Quantum phase estimation using path-symmetric entangled states
QQuantum phase estimation using path-symmetricentangled states
Su-Yong Lee , Chang-Woo Lee , Jaehak Lee , and Hyunchul Nha School of Computational Sciences, Korea Institute for Advanced Study, Hoegi-ro 85,Dongdaemun-gu, Seoul02455, Korea Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore,Singapore Department of Physics, Texas A&M University at Qatar, Education City, POBox 23874, Doha, Qatar * [email protected] ABSTRACT
We study the sensitivity of phase estimation using a generic class of path-symmetric entangled states | ϕ (cid:105)| (cid:105) + | (cid:105)| ϕ (cid:105) , where anarbitrary state | ϕ (cid:105) occupies one of two modes in quantum superposition. With this generalization, we identify the fundamentallimit of phase estimation under energy constraint that is characterized by the photon statistics of the component state | ϕ (cid:105) .We show that quantum Cramer-Rao bound (QCRB) can be indefinitely lowered with super-Poissonianity of the state | ϕ (cid:105) . Forpossible measurement schemes, we demonstrate that a full photon-counting employing the path-symmetric entangled statesachieves the QCRB over the entire range [ , π ] of unknown phase shift φ whereas a parity measurement does so in a certainconfined range of φ . By introducing a component state of the form | ϕ (cid:105) = √ q | (cid:105) + √ − q | N (cid:105) , we particularly show that anarbitrarily small QCRB can be achieved even with a finite energy in an ideal situation. This component state also providesthe most robust resource against photon loss among considered entangled states over the range of the average input energy N av > . Finally we propose experimental schemes to generate these path-symmetric entangled states for phase estimation. Introduction
It is a task of fundamental and practical interest to identify the ultimate precision in measuring an unknown parameter of asystem. In particular, estimating an unknown phase has many important applications such as the observation of gravitationalwaves and detection of weak signals or defects leading to the design of highly sensitive sensors . Typically, an interferometricscheme, e.g. Mach-Zehnder interferometer (MZI), is employed to measure an unknown phase φ present along one arm of theinterferometer and its sensitivity δ φ is investigated under the constraint of average input energy N av . If a coherent state is usedas an input to the MZI, it is well known that the phase sensitivity is given by δ φ ∼ / √ N av , referred to as shot-noise limit. Onthe other hand, the phase sensitivity can be further enhanced to δ φ ∼ / N av by using quantum resources such as squeezedstates and entangled states under ideal conditions. However, as photons carrying information undergo a loss mechanism inrealistic situations, it becomes a practically important question what kind of resources and measurement schemes can yieldphase sensitivity robust against loss.Here we consider the problem of local quantum estimation which aims at minimizing the variance of the estimator at a fixedvalue of a parameter . It is well known that NOON state provides Heisenberg limit (HL) in local quantum phase estimation .There arises a question what if the single-mode component state | N (cid:105) is replaced by an arbitrary state | ϕ (cid:105) . It is also known thatentanglement is not a crucial resource necessary to achieve Heisenberg limit under certain conditions , e.g. using separablesqueezed states . On the other hand, it is still an interesting and important issue to identify resource states, whether entangledor separable, which can give enhanced performance desirably under practical situations. We here address the problem ofestimating an unknown phase using a general class of two-mode entangled states as a probe, specifically