1. Introduction
Quantum state discrimination plays an important role in quantum information processing [1]. Since it is impossible to discriminate two nonorthogonal quantum states perfectly, it is then a key task in state discrimination to determine the maximal success probability. The state discrimination problems can be divided into two categories: ambiguous [2–7] and unambiguous quantum state discrimination [9–19]. The lower bound of ambiguous state discrimination is given by Helstrom primarily in [2]. This bound can be saturated theoretically [3–5]. The error-free unambiguous state discrimination plays a key role in various contexts in quantum information theory, which can also be optimized through different channels [15–21].
Especially, as a kind of physical implementation of state discrimination, the discrimination of nonorthogonal coherent-state signals has attracted much attention [22–36]. Since the coherent states are easy to generate and have the best achievable signal-to-noise ratio during the propagation of information [37], the discrimination of coherent states is of great importance in quantum optics. Nevertheless, concerning the existing experiment protocols in physical implementation, there remains a significant gap between the ideal bound and the possible optimal result of the experiment for both ambiguous and unambiguous discriminations.
In order to reduce the deficiency, the coherent light states are coupled with an auxiliary atom and discriminated by measuring the ancilla. This is equivalent to performing a positive operator-valued measure (POVM) on the light states according to Naimark’s theorem. The interaction between the field and the atom can be characterized via the Jaynes–Cummings (JC) model [37]. The general JC model is difficult to solve accurately. Under rotation wave approximation (RWA), where the non-RWA terms in the Hamiltonian are neglected, an exact solution of the JC model can be obtained. The light state discrimination implemented via the JC model under RWA is discussed in [37]. If the post-measured light state is not thoroughly destroyed, the remaining information can be used for another measurement. These sequential measurements are performed on a sequence of auxiliary atoms. This nondestructive scheme with sequential measurement has been successfully applied in [37] to reduce the failure probability and acquire results approaching the ideal bound.
In order to get more accurate and practical results, the non-RWA terms in the atom-field Hamiltonian should be reserved. Namely, we consider the JC model without RWA. This non-RWA JC model can be solved by a perturbation method [38, 39]. It also includes the virtual-photon field causing the Lamb shift and quantum fluctuations [38, 39]. it is of great importance in quantum optics, e.g., in ensuring the causality of an atom–field coupling system.
From a theoretical point of view, the JC model under RWA is the simplest model that describes the interaction between an atom and a light field. Thus, the calculation for state discrimination of RWA gives only a rough estimation of the protocol based on the JC model. The RWA can be employed if the coupling between the atom and the light field is quite weak. From an experimental viewpoint, the JC model can be simulated by Josephson charge qubit coupling to an electromagnetic resonator [40, 41], a superconducting quantum interference device coupled with a nanomechanical resonator [42, 43] and an LC resonator magnetically coupled to a superconducting qubit [44]. With the progress of experimental technology, these artificial atoms may interact with on-chip resonant circuits [40–45] very strongly, and the RWA cannot describe well the strong-coupling regime [46]. Hence, from the above two aspects, it is also very desirable to investigate state discrimination based on the model without RWA.
The present study aims to determine whether light states can be discriminated with a better result in a strong-coupling regime. Namely, we tend to optimize the coherent-state discrimination to a further step and find a minimum failure probability that is closer to the ideal lower bound in the framework of the non-RWA JC model than the existing results with RWA in [37]. In addition, we also intend to determine the roles played by QEOF, which are included in the JC model without RWA, in quantum state discrimination.
This paper is organized as follows. First, we present an optimal ambiguous discrimination of two coherent states via the JC model without RWA. We show that the minimum probability of failure is reduced and the ideal Helstrom bound can be approached and further compared with the results of the JC model with RWA. In addition, we study a protocol for the physical implementation of unambiguous discrimination of coherent states including nondestructive sequential measurements. We show that under the effect of the non-RWA term, the discrimination of light states can be enhanced. We illustrate the relation between such superiority and the number of sequential measurements. We present a summary in the last section.
2. Ambiguous discrimination
We first consider the ambiguous discrimination of the well-known coherent states ∣ψ1⟩ = ∣α⟩ and ∣ψ2⟩ = ∣ − α⟩ occurring with probabilities P1 and P2, respectively, where2 ). By a straightforward calculation, one can see that the Helstrom bound (minimum failure probability) ${P}_{A}=\tfrac{1}{2}(1-\sqrt{1-4{P}_{1}{P}_{2}{s}^{2}})$ with s = ∣⟨ψ1∣ψ2⟩∣ is saturated when ∣χ1⟩ = ∣χ2⟩ and ∣φ1⟩ = ∣φ2⟩.
