},{\hat{\rho }}_{\mathrm{tot}}]+\mathcal{O}(\delta t^2).$$

$${\hat{\Pi }}_{\mathrm{C}}{\mathcal{E}}_{\delta t}({\hat{\rho }}_{\mathrm{tot}}){\hat{\Pi }}_{\mathrm{C}}$$

$${\hat{\Pi }}_{\mathrm{E}}{\mathcal{E}}_{\mathrm{d}t}\left({\hat{\rho }}_{\mathrm{tot}}\right){\hat{\Pi }}_{\mathrm{E}}=\sum _{n=1}^N\sum _{k=1}^R{\hat{E}}_k^{(n)}{\hat{\rho }}_{\mathrm{tot}}[{\hat{E}}_k^{(n)}{]}^{†}+\mathcal{O}(\delta t^2)$$

$${\hat{\Pi }}_{\mathrm{C}}[{\hat{E}}_j^{(n)}{]}^{†}{\hat{E}}_k^{(m)}{\hat{\Pi }}_{\mathrm{C}}={\hat{\Pi }}_{\mathrm{C}}\left({\mu }_{jk}^{(nm)}-[{\lambda }_j^{(n)}{]}^{\ast }{\lambda }_k^{(m)}\right)\delta t.$$

$$\hat{H}=\left(\begin{array}{ccc}0& 0& i\\ 0& 0& 0\\ -i& 0& 0\end{array}\right).$$

$${\hat{L}}_1=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),{\hat{L}}_2=\left(\begin{array}{ccc}0& -i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right),\text{and}{\hat{L}}_3=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right).$$

$$|0_{\mathrm{L}}⟩={|\circlearrowleft ⟩}^{\otimes N}\text{and}|1_{\mathrm{L}}⟩={|\circlearrowright ⟩}^{\otimes N}.$$

$${\hat{H}}_{\perp }=\Vert {\hat{H}}_{\perp }\Vert \left(|{\psi }_{+}⟩⟨{\psi }_{+}|-|{\psi }_{-}⟩⟨{\psi }_{-}|\right)$$

$${\hat{\Pi }}_{\mathrm{C}}:={\left(|{\psi }_{+}⟩⟨{\psi }_{+}|\right)}^{\otimes N}+{\left(|{\psi }_{-}⟩⟨{\psi }_{-}|\right)}^{\otimes N},$$