those states representedby | ϕ (cid:105) a | (cid:105) b + | (cid:105) a | ϕ (cid:105) b . This generalized class of states is worth investigating comprehensively for a phase-estimation problem.First, NOON state extensively studied so far and the entangled-coherent state | α (cid:105) a | (cid:105) b + | (cid:105) a | α (cid:105) b known to give betterperformance than NOON state all belong to this class . Second, these states are all path-symmetric states according toRef. , which means, e.g., that a parity measurement can achieve the quantum Cram´er-Rao bound , opening a venue forpractical applications as well.The upper limit of quantum phase estimation is determined by quantum Fisher information (QFI) from an information-theoretic perspective . An arbitrarily large QFI with a finite energy is achievable, e.g., with a superposition of vacuum andsqueezed states and a specific form of superpositions of NOON states . It is also known that path-symmetric entangledpure states can achieve the QFI with photon counting experiments that resolve different photon numbers . As a specific a r X i v : . [ qu a n t - ph ] J u l easurement scheme, parity measurement was employed in trapped ion system and also in optical interferometry . Theparity measurement can be realized using photon number resolving detection. It may be practically demanding to observesub shot-noise limit due to required high-level of photon number resolution. Nevertheless, there were some experimentalprogresses overcoming this limit e.g. using silicon photo-multiplier detector or homodyne detection with sign-binningprocess , manifesting super-resolution of optical measurements in the shot-noise limit. For another estimation scheme, onecan use the whole photon-counting statistics on both output modes of MZI. This gives a classical Fisher information whoseinverse represents phase sensitivity, and it was also employed in experiment with a small-photon-number entangled state .In this paper, we introduce a generic class of path-symmetric entangled states and investigate their QFIs to obtain the limitof phase sensitivity for each resource state. In particular, we show that it is possible to achieve an arbitrarily large QFI usingthis class of states even with a finite energy in an ideal situation. An arbitrarily large QFI does not immediately lead to aninfinite precision of phase measurement in practice, requiring some prior information or an arbitrariliy many repetitions ofexperiment . Next, we consider two specific measurement schemes for phase estimation under a practical situation withloss, i.e. parity measurement at one output mode and photon-counting measurement at both output modes. We discuss how thephase sensitivity varies with the photon-counting statistics of the single-mode component state | ϕ (cid:105) under two measurementschemes. Finally, we propose experimental schemes to generate the path-symmetric entangled states under current technology. Results
Quantum Fisher Information of path-symmetric entangled states
We begin by introducing a path-symmetric entangled state, | ψ (cid:105) ab = (cid:112) ( + p ) ( | ϕ (cid:105) a | (cid:105) b + | (cid:105) a | ϕ (cid:105) b ) , (1)where an arbitrary single-mode state | ϕ (cid:105) can be expressed as | ϕ (cid:105) = ∑ ∞ n = e i θ n √ p n | n (cid:105) with ∑ ∞ n = p n = p = |(cid:104) | ϕ (cid:105)| isthe overlap between the vacuum | (cid:105) and the state | ϕ (cid:105) . The photon counting statistics (PCS) of a single-mode component | ϕ (cid:105) thus becomes a crucial factor for the performance of phase estimation. In particular, the Mandel Q -factor, Q M = (cid:104) ϕ | ∆ ˆ n | ϕ (cid:105) / (cid:104) ϕ | ˆ n | ϕ (cid:105) −
1, plays a dominant role in determining Quantum Fisher information (QFI) . The PCS with − ≤ Q M < Q M =