$\begin{eqnarray}| \pm \alpha \rangle =\displaystyle \sum _{n=0}^{\infty }{F}_{n}(\pm \alpha )| n\rangle ,\end{eqnarray}$
with ${F}_{n}(\pm \alpha )={{\rm{e}}}^{-| \alpha {| }^{2}/2}\tfrac{{\left(\pm \alpha \right)}^{n}}{\sqrt{n!}}.$ In order to discriminate ∣ψ1⟩ from ∣ψ2⟩ ambiguously, we couple the system with an ancilla state ∣k⟩ and perform a joint unitary transformation U [18], $\begin{eqnarray}\begin{array}{rcl}U| {\psi }_{1}\rangle | k\rangle & = & \sqrt{1-{r}_{1}}| {\chi }_{1}\rangle | 1\rangle +\sqrt{{r}_{1}}| {\phi }_{1}\rangle | 2\rangle ,\\ U| {\psi }_{2}\rangle | k\rangle & = & \sqrt{{r}_{2}}| {\chi }_{2}\rangle | 1\rangle +\sqrt{1-{r}_{2}}| {\phi }_{2}\rangle | 2\rangle ,\end{array}\end{eqnarray}$
where {∣1⟩, ∣2⟩} is an orthogonal basis of the ancilla. Through a von Neumann measurement {∣1⟩⟨1∣, ∣2⟩⟨2∣} on the ancilla, the states ∣ψi⟩ are identified with the error probability $\begin{eqnarray*}{P}_{\mathrm{err}}={P}_{1}{r}_{1}+{P}_{2}{r}_{2},\end{eqnarray*}$
where r1 and r2 satisfy $\langle {\psi }_{1}| {\psi }_{2}\rangle =\sqrt{(1-{r}_{1}){r}_{2}}\langle {\chi }_{1}| {\chi }_{2}\rangle \,+\sqrt{(1-{r}_{2}){r}_{1}}\langle {\phi }_{1}| {\phi }_{2}\rangle $ according to equation (The detailed physical system contains light fields (coherent states) interacting with a two-level atom prepared in the ground state ∣g⟩, which plays as the auxiliary qubit in state discrimination.
In order to get more accurate and practical results and determine the roles played by QEOF in state discrimination, we consider the JC model without RWA. Namely, we do not neglect the counterrotating terms in the Hamiltonian [38], which can be written as5 ) represent the virtual-photon process resulting from the system without RWA.
$\begin{eqnarray}H={H}_{0}+{H}_{{I}_{0}},\end{eqnarray}$
with $\begin{eqnarray}{H}_{0}=\omega {a}^{\dagger }a+\displaystyle \frac{1}{2}{\hslash }{\omega }_{0}{\sigma }_{z},\end{eqnarray}$
and $\begin{eqnarray}{H}_{{I}_{0}}=g({\sigma }_{+}a+{a}^{\dagger }{\sigma }_{-}+{a}^{\dagger }{\sigma }_{+}+a{\sigma }_{-}),\end{eqnarray}$
where &ohgr; is the frequency of the light field, &ohgr;0 is the frequency of the atomic transition, σ+ and σ− denote the atomic raising and lowering operators, a and a† are the field annihilation and creation operators, g is the atom–field coupling constant. We focus on the situation where the dipole coupling is on resonance, i.e., &ohgr; = &ohgr;0. The last two terms in equation (In the interaction picture, the interaction Hamilton of the atom–field coupling becomes7 ) into the Schrödinger equation, one has that
$\begin{eqnarray}{H}_{I}=g({\sigma }_{+}a+{a}^{\dagger }{\sigma }_{-})+{a}^{\dagger }{\sigma }_{+}{{\rm{e}}}^{2{\rm{i}}\omega t}+a{\sigma }_{-}{{\rm{e}}}^{-2{\rm{i}}\omega t}.\end{eqnarray}$
