Local and Non-Local Invasive Measurements on Two Quantum Spins Coupled via Nanomechanical Oscillations

Abstract

Symmetry plays the central role in the structure of quantum states of bipartite (or many-body) fermionic systems. Typically, symmetry leads to the phenomenon of quantum coherence and correlations (entanglement) inherent to quantum systems only. In the present work, we study the role of symmetry (i.e., quantum correlations) in invasive quantum measurements. We consider the influence of a direct or indirect measurement process on a composite quantum system. We derive explicit analytical expressions for the case of two quantum spins positioned on both sides of the quantum cantilever. The spins are coupled indirectly to each others via their interaction with a magnetic tip deposited on the cantilever. Two types of quantum witnesses can be considered, which quantify the invasiveness of a measurement on the systems’ quantum states: (i) A local quantum witness stands for the consequence on the quantum spin states of a measurement done on the cantilever, meaning we first perform a measurement on the cantilever, and subsequently a measurement on a spin. (ii) The non-local quantum witness signifies the response of one spin if a measurement is done on the other spin. In both cases the disturbance must involve the cantilever. However, in the first case, the spin-cantilever interaction is linear in the coupling constant $\mathsf{\Omega }$, where as in the second case, the spin-spin interaction is quadratic in $\mathsf{\Omega }$. For both cases, we find and discuss analytical results for the witness.

1. Introduction

Since the foundational development of quantum mechanics, the measurement process and the wave function collapse have been under discussion [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. One aspect is how to quantify and possibly modify the destructive effect of a certain measurement on a quantum state, an issue that is clearly of importance for quantum information transfer and processing. Here, we consider the “quantum witness” as the measure invasiveness in a combined quantum system. Our system described by the density operator ${\widehat{\rho }}_{AB}$ consists of two parties A and B that interact with each other either directly $A⟺B$ or indirectly through the third quantum system f, i.e., $A\stackrel{\mathsf{\Omega }}{⟺}f\stackrel{\mathsf{\Omega }}{⟺}B$. Here, $\mathsf{\Omega }$ is the coupling constant between f and A or B. Let us consider two types of measurements: (i) We perform a measurement on f, and then on A, in contrast to measuring A directly. (ii) We conduct a measurement on A and then on B. For Case (i), A interacts with f directly. Measurements done on f have a direct influence on the outcome of measurements done on A; this case is referred to as the local quantum witness. In Case (ii), A and B are interrelated via f; a measurement performed on A affects B through the quantum channel f, a case called the non-local quantum witness. One may wonder about the outcome when the character of the channel f changes from inherently quantum to purely classical ($f_q\to f_{cl}$); the latter situation is relevant to local operations and classical communication. As discussed below, in our system, we find that the non-local quantum witness mediated through the classical channel is not zero. Specifically, we consider two nitrogen vacancy (NV) centers described by the density operator $\equiv {\widehat{\rho }}_{AB}$ interacting via quantum modes of a nanomechanical oscillator f (see [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53] and the references therein) with a magnetic tip attached to its end.

Wave-particle duality is a fundamental principle of quantum mechanics postulating that even macroscopic objects have wave features. However, the manifestation of quantum behavior is inversely proportional to the mass of the particle $\lambda =\frac{2\pi \hslash }{mv}$, where $\lambda$ is the de Broglie wavelength, m and v are the mass and velocity of the particle, and ℏ is the Planck constant. Hence, mechanical motion, in general, should be viewed as classical unless the system is not cooled down to the temperature below $T\ll 2\pi \hslash \omega /k_B$. For example, for mode frequency $\omega =1$ kHz, the temperature should be lower than $T\ll 50$ nK. Fortunately, the resonant frequency scales inversely with the size of the system. Thus, by considering nanosized cantilevers and cooling the systems to low temperature, mechanical motion switches its character and crosses the threshold boundary between the classical and quantum world. The viable technological progress of the last two decades allowed the fabrication of hybrid spin nanomechanical systems. Nanoelectromechanical resonators (NAMR) are essential for both the understanding of fundamental questions and technological applications as well. The paradigmatic model of NAMR consists of the spin of the NV center interacting with a magnetic tip. The magnetic tip is attached to the end of the nano-cantilever (i.e., cavity field). The oscillation of the cantilever modulates the spin-tip interaction and activates the spin dynamics or vice versa; the highly sensitive cantilever can detect the spin dynamics. The magnetic force between a ferromagnetic tip and NV spin implemented in magnetic resonance imaging experiments allows achieving a single-spin sensitivity and is an essential experimental scheme for NAMR [52,53].

