On the connection between quantum nonlocality and phase sensitivity of two-mode entangled Fock state superpositions

Abstract

In two-mode interferometry, for a given total photon number N, entangled Fock state superpositions of the form $(|N-m\rangle _a|m\rangle _b+\mathrm{e}^{i (N-2m)\phi }|m\rangle _a|N-m\rangle _b)/\sqrt{2}$ have been considered for phase estimation. Indeed all such states are maximally mode-entangled and violate a Clauser–Horne–Shimony–Holt (CHSH) inequality. However, they differ in their optimal phase estimation capabilities as given by their quantum Fisher informations. The quantum Fisher information is the largest for the N00N state $(|N\rangle _a|0\rangle _b+\mathrm{e}^{i N\phi }|0\rangle _a|N\rangle _b)/\sqrt{2}$ and decreases for the other states with decreasing photon number difference between the two modes. We ask the question whether for any particular Clauser–Horne (CH) (or CHSH) inequality, the maximal values of the CH (or the CHSH) functional for the states of the above type follow the same trend as their quantum Fisher informations, while also violating the classical bound whenever the states are capable of sub-shot-noise phase estimation, so that the violation can be used to quantify sub-shot-noise sensitivity. We explore CH and CHSH inequalities in a homodyne setup. Our results show that the amount of violation in those nonlocality tests may not be used to quantify sub-shot-noise sensitivity of the above states.

Appendices

Appendix 1

Let $\rho$ be the density operator corresponding to the $(N-m)::m$ state of Eq. (1), i.e.,

$$\begin{aligned} \rho&=| (N-m)::m\rangle \langle (N-m)::m|\nonumber \\&=\frac{1}{2}\bigg [|N-m,m\rangle \langle N-m,m|+|m,N-m\rangle \langle m,N-m|\nonumber \\&\quad + \mathrm{e}^{-i(N-2m) \phi }|N-m,m\rangle \langle m,N-m|+\mathrm{e}^{i(N-2m) \phi }|m,N-m\rangle \langle N-m,m|\bigg ]. \end{aligned}$$

(22)

The logarithmic negativity $\mathcal {\varepsilon }$ of the state can be calculated using the absolute sum of the negative eigenvalues $\mathcal {N}=|\sum _{i}\lambda _i|$, $\lambda _i < 0$ of the partial transpose of the density operator $\rho ^{PT}$, as $\mathcal {\varepsilon }=\log {(1+2\mathcal {N})}$. The partial transpose of $\rho$ of Eq. (22) is given by

$$\begin{aligned} \rho ^{PT}&=| (N-m)::m\rangle \langle (N-m)::m|\nonumber \\&=\frac{1}{2}\bigg [|N-m,m\rangle \langle N-m,m|+|m,N-m\rangle \langle m,N-m|\nonumber \\&\quad + \mathrm{e}^{-i(N-2m) \phi }|N-m,N-m\rangle \langle m,m|+\mathrm{e}^{i(N-2m) \phi }|m,m\rangle \langle N-m,N-m|\bigg ]. \end{aligned}$$

(23)

Diagonalizing the off-diagonal terms, $\rho ^{PT}$ of Eq. (23) can be equivalently written as

$$\begin{aligned} \rho&=| (N-m):{:}m\rangle \langle (N-m)::m|\nonumber \\&=\frac{1}{2}\bigg [|N-m,m\rangle \langle N-m,m|+|m,N-m\rangle \langle m,N-m|+|\varphi _1\rangle \langle \varphi _1|-|\varphi _2\rangle \langle \varphi _2|\bigg ], \end{aligned}$$

(24)

where $|\varphi _1\rangle$ and $|\varphi _2\rangle$ are normalized two-mode states given by

$$\begin{aligned} |\varphi _1\rangle =\frac{1}{\sqrt{2}}\left( \mathrm{e}^{-i (N-m) \phi }|N-m,N-m\rangle +\mathrm{e}^{-i m \phi }|m,m\rangle \right) ,\nonumber \\ |\varphi _2\rangle =\frac{1}{\sqrt{2}}\left( \mathrm{e}^{-i (N-m) \phi }|N-m,N-m\rangle -\mathrm{e}^{-i m\phi }|m,m\rangle \right) . \end{aligned}$$

(25)

The eigenvalues of $\rho ^{PT}$ are $\{1/2,\ 1/2,\ 1/2,\ -1/2\}$. We notice that they are independent of the values of N and m. Thus, the logarithmic negativity of all $(N-m)::m$ states is $\log 2$.