The evolution of the system characterized by the state vector ∣$\Psi$(± α, t)⟩ is given by the Schrödinger equation, $\begin{eqnarray*}{\rm{i}}\displaystyle \frac{\partial }{\partial t}| {\rm{\Psi }}(\pm \alpha ,t)\rangle ={H}_{I}| {\rm{\Psi }}(\pm \alpha ,t)\rangle ,\end{eqnarray*}$
with its solution of the following form: $\begin{eqnarray}| {\rm{\Psi }}(\pm \alpha ,t)\rangle =\displaystyle \sum _{n=0}^{\infty }[{A}_{n}(\pm \alpha ,t)| e,n\rangle +{B}_{n}(\pm \alpha ,t)| g,n\rangle ].\end{eqnarray}$
Substituting ( $\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\displaystyle \frac{\partial }{\partial t}{A}_{n} & = & g({B}_{n+1}\sqrt{n+1}+{B}_{n-1}\sqrt{n}{{\rm{e}}}^{2{\rm{i}}\omega t}),\\ {\rm{i}}\displaystyle \frac{\partial }{\partial t}{B}_{n} & = & g({A}_{n-1}\sqrt{n}+{A}_{n+1}\sqrt{n+1}{{\rm{e}}}^{-2{\rm{i}}\omega t}).\end{array}\end{eqnarray}$
The solution ∣$\Psi$(± α, t)⟩ depends on the coefficients An(± α, t) and Bn(± α, t) given by $\begin{eqnarray}\begin{array}{rcl}{A}_{n}(\pm \alpha ,t) & = & C[{A}_{n}^{0}(\pm \alpha ,t)+{A}_{n}^{{\prime} }(\pm \alpha ,t)],\\ {B}_{n}(\pm \alpha ,t) & = & C[{B}_{n}^{0}(\pm \alpha ,t)+{B}_{n}^{{\prime} }(\pm \alpha ,t)],\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{A}_{n}^{0}(\pm \alpha ,t) & = & -\mathrm{isin}(\sqrt{n+1}{gt}){F}_{(n+1)}(\pm \alpha ),\\ {B}_{n}^{0}(\pm \alpha ,t) & = & \cos (\sqrt{n}{gt}){F}_{n}(\pm \alpha ),\\ {A}_{n}^{{\prime} }(\pm \alpha ,t) & = & g\displaystyle \frac{\sqrt{n}{F}_{n-1}(\pm \alpha )}{2}\left[\displaystyle \frac{1-{{\rm{e}}}^{{\rm{i}}{K}_{1}(n-1)t}}{{K}_{1}(n-1)}\right.\\ & & +\,\left.\displaystyle \frac{1-{{\rm{e}}}^{{\rm{i}}{K}_{2}(n-1)t}}{{K}_{2}(n-1)}\right],\\ {B}_{n}^{{\prime} }(\pm \alpha ,t) & = & g\displaystyle \frac{\sqrt{n+1}{F}_{n+2}(\pm \alpha )}{2}\left[\displaystyle \frac{1-{{\rm{e}}}^{-{\rm{i}}{K}_{2}(n+2)t}}{{K}_{2}(n+2)}\right.\\ & & -\,\left.\displaystyle \frac{1-{{\rm{e}}}^{-{\rm{i}}{K}_{1}(n+2)t}}{{K}_{1}(n+2)}\right],\end{array}\end{eqnarray}$
with ${K}_{1}(n)=2\omega +\sqrt{n}g$, ${K}_{2}(n)=2\omega -\sqrt{n}g$ and C as a normalization factor.For simplicity, we suppose that the terms $\tfrac{g\sqrt{n}}{{K}_{1}(n-1)}$, $\tfrac{g\sqrt{n}}{{K}_{2}(n-1)}$, $\tfrac{g\sqrt{n+1}}{{K}_{1}(n+2)}$ and $\tfrac{g\sqrt{n+1}}{{K}_{2}(n+2)}$ are not large so that the perturbation theory can be applied. An0 and Bn0 (${A}_{n}^{{\prime} }$ and ${B}_{n}^{{\prime} }$) in (9 ) are a zero-ordered (first-ordered) approximation corresponding to the solution of the JC model with (without) RWA in [38]. ${A}_{n}^{{\prime} }$ and ${B}_{n}^{{\prime} }$ represent the influence of the virtual-photon processes on An0 and Bn0 [38]. From equation (10 ), one can see that ${A}_{n}^{{\prime} }$ and ${B}_{n}^{{\prime} }$ remain infinitesimal only when the radiation field is not intensive. If all of the four terms exist, that is the non-RWA case; otherwise, we will return to the case in [37] with RWA if both ${A}_{n}^{{\prime} }$ and ${B}_{n}^{{\prime} }$ are neglected.