For observing quantum behavior, the cantilever must be cooled to low temperatures to avoid thermally excited vibrations. MRFM provides tools needed to read out the cantilever and determine and possibly control its quantum state. The read out procedure on the formal mathematical language can be interpreted in terms of the phonon number projective operators (i.e., phonons are particles characterizing the quantized motion of the cantilever). We note that our model corresponds to the zero temperature case and closed quantum systems. Therefore, phenomena typical for open quantum systems, i.e., dissipation and decoherence, are beyond our project’s scope. Concerning these issues, we refer to the works of Melkikh [54,55].

The spins of the NV centers are affected by a tip magnetization. In turn, the tip magnetization is strongly stabilized by intrinsic magnetic interactions (magnetic anisotropy) and hence remains basically static during the NV spin dynamics (cf. Figure 1. The quantized elastic modes of the cantilever are however decisive in that the cantilever oscillation modulates the spin-magnetic tip interaction. The modes of the cantilever are in turn modified by the coupling to the NV centers, which allows sensing the NV’s spin dynamics [52,53]. The goal is to study the invasiveness of a measurement done on the quantum cantilever f (see Figure 1) for the NV spin A. In particular, we monitor the invasiveness as a function of mean “phonon number” $u^2$ in the coherent state and as we vary the detuning between frequencies of the NV center and cantilever mode. The paper is organized as follows: In Section 2, we specify the model. In Section 3, we present the analytical solution of the model. In Section 4, we study the invasiveness of quantum measurements done on the cantilever and NV centers.

2. Model

We consider two NV spins embedded in a cavity at an equal distance from the cantilever. In the general case, the Hamiltonian of the system reads [51]:

$$\begin{array}{cc}\hfill \widehat{H}=& {\widehat{H}}_0+\widehat{V},\hfill \\ \hfill {\widehat{H}}_0=& {\omega }_0\left({\widehat{S}}_1^z+{\widehat{S}}_2^z\right)+{\omega }_{\mathrm{f}}^{\phantom{z}}{\widehat{a}}^{†}\widehat{a},\hfill \\ \hfill \widehat{V}=& \mathsf{\Omega }\left\{{\widehat{a}}^{†}\left({\widehat{S}}_1^{-}+{\widehat{S}}_2^{-}\right)+\widehat{a}\left({\widehat{S}}_1^{+}+{\widehat{S}}_2^{+}\right)\right\}.\hfill \end{array}$$

(1)

Here, $\mathsf{\Omega }$ is the coupling constant of the NV spin with the magnetic tip, ${\widehat{a}}^{†},\widehat{a}$ are the phononic creation and annihilation operators describing the cantilever oscillations with the frequency ${\omega }_{\mathrm{f}}$, and ${\omega }_0$ is the frequency of the NV spin. For a more detailed discussion of the parameters, we refer to [51] and the references therein.

The spins’ interaction with the magnetic tip $\widehat{V}$ leads to an effective indirect interaction between NV spins mediated by phonons. The experimental relevance of the model relies on the fabrication of cantilevers with frequencies comparable with the NVs’ spin transition frequencies. We consider the analytical solution in the resonant and highly off-resonant cases. These two situations are distinguished by the detuning of the cantilever’s oscillation with regard to the frequency of the NV spins. For strong detuning, we employ Fröhlich’s method [56] to infer the effective interaction ${\widehat{H}}_{\mathrm{eff}}^{\left(1\right)}$ and then the effective Hamiltonian ${\widehat{H}}_{\mathrm{eff}}$:

$$\begin{array}{cc}\hfill {\widehat{H}}_{\mathrm{eff}}^{\left(1\right)}=& \frac{\mathrm{i}}{2}\underset{-\infty }{\overset{0}{\int }}\mathrm{d}\tau \left[\widehat{V}\left(\tau \right),\widehat{V}\left(0\right)\right],\hfill \\ \hfill \widehat{V}\left(t\right)=& exp\left(-\mathrm{i}{\widehat{H}}_{0}t\right)\widehat{V}exp\left(\mathrm{i}{\widehat{H}}_{0}t\right),\hfill \end{array}$$

(2)

or explicitly [57]:

$${\widehat{H}}_{\mathrm{eff}}=H_0+\frac{{\mathsf{\Omega }}^2}{{\omega }_0-{\omega }_{\mathrm{f}}}\left\{1+{\widehat{S}}_1^{+}{\widehat{S}}_2^{-}+{\widehat{S}}_1^{-}{\widehat{S}}_2^{+}+\left({\widehat{S}}_1^z+{\widehat{S}}_2^z\right)\left(2{\widehat{a}}^{†}\widehat{a}+1\right)\right\}.$$

(3)

In what follows, we present analytic solutions for the model Equations (1) and (3).