0, and 0 < Q M < ∞ are sub-Poissonian, Poissonian, and super-Poissonian statistics, occurring e.g. with aphoton number, a coherent, and a squeezed vacuum state, respectively. We study the connection of the Mandel Q factor tothe sensitivity of phase estimation. As shown below, given a fixed amount of input energy, the QFI is enhanced with a largevariance of a single-mode component in the path-symmetric entangled state.For a pure state, QFI is given by F Q = ( (cid:104) ˙ ψ out | ˙ ψ out (cid:105) − |(cid:104) ˙ ψ out | ψ out (cid:105)| ) , where | ψ out (cid:105) = ˆ U φ | ψ (cid:105) and | ˙ ψ out (cid:105) = ∂ | ψ out (cid:105) / ∂ φ with ˆ U φ a phase shift operation . Whether the phase shift occurrs in only one arm or in both arms, one would expect anidentical result as long as the phase difference is the same. However, Ref. pointed out that two phase-shift configurations cangive different quantum Cramer-Rao bounds. This inconsistency can be resolved by considering a phase-averaged state as aninput , which also makes a practical sense particularly when one has no access to external phase reference. It turns out that theQCRB for the phase-averaged state of our consideration is the same as that of a pure-state occurrung when the phase-shiftoperation is applied to both of the arms. Applying a phase shift operation to two optical paths, | ψ out (cid:105) = e − i φ ( ˆ n a − ˆ n b ) / | ψ (cid:105) , theQFI is given by the variance of number difference for the input two-mode state, F Q = (cid:104) ψ | ( ˆ n b − ˆ n a ) | ψ (cid:105) − (cid:104) ψ | ( ˆ n b − ˆ n a ) | ψ (cid:105) , (2)where | ψ (cid:105) = | ψ (cid:105) ab , ˆ n a = ˆ a † ˆ a and ˆ n b = ˆ b † ˆ b . The average input energy is given by N av = (cid:104) ψ | ( ˆ n a + ˆ n b ) | ψ (cid:105) . Repeatingmeasurements m times, the QFI provides the quantum Cram´er-Rao bound (QCRB) as ( δ φ c ) ≥ / ( mF Q ) .The QFI of the state in Eq. (1) is characterized by the average input energy and the Mandel Q -factor of the single-modecomponent, F Q = (cid:104) ˆ n (cid:105) + p ( (cid:104) ˆ n (cid:105) + + Q M ) , (3)where (cid:104) ˆ n (cid:105) = (cid:104) ϕ | ˆ n | ϕ (cid:105) . For a single-shot measurement with a given energy, the corresponding QCRB is thus determined bythe Mandel Q -factor of the single-mode component. That is, Observation : Given a fixed amount of input energy, quantumCram´er-Rao bound is lowered with Q M , thus the super-Poissonian statistics of a single-mode component in the path-symmetricentangled state gives a best result. For comparison, let us consider three different states for | ϕ (cid:105) , i.e. a photon number state | N (cid:105) , a coherent state | α (cid:105) ,and a squeezed vacuum state | ξ (cid:105) , as representing sub-Poissonian, Poissonian, and super-Poissonian statistics, respectively.Furthermore, we introduce a superposition of single photon and N photon states, | ϕ (cid:105) = √ q | (cid:105) + √ − q | N (cid:105) . This last choice igure 1. Quantum Cram´er-Rao bound as a function of N av , using NOON (blue dashed curve),
AOOA (black dotted curve),
SOOS (red long-dashed curve), and
QOOQ (purple curves) states. The purple curves represent the
QOOQ states with N = A stands for a coherent state, S for a squeezed vacuum state, and Q for | ϕ (cid:105) = √ q | (cid:105) + √ − q | N (cid:105) .is motivated by the general QFI in Eq. (3): For a fixed energy (cid:104) ˆ n (cid:105) , the QFI depends on two parameters Q M and p . Wemay set p = Q M . For the case of | ϕ (cid:105) = √ q | (cid:105) + √ − q | N (cid:105) (no vacuum component), the average energy is expressed by (cid:104) ˆ n (cid:105) = q + ( − q ) N , which in turn gives q = N −(cid:104) ˆ n (cid:105) N − . Therefore, we may treat N as a single free parameter, which can contribute to an arbitrarily large Mandel- Q factorthereby enhancing QFI, under energy-constraint (cid:104) ˆ n (cid:105) .The first three states, | N (cid:105) , | α (cid:105) , and | ξ (cid:105) , exhibit sub-Poissonian, Poissonian, and super-Poissonian statistics, respectively. Thesuperposition state √ q | (cid:105) + √ − q | N (cid:105) may exhibit all kinds of photon counting statistics with the parameter N . For convenience,we call the corresponding entangled states as NOON state,