Appendix 2

The correlation function $\Pi (\alpha , \beta )$ of Eq. (16), for an $(N-m)::m$ state of Eq. (1), is given by

$$\begin{aligned}&\langle (N-m)::m|\hat{\Pi }(\alpha )\otimes \hat{\Pi }(\beta )|(N-m)::m\rangle \nonumber \\&\quad =\frac{1}{2}\bigg [\langle N-m|\hat{D}(\alpha )(-1)^{\hat{n}_a} \hat{D}(-\alpha )|N-m\rangle \langle m|\hat{D}(\beta )(-1)^{\hat{n}_b} \hat{D}(-\beta )|m\rangle \nonumber \\&\qquad +\langle m|\hat{D}(\alpha )(-1)^{\hat{n}_a} \hat{D}(-\alpha )|m\rangle \langle N-m |\hat{D}(\beta )(-1)^{\hat{n}_b} \hat{D}(-\beta )|N-m\rangle \nonumber \\&\qquad +\{\mathrm{exp}(i(N-2m)\phi )\langle N-m|\hat{D}(\alpha )(-1)^{\hat{n}_a} \hat{D}(-\alpha )|m\rangle \nonumber \\&\qquad \times \langle m |\hat{D}(\beta )(-1)^{\hat{n}_b} \hat{D}(-\beta )|N-m\rangle +\mathrm{c.c.}\}\bigg ], \end{aligned}$$

(26)

where we have used the fact that $\hat{D}(\alpha )^{\dagger }=\hat{D}(-\alpha )$. Denoting displaced Fock states by $\hat{D}(\alpha )|n\rangle =|\alpha ,n\rangle$, Eq. (26) can be rewritten as

$$\begin{aligned}&\langle (N-m)::m|\hat{\Pi }(\alpha )\otimes \hat{\Pi }(\beta )|(N-m)::m\rangle \nonumber \\&\quad =\frac{1}{2}\bigg [\langle -\alpha ,N-m|(-1)^{\hat{n}_a}|-\alpha ,N-m\rangle \langle -\beta , m|(-1)^{\hat{n}_b}|-\beta ,m\rangle \nonumber \\&\qquad +\,\langle -\alpha ,m|(-1)^{\hat{n}_a}|-\alpha ,m\rangle \langle -\beta , N-m|(-1)^{\hat{n}_b}|-\beta ,N-m\rangle \nonumber \\&\qquad +\,2{Re}\{\mathrm{exp}(i(N-2m)\phi )\langle -\alpha ,N-m|(-1)^{\hat{n}_a}|-\alpha ,m\rangle \langle -\beta , m|(-1)^{\hat{n}_b}|-\beta ,N-m\rangle \}\bigg ]. \end{aligned}$$

(27)

Using the number basis expansion of states of the form $|\alpha ,n\rangle$ as given in Ref. [39], one can show that

$$\begin{aligned} (-1)^{\hat{n}}|\alpha ,N-m\rangle =(-1)^{n}|-\alpha ,n\rangle . \end{aligned}$$

(28)

Therefore, Eq. (27) can be written as:

$$\begin{aligned}&\langle (N-m)::m|\hat{\Pi }(\alpha )\otimes \hat{\Pi }(\beta )|(N-m)::m\rangle \nonumber \\&\quad =\frac{(-1)^{N}}{2}\bigg [\langle -\alpha ,N-m|\alpha ,N-m\rangle \langle -\beta , m|\beta ,m\rangle \nonumber \\&\qquad +\,\langle -\alpha ,m|\alpha ,m\rangle \langle -\beta , N-m|\beta ,N-m\rangle \nonumber \\&\qquad +\,2{Re}\{\mathrm{exp}(i(N-2m)\phi )\langle -\alpha ,N-m|\alpha ,m\rangle \langle -\beta , m|\beta ,N-m\rangle \}\bigg ]. \end{aligned}$$

(29)

The inner product of displaced Fock states is given by [23, 40]

$$\begin{aligned} \langle -\alpha ,N-m|\alpha ,m\rangle =\mathrm{exp}(-2|\alpha |^2)&\sqrt{\frac{m!}{(N-m)!}}(2\alpha )^{N-2m}L_{N-m}^{N-2m}(4|\alpha |^2). \end{aligned}$$

(30)

Using Eq. (30) in Eq. (29), the correlation function $\Pi (\alpha , \beta )$ for the $(N-m)::m$ state is found to be of the form given in Eq. (17). Further, the two-mode Wigner function of the state can be written as

$$\begin{aligned} W(\alpha ,\beta )=\frac{4}{\pi ^2}\Pi (\alpha ,\beta ). \end{aligned}$$

(31)