Since the discrimination of light states is dependent on the ancilla—the atoms, a von Neumann measurement of the atomic states can induce the successful discrimination of the light states. Tracing over the variables of the field, the atomic state is acquired as
$\begin{eqnarray}\begin{array}{rcl}\rho (\pm \alpha ,t) & = & {\mathrm{Tr}}_{F}| \psi (\pm \alpha ,t)\rangle \langle \psi (\pm \alpha ,t)| \\ & = & \displaystyle \sum _{n=0}^{\infty }\left(\begin{array}{cc}{A}_{n}{A}_{n}^{* } & {A}_{n}{B}_{n}^{* }\\ {A}_{n}^{* }{B}_{n} & {B}_{n}{B}_{n}^{* }\end{array}\right).\end{array}\end{eqnarray}$
The error probability for discriminating the two coherent states ∣α⟩ and ∣−α⟩ is lower bounded by $\begin{eqnarray*}{P}_{\mathrm{err}}^{\min }=\mathop{\min }\limits_{t}\displaystyle \frac{1}{2}[1-2D(\alpha ,t)],\end{eqnarray*}$
with $\begin{eqnarray}\begin{array}{rcl}D(\alpha ,t) & = & \mathop{\max }\limits_{{\rm{\Pi }}}| \mathrm{Tr}{\rm{\Pi }}\,[{P}_{1}\rho (\alpha ,t)-{P}_{2}\rho (-\alpha ,t)]| \\ & & -\displaystyle \frac{1}{2}| {P}_{1}-{P}_{2}| .\end{array}\end{eqnarray}$
We call D(α, t) the trace distance, which is closely related to the successful probability of discriminating the two coherent states. The projective measurement operator for state discrimination can be written as Π(r, θ) = ∣φ⟩⟨φ∣, where $| \phi \rangle =\tfrac{1}{\sqrt{1+{r}^{2}}}(| g\rangle +{{\rm{e}}}^{{\rm{i}}\theta }r| e\rangle )$ with real parameters r and θ. In order to minimize the failure probability (to maximize D(α, t)), one should seek the optimal measurement and the parameter gt. Through the contour plot of D versus the parameters r and θ, one can easily find that the optimal D is obtained at θ = ±π/2 and ∣r∣ = 1, which is the same result as in [37] with RWA.Since it is difficult to have a large value of gt experimentally, we chose the scope of gt as 0 ≤ gt ≤ 10 [37]. In figure 1, we present the trace distance D[(α, t)] against gt. Compared with the result with RWA in [37], the global optimal successful probability for identifying the two light states is increased from 0.9896 (at gt = 0.3960) to 0.9960 (at gt = 0.3636). In addition, the result is enhanced obviously when the RWA is excluded for α = 2 rather than α = 0.5, 1.
Figure 1. Trace distance D[ρ(α, t), ρ(−α, t)] against gt for α = 2, θ = π/2, and r = 1. The solid and dotted lines correspond to the cases with and without RWA, respectively. |
The minimum error probability ${P}_{\mathrm{err}}^{\min }$ of discriminating the two coherent states versus the average photon numbers ∣α∣2 is shown in figure 2. The result of the JC model without RWA is superior to the one with RWA for ∣α∣2 > 3. Thus, the ideal Helstrom bound can be approached to a further step and one can acquire higher precision of state discrimination via adjusting the average photon number of coherent states to a proper value. Since the inner product of ∣α⟩ and ∣−α⟩ increases with the average photon number ∣α∣2, one concludes that as the inner product of states increases, the superiority of light state discrimination of the JC model without RWA versus the one with RWA is enhanced, obviously.
Figure 2. The minimum error probability for ambiguous discrimination of coherent states ∣α⟩ and ∣−α⟩ versus the average photon number ∣α∣2. Solid line: the ideal Helstrom bound; dotted (dotted-dashed) line: the minimum error probability for the discrimination of the two coherent states relying on the JC model with (without) RWA, respectively. |
The local maximum 0.0196 (0.0118) in the dotted-dashed (dotted) line in figure 2 can be acquired for the JC model without (with) RWA at gt = 0.58, ∣α∣2 = 2.35 (∣α∣2 = 1.65, gt = 8.43). These results, corresponding to the minimum optimal success probability, imply that the effect of the non-RWA term in equation (9 ) is impaired. One can draw an opposite conclusion for the model including RWA.
In contrast, with relative significance the local minimum of Perror, 0.0084 (0.0069) denoted by the dotted-dashed (dotted) line in figure 2, is acquired at gt = 8.41, ∣α∣2 = 1.22 (gt = 8.35, ∣α∣2 = 1.15) for the JC model without (with) RWA. From figure 2, it can be seen that the difference between the local minimum of Perror for the JC model with and without RWA is much smaller than the one for local maxima.
Depending on detailed physical systems, the ideal bound (e.g. the Helstrom bound for ambiguous discrimination) may not be always saturated. Some information may still be left in the post-measured states. The observer can perform a subsequent measurement to proceed with the discrimination operation. This is called the nondestructive scheme. The larger the result of the first measurement deviates from the ideal bound, the more superiority this nondestructive scheme has. Since such a state discrimination scheme with sequential measurement bases on the failure of the first measurement, it completely differs from the sequential state discrimination mentioned in [8] and [13, 19] for ambiguous and unambiguous discriminations, respectively, where the two measurements are performed by different observers and unrelated to each other.
Since unambiguous state discrimination requires zero errors, there exist additional constraints on the POVM operators. Then, the unambiguous discrimination tends to be more possibly failed than the ambiguous one, see theorem 1 in the Supplemental Material. Then, we will present the implementation of unambiguous state discrimination with sequential measurement performed by the Kennedy receiver as a special instance [37].