AOOA state ( A : a coherent state), SOOS state ( S : a squeezed vacuumstate), and QOOQ state ( Q : superposition ratio of | (cid:105) and | N (cid:105) states). The corresponding QFIs are readily derived as a functionof the average photon number (cid:104) ˆ n (cid:105) , as shown in Fig. 1. For QOOQ state, its F Q is given by q + ( − q ) N = ( N + ) (cid:104) ˆ n (cid:105) − N ,which shows a Heisenberg-limit scaling O ( (cid:104) ˆ n (cid:105) ) with respect to input energy (cid:104) ˆ n (cid:105) while its multiplicative constant N + N . Therefore, it can take an arbitrarily large value by letting the free parameter N → ∞ even underany finite energy (cid:104) ˆ n (cid:105) . For the other states, the ratio of F Q / N av has the order NOON < AOOA < SOOS . In Fig. 1, we show theQCRB as a function of N av . We illustrate the case of QOOQ state with N = N = N = SOOS state. However
QOOQ state with N =
100 beats
SOOS state in all ranges of N av > N in QOOQ state is, the wider the range of N av of beating all other states is. The plot clearly shows thatQCRB can be arbitrarily lowered with super-Poissonianity of the single-mode component in the path-symmetric entangledstate. Phase-averaged state:
QFI addresses how sensitively an input state can change under the variation of the phase parameter φ . It may give a different result depending on the actual configuration of phase shift, namely, whether the phase operation actson two modes e − i φ ( ˆ n a − ˆ n b ) / or on a single mode e i φ ˆ n b , even though the phase difference φ is the same. This inconsistency maybe addressed by averaging the two modes of an input state over a common phase θ with an additional reference beam . Here,applying the phase-averaged method to the path-symmetric entangled state in Eq. (1), we obtain the phase-averaged state in aform of NOON -state mixtures, ρ m = + p ∞ ∑ n = p n | noon (cid:105) ab (cid:104) noon | , (4)where | noon (cid:105) ab = | n (cid:105) a | (cid:105) b + | (cid:105) a | n (cid:105) b √ . Then, we find that the corresponding QFI becomes equivalent to Eq. (3) irrespective of the igure 2. Mach-Zehnder interferometer for phase estimation, for which an input state is the path-symmetric entangled state inEq. (1). PS stands for a phase shifter, exp [ i φ ˆ n b ] or exp [ − i φ ( ˆ n a − ˆ n b )] , and M for mirror. BS is a 50:50 beam splitter. T is thetransmission rate of the beam splitters.phase shift operations, e − i φ ( ˆ n a − ˆ n b ) / or e i φ ˆ n b , thus the above Observation also applies to the phase-averaged state. Note that theQFI of a mixed state can be obtained by a diagonalization of the mixed state . Measurement setup I: parity measurement on either of two output modes
From now on, we consider some concrete measurement schemes for the phase estimation under a loss mechanism. As a firstscheme, we consider a parity measurement in the Mach-Zehnder (MZ) interferometer in Fig. 2, of which the phase sensitivityis given by ( ∆ φ p ) = − (cid:104) ˆ Π b (cid:105) ( ∂ (cid:104) ˆ Π b (cid:105) / ∂ φ ) = − (cid:104) ˆ µ ab (cid:105) ( ∂ (cid:104) ˆ µ ab (cid:105) / ∂ φ ) . (5) (cid:104) ˆ Π b (cid:105) = (cid:104) ( − ) ˆ n b (cid:105) represents a parity measurement on mode b after the beam splitter, which can be expressed by (cid:104) ˆ µ ab (cid:105) = ∑ ∞ L = ∑ LM = | L − M (cid:105) a (cid:104) M | ⊗ | M (cid:105) b (cid:104) L − M | including all the phase-carrying off-diagonal terms in the