Our description of the problem is quite general. However, without loss of generality, we specify the values of the parameters relevant to the NV centers: Cantilever of mass $m=6\times {10}^{-17}$ kg performs oscillations of the frequency $\frac{{\omega }_r}{2\pi }=0.1-10$ MHz. The cavity mode $\frac{{\omega }_{\mathrm{f}}}{2\pi }=5$ MHz, and the coupling between cantilever and NV centers $\frac{\mathsf{\Omega }}{2\pi }=100$ kHz. The amplitude of the zero point fluctuations $a_0=\sqrt{\hslash /2m{\omega }_r}\approx 5\times {10}^{-3}$ m defines the characteristic energy scale of the problem $m{\omega }_r^2{a}_0^2\approx {10}^{-9}$ J, and the time scale is of the order of microseconds.

3. The Analytical Solution

The model Equations (1) and (3) admit an exact analytical solution through the following ansatz:

$$\begin{array}{cc}\hfill \left|\varphi \right(t)\rangle =& \sum _{n=0}^{\infty }\left\{{\mathsf{\Gamma }}_{n+1,eg}\left(t\right)\right|n+1,eg\rangle +{\mathsf{\Gamma }}_{n+1,ge}\left(t\right)|n+1,ge\rangle +{\mathsf{\Gamma }}_{n+2,gg}\left(t\right)|n+2,gg\rangle +{\mathsf{\Gamma }}_{n,ee}\left(t\right)|n,ee\rangle \}.\hfill \end{array}$$

(4)

Here, $|n\rangle$ is the state of the field with n phonons [49], and $|e\rangle \equiv |1\rangle ,|g\rangle \equiv |0\rangle$ corresponds to the excited (ground) spin states of the NV centers. The explicit values of the coefficients ${\mathsf{\Gamma }}_{n+1,eg}\left(t\right),{\mathsf{\Gamma }}_{n+1,ge}\left(t\right),{\mathsf{\Gamma }}_{n+2,gg}\left(t\right),{\mathsf{\Gamma }}_{n,ee}\left(t\right)$ are presented in theAppendix A. In what follows, we utilize these coefficients and construct the density matrix $\widehat{\varrho }=\left|\phantom{\varphi \left(t\right)}\varphi \left(t\right)\right.〉\left.〈\phantom{\varphi \left(t\right)}\varphi \left(t\right)\right|$ corresponding to the model Equations (1) and (3).

4. The Invasive Measurement

In a magnetic resonance force microscopy experiment, the information about the state of the NV spin is read out through the magnetic tip. A measurement done on the spin A may perturb the state of the spin A directly and simultaneously may alter the state of the spin B indirectly through the quantized cantilever field. This is a case of non-local invasiveness and the non-local quantum witness. On the other hand, a measurement conducted directly on the phonon (cantilever) subsystem may influence the state of the NV spins directly (local witness). We start with studying the local witness.

4.1. Local Invasive Measurement

Suppose the measured phonon number is n. The measurement done on the subsystem f projects the wave function $\left|\varphi \right(t)\rangle$ (see Equation (4)) to the post-measurement state:

$$\begin{array}{cc}\hfill & |\mathsf{\Phi }\left(t\right)\rangle =\frac{\left({\mathsf{\Pi }}_n^{\mathrm{f}}⨂I^{\mathrm{s}}\right)\left|\varphi \left(t\right)\right.〉}{\sqrt{\left.〈\phantom{\varphi \left(t\right)}\phantom{{\mathsf{\Pi }}_n^{\mathrm{f}}⨂I^{\mathrm{s}}}\varphi \left(t\right)\right|{\mathsf{\Pi }}_n^{\mathrm{f}}⨂I^{\mathrm{s}}\left|\varphi \left(t\right)\phantom{\varphi \left(t\right)}\phantom{{\mathsf{\Pi }}_n^{\mathrm{f}}⨂I^{\mathrm{s}}}\right.〉}}.\hfill \end{array}$$