3. Unambiguous state discrimination with sequential measurement
In unambiguous discrimination of N (N ≥ 2) nonorthogonal states, the projection measurements are applied to an extended space of at most 2N − 1 dimensions [47]. Thus, the discrimination of coherent states ∣α⟩ and ∣− α⟩ can be implemented via the JC model including three-level atoms, see the general scheme shown in the Supplemental Material. One may also tend to identify one of the states more successfully while ignoring the other state which can be realized via a displacement operator. Such asymmetric discrimination, called the ‘Kennedy receiver scheme’ [37], can be implemented physically via a two-level ancilla interacting with the light field via the JC model without RWA. Below we present the scheme for nondestructive implementation of the Kennedy receiver via the JC model without RWA, which unambiguously discriminates the coherent states.
Before interacting with the auxiliary atoms, we let the light fields in states ∣2α⟩ and ∣0⟩, obtained by applying the displacement operator D(α) to states ∣α⟩ and ∣−α⟩, respectively, interact with a sequence of atoms in ground state ∣g⟩. Suppose that the frequency and polarization of the coherent states match that of the atomic transition so that the atom can be excited from the ground state ∣g⟩ to the state ∣e⟩. We can discriminate the states ∣2α⟩ and ∣0⟩ unambiguously via a von Neumann measurement on the atomic states with respect to the basis {∣g⟩, ∣e⟩}.
Since the atomic level transition is prohibited for the vacuum state ∣0⟩ [37], the state ∣0⟩ is bound to be neglected. Then, the whole system evolves through the following unitary operator U such that7 ) with the parameters ±α replaced by 2α.
$\begin{eqnarray}\begin{array}{rcl}U| 2\alpha \rangle | g\rangle & = & D(2\alpha ,t)| \psi \rangle | e\rangle +E(2\alpha ,t)| \phi \rangle | g\rangle ,\\ U| 0\rangle | g\rangle & = & | 0\rangle | g\rangle ,\end{array}\end{eqnarray}$
with the functions D(2α, t) and E(2α, t) satisfying $\begin{eqnarray*}| D(2\alpha ,t){| }^{2}=\displaystyle \sum _{n=0}^{\infty }| {A}_{n}(2\alpha ,t){| }^{2},\end{eqnarray*}$
$\begin{eqnarray*}| E(2\alpha ,t){| }^{2}=\displaystyle \sum _{n=0}^{\infty }| {B}_{n}(2\alpha ,t){| }^{2},\end{eqnarray*}$
where the functions An(2α, t) and Bn(2α, t) are of the same form as in equation (Then, we can identify the state ∣2α⟩ through a von Neumann measurement ∣e⟩⟨e∣ on the auxiliary system. If the atom is found to be in the exited state (atomic level transition is successful), the discrimination is successful. The post-measured light states ∣ψ⟩ and ∣φ⟩ correspond to ∣e⟩ and ∣g⟩, respectively.
If the atom is found to be in the ground atomic state ∣g⟩, we perform a sequential measurement on the ancilla to identify the post-measured state ∣φ⟩. Suppose that the optimal success probability of the first measurement, denoted by the excitation probability of the atom, is ${P}_{A}={\max }_{t}\,| D(2\alpha ,t){| }^{2}$, which is acquired at t = t0. By straightforward calculation we have
$\begin{eqnarray*}| \phi \rangle =\displaystyle \frac{1}{\sqrt{\displaystyle \sum _{n=0}^{\infty }| {B}_{n}(2\alpha ,{t}_{0}){| }^{2}}}\displaystyle \sum _{n=0}^{\infty }{B}_{n}(2\alpha ,{t}_{0})| n\rangle .\end{eqnarray*}$
The state of the atom-field system is given by
$\begin{eqnarray*}| {\rm{\Psi }}(t^{\prime} )\rangle =\displaystyle \sum _{n=0}^{\infty }[{\tilde{A}}_{n}(2\alpha ,t^{\prime} )| e,n-1\rangle +{\tilde{B}}_{n}(2\alpha ,t^{\prime} )| g,n\rangle ],\end{eqnarray*}$