two modes before the beamsplitter. With the path-symmetric entangled states, the resulting ∆ φ p does not depend on the type of phase-shift operation,exp [ i φ ˆ n b ] or exp [ − i φ ( ˆ n a − ˆ n b )] , before the measurements.The parity measurement for the state in Eq. (1) gives (cid:104) ˆ µ ab (cid:105) = + p (cid:34) p + ∞ ∑ n = p n cos ( n φ ) (cid:35) , (6)and the corresponding phase sensitivity can be obtained by using the above expression of ∆ φ p . In the case of cos ( n φ ) ≈ ( ∆ φ p ) = / F Q = ( δ φ c ) , by L’Hˆopital’s rule. It means that the information-theoretic optimal bound can be achieved by the path-symmetric entangled resource going through a phase-shift at φ ≈ π l / n ( l = , , , ... ). On the other hand, notably, only NOON state achieves the optimal bound for the entire range [ , π ] of phaseshifts.For a realistic situation with loss, we assume that the entangled state in Eq. (1) goes through a beam splitter acting on eachmode with same transmission rate T after a phase shift operation, as shown in Fig. 2. After the loss channel in the two opticalpaths, the parity measurement gives (cid:104) ˆ µ ab (cid:105) T = + p ∞ ∑ n = p n [ R n + T n cos ( n φ )] , (7)where R = − T . From the considered single-mode components | ϕ (cid:105) = | N (cid:105) , | α (cid:105) , | ξ (cid:105) , and √ q | (cid:105) + √ − q | N (cid:105) , we find thatsub-Poissonianity of | ϕ (cid:105) gives a better result robust against the loss channel ( T = .
9) than (super) Poissonianity, in constrast tothe QFI under an ideal situation ( T = ( ∆ φ p ) SNL = / N av is beaten in awider range of phase-shift by NOON state than the other path-symmetric states. It shows a trend between the photon countingstatistics of | ϕ (cid:105) and the performance of phase estimation under the loss channel. When the photon counting statistics changesfrom super-Poissonian to sub-Poissonian, the corresponding path-symmetric entangled state shows better performance, with therobustness in the order of QOOQ ( N = ) < SOOS < QOOQ ( N = ) < AOOA < NOON . igure 3. (a) Phase sensitivity ∆ φ p achieved via parity measurement and (b) Phase sensitivity 1 / F given by Fisherinformation F , under a loss channel ( T : transmittance rate) as a function of an unknown phase φ , using NOON (blue dashedcurve),
AOOA (black dotted curve),
SOOS (red long-dashed curve), and
QOOQ (purple solid curve) states. The shot-noise limit(green dashed line) is given by ( ∆ φ p ) SNL = / N av . The average input photon number is set to be N av = Measurement setup II: photon counting on both output modes
As the second measurement scheme, we consider photon counting on both output modes in the MZI. The phase sensitivity isthen given by classical Fisher information (FI) F = ∑ n a , n b p ( n a , n b | φ ) (cid:20) ∂ p ( n a , n b | φ ) ∂ φ (cid:21) , (8)where p ( n a , n b | φ ) is a probability of detecting n a photons on mode a and n b photons on mode b for a given phase φ . For m measurements, the FI provides Cram´er-Rao bound (CRB), ( δ φ ) ≥ / ( mF ) , and we are here interested in a single-shotmeasurement ( m = p ( n a , n b | φ ) = p n − n + p n ! n a ! n b ! [ + ( − ) n a cos ( n φ )] , (9)where n = n a + n b . Remarkably, the corresponding Fisher information is the same as the quantum Fisher information of Eq. (3)and the path-symmetric entangled states achieve the optimal bound for the entire range [ , π ] of unknown phase shifts in anideal situation without loss.On the other hand, when the state goes through loss in the two optical paths, the photon-counting probability is given by p ( n a , n b | φ ) T = p n ( T / ) n + p n ! n a ! n b ! [ + ( − ) n a cos ( n φ )] + + p (cid:18) T (cid:19) n n a ! n b ! ∞ ∑ k = p n + k ( n + k ) ! k ! R k , (10)where n = n a + n b and R = − T . In Fig. 3 (b), we see that QOOQ state with N = T = .