(5)

The post-measurement density matrix is given by:

$$\begin{array}{cc}\hfill & {\widehat{\varrho }}_{\mathrm{post}}=\left|\phantom{\mathsf{\Phi }\left(t\right)}\mathsf{\Phi }\left(t\right)\right.〉\left.〈\phantom{\mathsf{\Phi }\left(t\right)}\mathsf{\Phi }\left(t\right)\right|=\frac{{\mathsf{\Pi }}_m^{\mathrm{f}}\widehat{\varrho }{\mathsf{\Pi }}_m^{\mathrm{f}}}{\mathrm{Tr}\left({\mathsf{\Pi }}_m^{\mathrm{f}}\widehat{\varrho }{\mathsf{\Pi }}_m^{\mathrm{f}}\right)}.\hfill \end{array}$$

(6)

Here, $I^{\mathrm{s}}$ is the identity operator acting on the spin subsystem, and ${\mathsf{\Pi }}_n^{\mathrm{f}}=|n\rangle \langle n|$ is the phonon projection operator.

To quantify the quantum witness $\mathcal{W}$, we note that the system evolution is describable by an arbitrary positive, trace-preserving map $\widehat{\mathcal{F}}$. The projector measurement operators for the spins we define as follows: ${\mathsf{\Pi }}_{j\alpha }^{\mathrm{s}}=|\alpha \rangle \langle \alpha |$. Here, $j=1,2$ enumerates the spins, and $\alpha =g,e$ stands for the ground and excited states. Then, for the (in the above sense) direct spin measurement probability, we deduce:

$$\begin{array}{c}\hfill {\mathcal{P}}_{\widehat{\varrho }}^{j\alpha }=\mathrm{Tr}\left\{{\mathsf{\Pi }}_{j\alpha }^{\mathrm{s}}\widehat{\mathcal{F}}\left[\widehat{\varrho }\right]\right\}.\end{array}$$

(7)

The indirect measurements probability reads:

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\widehat{\varrho }}^{j\alpha }=\mathrm{Tr}\left\{{\mathsf{\Pi }}_{j\alpha }^{\mathrm{s}}\widehat{\mathcal{F}}\left[{\widehat{\varrho }}_{\mathrm{post}}\right]\right\}.\end{array}$$

(8)

We note that in Equation (8), the first measurement (blind measurement) is done on the field subsystem. The trace preserving positive maps $\widehat{\mathcal{F}}\left[\widehat{\varrho }\right]={\displaystyle \sum _{\alpha }}{\widehat{L}}_{\alpha }\widehat{\varrho }{\widehat{L}}_{\alpha }^{†}$ and $\widehat{\mathcal{F}}\left[{\widehat{\varrho }}_{\mathrm{post}}\right]={\displaystyle \sum _{\alpha }}{\widehat{L}}_{\alpha }{\widehat{\varrho }}_{\mathrm{post}}{\widehat{L}}_{\alpha }^{†}$ are defined in terms of the Kraus operators $L_1=\sqrt{{\mu }_1}{|e\rangle \langle g|}_1\otimes {|g\rangle \langle e|}_2$ and $L_2=\sqrt{{\mu }_2}{|g\rangle \langle e|}_1\otimes {|e\rangle \langle g|}_2$, ${\mu }_1+{\mu }_2=1$, see Appendix B.

For the system Equations (7) and (8), the quantum witness [11] is defined as follows:

$$\begin{array}{c}\hfill {\mathcal{W}}_{\widehat{\varrho }}(j,\alpha )=\left|{\mathcal{P}}_{\widehat{\varrho }}^{j\alpha }-{\mathcal{Q}}_{\widehat{\varrho }}^{j\alpha }\right|.\end{array}$$

(9)