where $\begin{eqnarray}\begin{array}{rcl}{\tilde{A}}_{n}(2\alpha ,t^{\prime} ) & = & C^{\prime} [{\tilde{A}}_{n}^{0}(2\alpha ,t^{\prime} )+{\tilde{A}}_{n}^{{\prime} }(2\alpha ,t^{\prime} )],\\ {\tilde{B}}_{n}(2\alpha ,t^{\prime} ) & = & C^{\prime} [{\tilde{B}}_{n}^{0}(2\alpha ,t^{\prime} )+{\tilde{B}}_{n}^{{\prime} }(2\alpha ,t^{\prime} )],\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{\tilde{A}}_{n}^{0}(2\alpha ,t^{\prime} ) & = & -\mathrm{isin}(\sqrt{n+1}{gt}^{\prime} ){B}_{(n+1)}(2\alpha ,{t}_{0}),\\ {\tilde{B}}_{n}^{0}(2\alpha ,t^{\prime} ) & = & \cos (\sqrt{n}{gt}^{\prime} ){B}_{n}(2\alpha ,{t}_{0}),\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{\tilde{A}}_{n}^{{\prime} }(2\alpha ,t^{\prime} ) & = & \displaystyle \frac{\sqrt{n}{{gB}}_{n-1}(2\alpha ,{t}_{0})}{2}\left[\displaystyle \frac{1-{{\rm{e}}}^{{\rm{i}}{K}_{1}(n-1)t^{\prime} }}{{K}_{1}(n-1)}\right.\\ & & \left.+\displaystyle \frac{1-{{\rm{e}}}^{{\rm{i}}{K}_{2}(n-1)t^{\prime} }}{{K}_{2}(n-1)}\right],\\ {\tilde{B}}_{n}^{{\prime} }(2\alpha ,t^{\prime} ) & = & \displaystyle \frac{\sqrt{n+1}{{gB}}_{n+2}(2\alpha ,{t}_{0})}{2}\left[\displaystyle \frac{1-{{\rm{e}}}^{-{\rm{i}}{K}_{2}(n+2)t^{\prime} }}{{K}_{2}(n+2)}\right.\\ & & \left.-\displaystyle \frac{1-{{\rm{e}}}^{-{\rm{i}}{K}_{1}(n+2)t^{\prime} }}{{K}_{1}(n+2)}\right],\end{array}\end{eqnarray}$
${K}_{1}(n)=2\omega +\sqrt{n}g$, ${K}_{2}(n)=2\omega -\sqrt{n}g$, and $C^{\prime} $ is a normalization factor.The subsequent measurement identifies ∣φ⟩ with an optimal successful probability ${P}_{B}=\mathop{\max }\limits_{t^{\prime} }{\sum }_{n=0}^{\infty }| {\tilde{A}}_{n}(2\alpha ,t^{\prime} ){| }^{2}$. Then, see theorem 2 in the Supplemental Material, the total failure probability of these two sequential measurements is given by
$\begin{eqnarray*}{Q}_{\mathrm{SM}}=1-{P}_{1}[{P}_{A}+(1-{P}_{A}){P}_{B}].\end{eqnarray*}$
The minimum failure probability with only one measurement and two sequential measurements are shown as functions of the average photon number ∣α∣2 in figures 3(a) and (b), respectively. The ideal Kennedy bound,
$\begin{eqnarray}{Q}_{\min }={P}_{1}+{P}_{2}{{\rm{e}}}^{-4| \alpha {| }^{2}},\end{eqnarray}$
which is calculated in the Supplemental Material, is also shown in figure 3.
Figure 3. The minimum failure probability Qmin of unambiguous discrimination of coherent states ∣2α⟩ and ∣0⟩ against the average photon number ∣α∣2 with only the first measurement in (a) and two sequential measurements in (b). Solid line: the ideal Kennedy bound; dotted (dotted-dashed) line: the error probability of the JC model discrimination of the two coherent states with (without) RWA, respectively. |
It is seen that for the unambiguous state discrimination implemented via the JC model without RWA, the result is superior to the one with RWA except for the cases where the average photon numbers ∣α∣2 approach 2. Compared with the results of optimal ambiguous state discrimination, the superiority of the non-RWA JC model versus the one with RWA is more obvious, see figures 2 and 3(a). In figure 3(a), there exist two local minima 0.5507 (at ∣α∣2 = 1.15, gt = 0.70) and 0.5142 (at ∣α∣2 = 3.56, gt = 0.38) denoted by the dotted-dashed line. The local maximum also implies the weakening of the non-RWA term under the framework of unambiguous state discrimination with only one measurement. In the contrary, with respect to the local minimum 0.5142, the effect of the non-RWA term is strengthened, in coinciding with the global minimum of ${Q}_{\min }$. It is also shown that the minimum failure probability ${Q}_{\min }$ for the non-RWA JC model will increase asymptotically with ∣α∣2 when the average photon number is larger than the value corresponding to the global minimum. The failure probability will surpass the one with RWA again as ∣α∣2 > 4.07.
As the number of measurements increases, from figure 3(b) we have that the superiority of the results for the JC model without RWA is further enhanced. Namely, QEOF in the JC model without RWA may enhance the discrimination of quantum states, especially for the protocol with sequential measurements.
The protocol based on sequential measurement includes an initial pure ancilla state, which incurs additional resource overhead. We consider the cost of resources via the purity of the auxiliary states $F(\rho )=\mathrm{Tr}({\rho }^{2})$, see figure 4. It is shown that the result of the non-RWA JC model has a larger purity of the ancilla than the one with RWA. Consequently, one can conclude that unambiguous state discrimination implemented via the non-RWA JC model is beneficial to saving resource costs.