9) than the other states over the entire range [ , π ] of phase shifts. Typically, a high photon number state of asingle-mode component is more vulnerable to the loss channel and the robustness of the path-symmetic entangled statesshows the ordering QOOQ ( N = ) < AOOA < SOOS < NOON < QOOQ ( N = ) . They are ordered according to the sizeof SNL-beating range of phase-shift angle. Compared with the parity measurement scheme under the same degree of loss, thepath-symmetric entangled states appear less vulnerable in this second scheme based on the full photon-counting statistics.Note that both NOON and
QOOQ states show performance not only more robust against loss than other states but alsoindependent of (insensitive to) phase shift. In Fig. 4 (a), we show that
QOOQ state can provide a better resource against lossover the whole range of input energy. For each transmittance rate T , we optimize N of QOOQ states √ q | (cid:105) + √ − q | N (cid:105) forthe best phase sensitivity (red dotted curve). We see that under the loss of considered range T = . ∼ QOOQ state with N = T = . QOOQ state with N = NOON state over the range of the average input energy N av >
1. Thus,
QOOQ state with N = QOOQ states. Furthermore, in Fig. 4 (b), we show the result of phase sensitivity by randomly sampling single-mode component statesin a form | ϕ (cid:105) = ∑ n = √ p n | n (cid:105) , where we still find QOOQ state with N = igure 4. (a) Phase sensitivity 1 / F given by Fisher information F under a loss channel ( T : transmittance rate) as a function of N av , using NOON state (blue dashed curve),
QOOQ state with N = QOOQ states with an optimized N for each T (red dotted curve). At T = . , . , . , .
8, the
QOOQ states are optimized with N = ∼ , ∼ , ∼ , ∼
4, respectively. (b) Phase sensitivity 1 / F given by Fisher information F under a loss channel( T = .
9) as a function of N av , using NOON state (blue dashed curve),
QOOQ state with N = ∑ n = √ p n | n (cid:105) are sampled with a total number of 37 ,
132 (grey dots).
State generation scheme
In this section, we propose how to generate a resource state of Eq. (1) in experiment. The class of entangled states, | ϕ (cid:105) a | (cid:105) b + | (cid:105) a | ϕ (cid:105) b , may be generated by optical Fredkin gate utilizing a cross-Kerr nonlinearity and beam splitters. However,the cross-Kerr nonlinearity achieved in laboratory is rather limited, which can limit a successful generation of the resourcestates. Instead, we propose a scheme using a polarization entangled resource and conditional phase shift (CPS) operations.The CPS operation was implemented in a superconducting transmon qubit . Specifically we are interested in the generationof SOOS and
QOOQ states. The generation of high
NOON state ( N >
4) was proposed theoretically by the superposition ofsingle (two)-photon operations or experimentally by post-selection after merging coherent laser light and down-convertedphoton . An AOOA state may be generated by injecting a cat state and a coherent state into a beam splitter . However it is notknown how to generate a SOOS state or a
QOOQ state with currently available techniques.First, we propose a generation scheme for