$${\hat{H}}_{\mathrm{eff}}=\sum _{n=1}^N{\hat{H}}_{\perp }^{(n)},$$

$${\Delta }^2\omega \ge 1/N^2{T}^2\Vert \hat{H}\Vert .$$

Full text

Introduction— Quantum metrology is one of the three main streams in the thriving field of quantum information besides quantum communication and quantum computation Grumbling and Horowitz (2018). In the ideal noiseless condition, quantum resources such as entanglement can enhance the estimation precision of parameters to breach the shot-noise limit (SNL) $1/\sqrt{NT}$ and reach the ultimate Heisenberg limit (HL) $1/NT$, with $T$ being the total probing time and $N$ the number of probes employed Giovannetti et al. (2004, 2006); Demkowicz-Dobrzański et al. (2015); Degen et al. (2017); Pezzè et al. (2018); Kwon et al. (2019). The advantage provided by quantum resources ensures a promising future of applications as diverse as spectroscopy Bollinger et al. (1999), accurate clock construction Kessler et al. (2014); Derevianko and Katori (2011); Zhang and Ye (2016), gravitational wave detection Collaboration (2013); Schnabel et al. (2010), fundamental biology research and medicine development Taylor and Bowen (2016), and et. al Giovannetti et al. (2004); Demkowicz-Dobrzański et al. (2015); Degen et al. (2017); Pezzè et al. (2018). However, in the presence of uncontrollable environmental noise, such an advantage would be diminished or destroyed in most cases Huelga et al. (1997); Escher et al. (2011); Chin et al. (2012); Demkowicz-Dobrzański et al. (2015, 2012); Degen et al. (2017). As a result, extensive efforts have been made by many groups to reduce or even neutralize the detrimental effect of the noise, which include: a) searching for noise-resilient input states such as the spin-squeezed states and Dicke states Sørensen et al. (2001); Dunningham et al. (2002); Ma et al. (2011); Duan (2011); Zhang and Duan (2013); Pezzé and Smerzi (2013), b) designing schemes to harness the entanglement with noiseless ancilla systems Demkowicz-Dobrzański and Maccone (2014) or take advantage of the specific geometry of the noise Chaves et al. (2013); Brask et al. (2015), c) designing noise-resilient control sequences via machine learning Hentschel and Sanders (2010, 2011); Lovett et al. (2013); Palittapongarnpim et al. (2017); Lumino et al. (2018), and d) adopting noise-suppression techniques from the fields of quantum computation and quantum communication such as dynamical decoupling (DD) Viola et al. (1999a, b); Tan et al. (2013); Sekatski et al. (2016), quantum error correction (QEC) Unden et al. (2016); Reiter et al. (2017); Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018); Layden and Cappellaro (2018); Layden et al. (2019); Cappellaro et al. (2019); Nielsen and Chuang (2010) and quantum teleportation Matsuzaki et al. (2018). Each routine has its own advantages. For DD and QEC which are among the most powerful tools developed for fault-tolerant quantum computation and communication, we have a good reason to expect their prowess in quantum metrology against environment noise. Their adoption to quantum metrology allows us to take advantage of the fruitful results designated originally for practical quantum computation and communication Nielsen and Chuang (2010).

Here we focus on the QEC scheme. In the presence of general Makovian noise, it has been shown by explicitly constructing QEC code $C_{\mathrm{O}}$ such that HL can be achieved with the help of noiseless ancilla. This is possible if the probe Hamiltonian has non-vanishing component perpendicular to the Markovian noise. It is assumed that the ancilla systems are neither affected by the parameter field, for instance the magnetic field, nor any environment noise. If the ancilla systems suffer from Markovian noise also, one has to apply an inner code $C_{\mathrm{I}}$ to obtain effectively noiseless ancilla systems. The HL can be achieved by concatenating $C_{\mathrm{I}}$ and $C_{\mathrm{O}}$ Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). This kind of scheme is very suitable for the situation where there are at least two types quantum systems of which one species has a much longer coherence time than others. A typical example is the nitrogen-vacancy (NV) center where the decoherence rate of the nuclear spin is very small compared with that of the electron spin Maze et al. (2008); Liu et al. (2015); Unden et al. (2016). In the case such as the superconducting qubit Naghiloo et al. (2017) and trapped ion system Reiter et al. (2017) where there is only one kind of quantum system at disposal, the employment of the inner code $C_{\mathrm{I}}$ seems inevitable which can bring further complexity. Devising an ancilla-free QEC scheme for quantum metrology is thus very important. Given commuting Markovian noise, it has been further elaborated that there exists an ancilla-free QEC scheme ensuring the HL if part of the probe Hamiltonian is perpendicular to the noise. The commuting Markovian noise covers common noises such as the lossy bosonic  channel and dephasing Layden et al. (2019). We generalize the result of Ref. Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018); Layden and Cappellaro (2018); Layden et al. (2019) by showing that the HL can recovered by a QEC scheme for qubit probes affected by general Markovian noise without any ancilla. One can thus exorcise commuting as well as non-commuting noises for example the general Rank-one Pauli channel which includes dephasing as well as bit flit channel Wilde (2013).