Figure 4. The purity F(ρ) of the auxiliary state against gt for ∣α∣2 = 3.56 (optimal global minimum of ${Q}_{\min }$). Solid (dotted) line corresponds to the result with (without) RWA. |
4. Conclusion
In summary, we have investigated the physical implementations of both the ambiguous and unambiguous state discriminations via JC interaction between the light field and atoms without RWA. It has been shown that for ambiguous state discriminations, only for larger phonon numbers the results for the JC model without RWA are superior to the one with RWA. For unambiguous state discrimination, we adopted the so-called Kennedy receiver protocol including sequential measurements. Compared with the results of ambiguous discrimination, the superiority of unambiguous state discrimination without RWA is enhanced compared with the one with RWA. This superiority becomes more obvious as the number of sequential measurements increases. These results may be attributed to the fact that the non-RWA term in equation (5 ) brings about the virtual-photon processes including QEOF in the JC model without RWA, which play a critical role in enhancing the discrimination of quantum states, especially for the protocol with sequential measurements. We also find that unambiguous state discrimination implemented via the non-RWA JC model is beneficial to saving resource costs. Since the non-RWA JC model always corresponds to a strong-coupling regime that is closer to a real practical physical system with respect to the actual experimental setups [40–46], it is desirable that our results can be verified experimentally.
Acknowledgments
This work was funded by the NSF of China (Grant Nos. 11675119, 12075159, 11575125, 12171044), Shanxi Education Department Fund (2020L0543), Beijing Natural Science Foundation (Z190005), Academy for Multidisciplinary Studies, Capital Normal University, the Academician Innovation Platform of Hainan Province, and the Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (No. SIQSE202001).
General schemes for unambiguous state discrimination
(1) Unambiguous state discrimination with one measurement.
In order to discriminate the state ∣ψ1⟩ from ∣ψ2⟩ (0 < ⟨ψ1∣ψ2⟩ = s < 1) unambiguously, we couple the system with a three-level ancilla state ∣0b⟩ and perform a joint unitary transformation Ub [19] such that
$\begin{eqnarray}{U}_{b}| {\psi }_{i}\rangle | {0}_{b}\rangle =\sqrt{{q}_{i}^{b}}| {\chi }_{i}\rangle | 0{\rangle }_{b}+\sqrt{1-{q}_{i}^{b}}| {\phi }_{i}\rangle | i{\rangle }_{b},\end{eqnarray}$
with 0 < ⟨χ1∣χ2⟩ = t ≤ 1, and perform a von Neumann measurement with respect to the basis {∣0⟩, ∣1⟩, ∣2⟩} on the auxiliary qutrit.The discrimination is successful if the ancilla collapses to the states ∣1⟩b or ∣2⟩b, and failed if the outcome state is ∣0⟩b. Hence, the average failure probability is given by
$\begin{eqnarray}{Q}^{(1)}={P}_{1}{q}_{1}^{b}+{P}_{2}{q}_{2}^{b},\end{eqnarray}$
with the constraint q1q2 = s2/t2 as Ub is unitary. The lower bound of the average failure probability is then given by $\begin{eqnarray}{Q}_{\min }^{(1)}=2\sqrt{{P}_{1}{P}_{2}}s,\end{eqnarray}$
which is saturated at t = 1 (optimal measurement) and ${q}_{1}^{b}=\sqrt{{P}_{2}/{P}_{1}}s$ under the condition that $0\lt s\lt \sqrt{{P}_{1}/{P}_{2}}$. For $\sqrt{{P}_{1}/{P}_{2}}\leqslant s\lt 1$, this lower bound reduces to P1 + P2s2 achieved at qb = 1, t = 1 with one state ∣$\Psi$1⟩ bound to be neglected.In the physical implementation of unambiguous state discrimination performed by Kennedy receiver mentioned in the main text, only two-level ancilla is allowed. Thus, one state has to be ignored, which implies that ${q}_{1}^{b}=1$. Hence, the minimum failure probability can be acquired via the following relations:
$\begin{eqnarray}\mathrm{minimize}:\,{Q}^{(2)}={P}_{1}+{P}_{2}{q}_{2}^{b},\end{eqnarray}$
$\begin{eqnarray}\mathrm{subject}\,\mathrm{to}:\,{q}_{2}^{b}\in [{s}^{2}/{t}^{2},1].\end{eqnarray}$
Thus, the lower bound ${Q}_{\min }^{(2)}={P}_{1}+{P}_{2}{s}^{2}$ is saturated at t = 1 and ${q}_{2}^{b}={s}^{2}$, which is of the same form as the one with three-level ancilla for $\sqrt{{P}_{1}/{P}_{2}}\leqslant s\lt 1$. Then, by setting ∣ψ1⟩ = ∣α⟩ and ∣ψ2⟩ = ∣−α⟩, the relation ${Q}_{\min }={P}_{1}+{P}_{2}{{\rm{e}}}^{-4| \alpha {| }^{2}}$ can be acquired easily. Comparing the results of ambiguous and unambiguous state discrimination, we have the following theorem.The minimum failure probability of unambiguous state discrimination is superior to the one of ambiguous state discrimination.