Let the measured phonon number be m, then the post-measurement density matrix Equation (6) takes the form:

$$\begin{array}{cc}\hfill {\widehat{\varrho }}_{\mathrm{post}}=& Z^{-1}\left\{|{\mathsf{\Gamma }}_{m,gg}{\left(t\right)|}^2\left|\phantom{m,gg}m,gg\right.〉\left.〈\phantom{m,gg}m,gg\right|+|{\mathsf{\Gamma }}_{m,eg}{\left(t\right)|}^2\left|\phantom{m,eg}m,eg\right.〉\left.〈\phantom{m,eg}m,eg\right|+{\left|{\mathsf{\Gamma }}_{m,ge}\left(t\right)\right|}^2\left|\phantom{m,ge}m,ge\right.〉\left.〈\phantom{m,ge}m,ge\right|\right.\hfill \\ \hfill & \phantom{Z^{-1}0}+|{\mathsf{\Gamma }}_{m,ee}{\left(t\right)|}^2\left|\phantom{m,ee}m,ee\right.〉\left.〈\phantom{m,ee}m,ee\right|+{\mathsf{\Gamma }}_{m,ge}\left(t\right){\mathsf{\Gamma }}_{m,eg}^{*}\left(t\right)\left|\phantom{m,eg}m,ge\right.〉\left.〈\phantom{m,ge}m,eg\right|+{\mathsf{\Gamma }}_{m,ge}\left(t\right){\mathsf{\Gamma }}_{m,gg}^{*}\left(t\right)\left|\phantom{m,gg}m,ge\right.〉\left.〈\phantom{m,ge}m,gg\right|\hfill \\ \hfill & \phantom{Z^{-1}0}+{\mathsf{\Gamma }}_{m,ge}\left(t\right){\mathsf{\Gamma }}_{m,ee}^{*}\left(t\right)\left|\phantom{m,ee}m,ge\right.〉\left.〈\phantom{m,ge}m,ee\right|+{\mathsf{\Gamma }}_{m,eg}\left(t\right){\mathsf{\Gamma }}_{m,gg}^{*}\left(t\right)\left|\phantom{m,gg}m,eg\right.〉\left.〈\phantom{m,eg}m,gg\right|\hfill \\ \hfill & \left.\phantom{Z^{-1}0}+{\mathsf{\Gamma }}_{m,eg}\left(t\right){\mathsf{\Gamma }}_{m,ee}^{*}\left(t\right)\left|\phantom{m,ee}m,eg\right.〉\left.〈\phantom{m,eg}m,ee\right|+{\mathsf{\Gamma }}_{m,gg}\left(t\right){\mathsf{\Gamma }}_{m,ee}^{*}\left(t\right)\left|\phantom{m,ee}m,gg\right.〉\left.〈\phantom{m,gg}m,ee\right|+\mathrm{h}.\mathrm{c}.\right\},\hfill \\ \hfill Z_m=& |{\mathsf{\Gamma }}_{m,gg}{\left(t\right)|}^2+|{\mathsf{\Gamma }}_{m,eg}{\left(t\right)|}^2+|{\mathsf{\Gamma }}_{m,ge}{\left(t\right)|}^2+{\left|{\mathsf{\Gamma }}_{m,ee}\left(t\right)\right|}^2.\hfill \end{array}$$

(10)

The explicit analytical expressions of ${\mathcal{P}}_{\widehat{\varrho }}^{j\alpha },{\mathcal{Q}}_{\widehat{\varrho }}^{j\alpha }$ are presented in theAppendix C.

After some calculations, for the measured phonon number m, we derive the analytical expression for the local quantum witness:

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\widehat{\varrho }}^L(1,e)& =\left|{\mathcal{P}}_{\widehat{\varrho }}^{1,e}-{\mathcal{Q}}_{\widehat{\varrho }}^{1,e}\right|={\mu }_1\left|\sum _{n=0}^{\infty }{\left|{\mathsf{\Gamma }}_{n,ge}\left(t\right)\right|}^2-\frac{|{\mathsf{\Gamma }}_{m,ge}{\left(t\right)|}^2}{Z_m}\right|.\hfill \end{array}$$

(11)

Here, $Z_m=2\left(|{\mathsf{\Gamma }}_{m,ge}{\left(t\right)|}^2+{\left|{\mathsf{\Gamma }}_{m,gg}\left(t\right)\right|}^2\right)$.