Quantum Metrology under general Markovian noise.—Consider the quantum probe with Hamiltonian $\omega\hat{H}$ suffering from Markovian noise described by the Lindblad operator $\hat{L}_{k}$ such that its state evolves homogeneously in time

[TABLE]

with $R$ being the rank of the noise. The information of both Hamiltonian $\hat{H}$ and Markovian noise is available to us. Our task is to make an estimation $\omega_{\mathrm{est}}$ of $\omega$ as precisely as possible. It has been shown that such a time-homogeneous Markovian noise can render the estimation precision from HL to SNL Escher et al. (2011); Demkowicz-Dobrzański et al. (2012). However, scuh a disastrous setback can be avoid. It has been shown that Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018); Layden et al. (2019)

given HNLS being true, HL can be restored by QEC with the help of noiseless ancilla systems; 2. 2.

given commuting Markovian noise and HNLS being true, HL can be restored by QEC without any noiseless ancilla; 3. 3.

if HNLS fails, HL can not be restored by QEC even with ancilla.

If the Hamiltonian $\hat{H}$ can not be linearly spanned by the identity operator $\openone$, Lindblad operators $\hat{L}_{j}$, $\hat{L}_{j}^{\dagger}$ and their second-order products $\hat{L}_{j}^{\dagger}\hat{L}_{k}$ for $j,k=1,\ldots,R$, it was coined as the Hamiltonian-not-in-Lindblad span (HNLS) condition denoted concisely as

[TABLE]

Here $\hat{L}_{j}^{\mathrm{H}}:=(\hat{L}_{j}+\hat{L}_{j}^{\dagger})/2$ and $\hat{L}_{j}^{\mathrm{AH}}:=(\hat{L}_{j}-\hat{L}_{j}^{\dagger})/2i$ denote the Hermitian and anti-Hermitian components of $\hat{L}_{j}$ respectively and the same definition to $(\hat{L}_{j}^{\dagger}\hat{L}_{k})^{\mathrm{H}}$ and $(\hat{L}_{j}^{\dagger}\hat{L}_{k})^{\mathrm{AH}}$. $\mathbb{S}$ is thus an Euclidean space. Further, the Markovian noise is called commuting if the Hamiltonian $\hat{H}$ and Lindblad operators $\hat{L}_{j}$s are mutually commuting, namely $[{\hat{H},\hat{L}_{j}}]=[{\hat{L}_{j},\hat{L}_{k}}]=0$. In the following, we will illustrate for qubit systems that if HNLS holds, HL can be restored via QEC without ancilla, regardless whether the Markovian noise is commuting or not.

QEC quantum metrology without ancilla.—Suppose we employ $N$ probes which evolve independently according to (1). The Hamiltonian for the first probe is $\hat{H}^{(1)}:=\hat{H}\otimes\openone^{\otimes{N-1}}$ with the corresponding Lindblad operators being $\hat{L}_{k}^{(1)}:=\hat{L}_{k}\otimes\openone^{\otimes{N-1}}$, and so forth. The total Hamiltonian reads

[TABLE]

A QEC code $\mathcal{C}$ can render the noisy quantum channel to effectively noiseless if the projection onto code space $\hat{\Pi}_{\mathrm{C}}$ satisfies the QEC condition Ekert and Macchiavello (1996); Bennett et al. (1996); Knill and Laflamme (1997); Nielsen and Chuang (2010)

[TABLE]

for every $n,m=1,\ldots,N$ and $j,k=1,\ldots,S$. The parameter-imprinting unitary channel can survive the QEC procedure $\mathcal{C}$ if

[TABLE]

These conditions (4) and (5) are very similar to their counter part in the situation ancilla systems are plentiful Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). And so is the proof which we outline in the following.

Let us divide the total probing time $T$ into $D$ pieces such that we carry out a QEC procedure via fast control after every $\delta{t}{:=}T/D$ evolution

[TABLE]

where

[TABLE]

and

[TABLE]

There exists a QEC code $\mathcal{C}$ capable of rendering $\mathcal{E}_{\delta{t}}^{\otimes{N}}$ to an effective unitary evolution generated by the effective total Hamiltonian $\hat{H}_{\mathrm{eff}}:=\hat{\Pi}_{\mathrm{C}}\hat{H}_{\mathrm{tot}}\hat{\Pi}_{\mathrm{C}}$

[TABLE]