The difference in minimum failure probability between ambiguous and unambiguous discrimination satisfies
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}Q & = & {Q}_{\min }^{(1)}-{P}_{\mathrm{err}}\\ & = & 2\sqrt{{P}_{1}{P}_{2}}s-\displaystyle \frac{1}{2}(1-\sqrt{1-4{P}_{1}{P}_{2}{s}^{2}})\\ & = & {Q}_{\min }^{(1)}+\displaystyle \frac{1}{2}\sqrt{1-{Q}_{\min }^{(1)2}}-\displaystyle \frac{1}{2}\\ & \geqslant & \sqrt{\displaystyle \frac{1}{4}+\displaystyle \frac{3}{4}{Q}_{\min }^{(1)2}}-\displaystyle \frac{1}{2}\geqslant 0.\end{array}\end{eqnarray}$
Therefore, we have $\begin{eqnarray*}{Q}_{\min }^{(2)}={P}_{1}+{P}_{2}{s}^{2}\geqslant 2\sqrt{{P}_{1}{P}_{2}}s={Q}_{\min }^{(1)}\geqslant {P}_{\mathrm{err}}.\end{eqnarray*}$
(2) Unambiguous state discrimination with sequential measurement.
Since the ideal bound for state discrimination cannot always be saturated in detailed physical implementation, the state discrimination operation in equation (A1 ) is not always optimal (e.g., the general case ⟨φ1∣φ2⟩ = t < 1). If the discrimination succeeds via the first measurement, this procedure ends. Otherwise, we couple the system with another three-level ancilla state ∣0c⟩ and continue to discriminate the post-measured state via another unitary operation Uc,A8 ), we have the following theorem.
$\begin{eqnarray}{U}_{c}| {\chi }_{i}\rangle | {0}_{c}\rangle =\sqrt{{q}_{i}^{c}}| \chi ^{\prime} \rangle | 0{\rangle }_{c}+\sqrt{1-{q}_{i}^{c}}| \phi {{\prime} }_{i}\rangle | i{\rangle }_{c},\end{eqnarray}$
and a subsequent von Neumann measurement on the auxiliary qutrit. Then, the success probability with respect to the first (second) measurement PA (PB) is given by $\begin{eqnarray}\begin{array}{rcl}{P}_{A} & = & 1-{Q}^{(1)}={P}_{1}(1-{q}_{1}^{b})+{P}_{2}(1-{q}_{2}^{b}),\\ {P}_{B} & = & {P}_{1}^{0}(1-{q}_{1}^{c})+{P}_{2}^{0}(1-{q}_{2}^{c}),\end{array}\end{eqnarray}$
where ${P}_{i}^{0}=\tfrac{{P}_{i}{q}_{i}^{b}}{{P}_{1}{q}_{1}^{b}+{P}_{2}{q}_{2}^{b}}$ (i = 1, 2) is the conditional prior probability of the post-measured states corresponding to the failure result of the first measurement. Hence, according to equation (The total failure probability ${Q}^{\mathrm{SM}}$ of the nondestructive scheme with sequential measurement is given by
$\begin{eqnarray}\begin{array}{rcl}{Q}^{\mathrm{SM}} & = & {P}_{1}{q}_{1}^{b}{q}_{1}^{c}+{q}_{2}^{b}{q}_{2}^{c}\\ & = & 1-{P}_{1}[{P}_{A}+(1-{P}_{A}){P}_{B}].\end{array}\end{eqnarray}$
In [20], the two measurements are performed by different observers. Since the latter measurement occurs on the premise that the first one fails, the classical communications between the two observers are needed. However, in our scheme realized via nondestructive implementation mentioned in the main text, the classical communications are not necessary. Then, according to the constraints $0\lt {q}_{1}^{c},{q}_{2}^{c}\leqslant 1$, ${q}_{1}^{c}{q}_{2}^{c}\leqslant {t}^{2}$ and the relation (A9 ), we have the following corollary.
If the first measurement is optimal (t = 1), we have ${Q}^{\mathrm{SM}}={Q}^{(1)}$. Otherwise, if $t\lt 1$ we have ${Q}^{\mathrm{SM}}\lt {Q}^{(1)}$.