For $u\gg 1$ (meaning the large mean phonon number), we use the saddle-point method and perform summation over the field states to get an analytical expression for the quantum witness in resonant case ${\omega }_0={\omega }_{\mathrm{f}}$:

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\widehat{\varrho }}^L(1,e)\approx {\mu }_1{u}^2& \left|\frac{\left[1+\mathrm{erf}\left(u/\sqrt{2}\right)\right]}{4(2u^2+1)}{sin}^2\left(\sqrt{4u^2+2}\mathsf{\Omega }t\right)\right.\hfill \\ \hfill & \left.\phantom{|}-\frac{{sin}^2\left(\sqrt{4m+2}\mathsf{\Omega }t\right)}{2u^2{sin}^2\left(\sqrt{4m+2}\mathsf{\Omega }t\right)+4\frac{1+2m}{{(1-2m)}^2}{\left[m-1+mcos\left(\sqrt{4m-2}\mathsf{\Omega }t\right)\right]}^2}\right|\hfill \end{array}$$

(12)

For the off-resonant Hamiltonian Equation (3), the $\left|n,gg\right.〉$ state is decoupled. Therefore, one has to start from an excited state for the NV spins. Starting from the $\left|ge\right.〉$ and a coherent phonon state, we obtain:

$$\begin{array}{cc}\hfill & {\mathcal{W}}_{\widehat{\varrho }}^L(1,e)=\frac{{\mu }_1}{2}\left|cos\left(\frac{2{\mathsf{\Omega }}^2}{{\omega }_0-{\omega }_{\mathrm{f}}}t\right)\right|.\hfill \end{array}$$

(13)

A plot of the quantum witness is presented in Figure 2. From Figure 2, we see that the saddle-point method smears out slow modulation of the amplitude of the quantum witness that we observe in the exact solution. The modulation of the witness strength amplitude is inherently a quantum effect related to the entanglement between the cavity field and NV centers. The saddle-point, which is valid for $u^2\gg 1$, partially eliminates the entanglement effect. The witness is finite, but smaller than the maximum witness observed for the entangled states. The limit $u^2\to \infty$ for the first term in Equation (12) leads to a fast oscillation between zero and $1/4$, while the second term results in $1/2$. In this case, the resonant Equation (12) and non-resonant Equation (13) results are of the same order.

4.2. Non-Local Invasive Measurement

The protocol of non-local invasive measurement is as follows: At the moment of time $t=\tau$, quantum measurement is done on the spin A through the positive operator valued measure (POVM) projection operator ${\mathsf{\Pi }}_m^{\mathrm{A}}=|m\rangle \langle m|$. Thus, the post-measurement density matrix has the form:

$$\begin{array}{cc}\hfill & {\widehat{\varrho }}_S^{\mathrm{post}}=\frac{{\mathsf{\Pi }}_m^{\mathrm{A}}\widehat{\varrho }\left(t\right){\mathsf{\Pi }}_m^{\mathrm{A}}}{\mathrm{Tr}\left({\mathsf{\Pi }}_m^{\mathrm{A}}\widehat{\varrho }\left(t\right){\mathsf{\Pi }}_m^{\mathrm{A}}\right)}.\hfill \end{array}$$

(14)

Combining pre- and post-measurement density matrices, we find the density matrix at an arbitrary moment of time:

$$\begin{array}{cc}\hfill & {\widehat{\varrho }}_1\left(t\right)=\mathsf{\Theta }[\tau -t]\widehat{\varrho }+\mathsf{\Theta }[t-\tau ]{\widehat{\varrho }}_S^{\mathrm{post}}.\hfill \end{array}$$

(15)

Here, the explicit form of the post-measurement density matrix reads:

$$\begin{array}{}\begin{array}{}\hfill {\widehat{\varrho }}_S^{\mathrm{post},e}=Z_e^{-1}\sum _{n,m=0}^{\infty }\left\{{\mathsf{\Gamma }}_{n+1,eg}\left(t\right){\mathsf{\Gamma }}_{m+1,eg}^{*}\left(t\right)\left|\phantom{m+1,eg}n+1,eg\right.〉\left.〈\phantom{n+1,eg}m+1,eg\right|+{\mathsf{\Gamma }}_{n+1,eg}\left(t\right){\mathsf{\Gamma }}_{m,ee}^{*}\left(t\right)\left|\phantom{m,ee}n+1,eg\right.〉\left.〈\phantom{n+1,eg}m,ee\right|\right.\end{array}\\ \left.\phantom{Z_e^{-1}\sum _{n,m=0}^{\infty }\{}+{\mathsf{\Gamma }}_{n,ee}\left(t\right){\mathsf{\Gamma }}_{m+1,eg}^{*}\left(t\right)\left|\phantom{m+1,eg}n,ee\right.〉\left.〈\phantom{n,ee}m+1,eg\right|+{\mathsf{\Gamma }}_{n,ee}\left(t\right){\mathsf{\Gamma }}_{m,ee}^{*}\left(t\right)\left|\phantom{m,ee}n,ee\right.〉\left.〈\phantom{n,ee}m,ee\right|\right\},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill Z_e=& \sum _{n=0}^{\infty }\left(|{\mathsf{\Gamma }}_{n,ee}{\left(t\right)|}^2+{\left|{\mathsf{\Gamma }}_{n,eg}\left(t\right)\right|}^2\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\widehat{\varrho }}_S^{\mathrm{post},g}=& Z_g^{-1}\sum _{n,m=0}^{\infty }\left\{{\mathsf{\Gamma }}_{n+1,ge}\left(t\right){\mathsf{\Gamma }}_{m+1,ge}^{*}\left(t\right)\left|\phantom{m+1,ge}n+1,ge\right.〉\left.〈\phantom{n+1,ge}m+1,ge\right|\right.\hfill \\ \hfill & \phantom{Z_e^{-1}\sum _{n,m=0}^{\infty }\{}+{\mathsf{\Gamma }}_{n+1,ge}\left(t\right){\mathsf{\Gamma }}_{m+2,gg}^{*}\left(t\right)\left|\phantom{m+2,gg}n+1,eg\right.〉\left.〈\phantom{n+1,eg}m+2,gg\right|\hfill \\ \hfill & \phantom{Z_e^{-1}\sum _{n,m=0}^{\infty }\{}+{\mathsf{\Gamma }}_{n+2,gg}\left(t\right){\mathsf{\Gamma }}_{m+1,ge}^{*}\left(t\right)\left|\phantom{m+1,ge}n+2,gg\right.〉\left.〈\phantom{n+2,gg}m+1,ge\right|\hfill \\ \hfill & \left.\phantom{Z_e^{-1}\sum _{n,m=0}^{\infty }\{}+{\mathsf{\Gamma }}_{n+2,gg}\left(t\right){\mathsf{\Gamma }}_{m+2,gg}^{*}\left(t\right)\left|\phantom{m+2,gg}n+2,gg\right.〉\left.〈\phantom{n+2,gg}m+2,gg\right|\right\},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill Z_g=& \sum _{n=0}^{\infty }\left(|{\mathsf{\Gamma }}_{n,gg}{\left(t\right)|}^2+{\left|{\mathsf{\Gamma }}_{n,ge}\left(t\right)\right|}^2\right).\hfill \end{array}$$

$\mathsf{\Theta }[\dots ]$ is the Heaviside step function.

The non-local quantum witness is defined as:

$${\mathcal{W}}_{\widehat{\varrho }}^{NL}{(1,e)}_g=|{\mathcal{P}}_{{\widehat{\varrho }}_{\mathrm{post},g}}^{1,e}-{\mathcal{Q}}_{{\widehat{\varrho }}_{\mathrm{post},g}}^{1,e}|.$$

(18)

Using the saddle-point approximation for Equation (18) and taking into account Equation (12), we conclude that for the case $m\approx u^2$, ${\mathcal{W}}^{NL}\approx {\mathcal{W}}^L$ follows. Therefore, the invasiveness of the measurement is not enhanced in the case of indirect measurements.

5. Conclusions

The issue of the influence of a direct or indirect measurement on a composite quantum system can be quantified by the local and nonlocal quantum witness. For two effective quantum spins coupled to each other indirectly via the quantum oscillations of a cantilever, we derived an explicit expression for the local quantum witness ${\mathcal{W}}^L$ that measures the invasiveness of a measurement done on the cantilever on the quantum states and hence on a subsequent measurement done on spins. The non-local quantum witness ${\mathcal{W}}^{NL}$ delivers information on the reaction of the states of one spin when a measurement is performed on the other spin. Both spins are coupled only via the cantilever whose states can be steered into the classical regime. In both cases, invasiveness is the same ${\mathcal{W}}^L={\mathcal{W}}^{NL}$. The obtained result confirms that the quantum channel mediates in both cases disturbances from the first to the second quantum spins. We analyzed the resonant solution comparing the exact result with the saddle-point method (Figure 2). We saw that the saddle-point method smeared out the slow modulation of the amplitude of the quantum witness that we observed in the exact solution. The modulation of the witness strength amplitude is inherently a quantum effect related to the entanglement between the cavity field and NV centers. The saddle-point, which is valid for $u^2\gg 1$, partially eliminates the entanglement effect. The witness is finite, but smaller than the maximum witness observed for the entangled states.