One can verify this by using the three conditions presented in (4) and (5) directly. Firstly, the diagnosis measurement would collapse the probes onto either

[TABLE]

if everything is all right, or

[TABLE]

with $\hat{E}_{k}^{(n)}:=\hat{\Pi}_{\mathrm{E}}\hat{L}_{k}^{(s)}\hat{\Pi}_{\mathrm{C}}\sqrt{\delta{t}}$ if error has been detected. There exist a QEC code to correct error (11) according to the QEC condition Ekert and Macchiavello (1996); Bennett et al. (1996); Knill and Laflamme (1997); Nielsen and Chuang (2010)

[TABLE]

Example.— Consider the Hamiltonian coupled to the parameter $\omega$ is

[TABLE]

The Markovian noise infesting the probe is described by

[TABLE]

It has been condemned beyond saving by any ancilla-free QEC in the Supplementary Material of Ref. Layden et al. (2019) when only one probe is available. The interesting part is that HNLS holds in this scenario. This condemnation can be avoid if at least $3$ probes are at our disposal. One can verify that the three conditions presented in (4) and (5) can be satisfied by the QEC code with logic states

[TABLE]

Here $\ket{\circlearrowleft}:=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&i\end{pmatrix}^{\mathrm{T}}$ and $\ket{\circlearrowright}:=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&-i\end{pmatrix}^{\mathrm{T}}$ are eigenvectors of $\hat{H}$ belonging to eigenvalue $\pm 1$ respectively. With $\ket{\Psi}=\frac{1}{\sqrt{2}}(\ket{0}_{\mathrm{L}}+\ket{1}_{\mathrm{L}})$ fed, the optimal precision is $\Delta{\omega}^{2}=1/4N^{2}T^{2}$ which is identical to the ultimate precision in the noiseless case Giovannetti et al. (2006).

Qubit probes.—Suppose $\hat{H}_{\perp}$ is the component of $\hat{H}$ perpendicular to $\mathbb{S}$. Its spectrum decomposition is

[TABLE]

with $\braket{\psi_{\pm}}{\psi_{\pm}}=1$, $\braket{\psi_{+}}{\psi_{-}}=0$ and $\|\bullet\|$ being the operator norm. An explicit QEC code that can eradicate the error (11) is

[TABLE]

when at least $3$ probes has been deployed. QEC condition (4) can be ensured, since $\braket{\psi_{+}}{\hat{L}_{j}}{\psi_{+}}=\braket{\psi_{-}}{\hat{L}_{j}}{\psi_{-}}$ and $\braket{\psi_{+}}{\hat{L}_{j}^{\dagger}\hat{L}_{k}}{\psi_{+}}=\braket{\psi_{-}}{\hat{L}_{j}^{\dagger}\hat{L}_{k}}{\psi_{-}}$. And the effective total Hamiltonian after QEC would be

[TABLE]

which is non-vanishing as long as HNLS holds. The optimal input state is $\ket{\psi_{\mathrm{opt}}}=\frac{1}{\sqrt{2}}(\ket{\psi_{+}}^{\otimes{N}}+\ket{\psi_{-}}^{\otimes{N}})$ and the corresponding ultimate precision is

[TABLE]

Thus HL is restored in both the sense of probing time $T$ as well as the number of probes.

Conclusion and discussion.—When noiseless ancilla is inaccessible as in many situations, we show for qubit probes that HNLS is the necessary and sufficient condition for restoring HL via QEC. This is true for any Markovian noise whether commuting or not. Our QEC code works for $N\geq 3$. In stark contrast to the traditional QEC Nielsen and Chuang (2010), the additional probing qubits are not merely redundant price paid for fighting the detrimental Markovian noise. They would also do the parameter probing job at the same time. As a result HL is restored in the sense of both time consumed and number of qubits employed. We thus has partially resolved the open question raised previously in Ref. Layden et al. (2019). Our analysis also indicates that the parallel scheme can provide more possibility for QEC construction as can be seen from the example presented in (13) and (14).