SOOS state, which is implemented by a polarization entangled state | H (cid:105) u | V (cid:105) d + | V (cid:105) u | H (cid:105) d and a conditional phase shift (CPS) operation ˆ C = I ⊗ | H (cid:105)(cid:104) H | + e ix ˆ a † ˆ a ⊗ | V (cid:105)(cid:104) V | . Applying the CPS operation to targetand ancillary qubits, the output qubits are given byˆ C a , u ˆ C b , d | ξ (cid:105) a | ξ (cid:105) b ⊗ ( | H (cid:105) u | V (cid:105) d + | V (cid:105) u | H (cid:105) d ) = | ξ (cid:105) a | ξ (cid:48) (cid:105) b | H (cid:105) u | V (cid:105) d + | ξ (cid:48) (cid:105) a | ξ (cid:105) b | V (cid:105) u | H (cid:105) d , (11)where ξ = re i θ and ξ (cid:48) = re i ( θ + x ) . Detecting the ancillary qubits to be in | + (cid:105) = ( | H (cid:105) + | V (cid:105) ) / √
2, we obtain an entangledsqueezed vacuum states, | ESV (cid:105) = | ξ (cid:105) a | ξ (cid:48) (cid:105) b + | ξ (cid:48) (cid:105) a | ξ (cid:105) b . Then, applying a single-mode squeezing operation on each mode, weobtain SOOS state with x = π / S ( ξ ) a ˆ S ( ξ ) b | ESV (cid:105) = | ξ (cid:105) a | (cid:105) b + | (cid:105) a | ξ (cid:105) b . (12)In Fig. 5 (a), we show the setup for the SOOS state. Two single-mode squeezed vacuum states | ξ (cid:105) a | ξ (cid:105) b interact withancillary qubits ( | H (cid:105) u | V (cid:105) d + | V (cid:105) u | H (cid:105) d ) under the CPS operation, where the ancillary qubits are generated by spontaneousparametric down conversion in a nonlinear PPKTP crystal. After the detection of the ancillary qubits, the SOOS state isproduced by the local operation ˆ S ( ξ ) on each output mode. In general, when the single-mode component of the path-symmetricentangled state is decomposable with unitary operations, i.e. | ϕ (cid:105) = ˆ U † e ix ˆ n ˆ U | (cid:105) , the generation scheme can be given by1 √ ( | H (cid:105) a | V (cid:105) b + | V (cid:105) a | H (cid:105) b ) ˆ U ˆ U | (cid:105) | (cid:105) CPS −−→ √ ( ˆ U | (cid:105) e ix ˆ n ˆ U | (cid:105) + e ix ˆ n ˆ U | (cid:105) ˆ U | (cid:105) ) ˆ U †1 ˆ U †2 −−−→ √ ( | (cid:105) ˆ U †2 e ix ˆ n ˆ U | (cid:105) + ˆ U †1 e ix ˆ n ˆ U | (cid:105) | (cid:105) ) , (13)where ˆ n i ( i = ,
2) is the photon number operator of mode i . igure 5. (a) Generation of SOOS state via a conditional phase shift (CPS) operation. Each single-mode squeezed vacuumstate is injected into a CPS operation setup. Ancillary qubits are entangled in polarization ( H , V ) and path ( u , d ) modes. LOstands for local squeezing operation ˆ S ( ξ ) . (b) Generation of QOOQ state via a successive application of the coherentoperation ˆ a + α e i φ k with φ k = π k / N . NOON state is first injected into the coherent operation setups. The coherent operationˆ a + α e i φ k is implemented with three beam splitters. Two beam splitters are highly reflective whereas one beam splitter is highlytransmissive. The phase φ k is a control parameter of strong coherent light.Second, we propose a generation scheme for QOOQ state, which is obtained by a successive application of a coherentoperation ˆ a + c with c = α e i π k / N . Based on the transformation ˆ a + c = ˆ D ( − c ) ˆ a ˆ D ( c ) , the coherent operation is obtained by asequential operation of displacement and photon subtraction operations. In Fig. 5 (b), we show that the coherent operationcan be implemented with three beam splitters. The displacement operation is achieved with strong coherent light and a beamsplitter with high reflectance . The photon subtraction operation is attained with a single-photon detection and a beam splitterwith high transmittance . Applying N set of the coherent operations to an N + | N + (cid:105) ), the outputstate is given by N ∏ k = ( ˆ a + α e i π kN ) | N + (cid:105) = (cid:112) ( N + ) ! | (cid:105) + ( − ) N − α N | N + (cid:105) . (14)For instance, initially preparing a NOON state, | N + ,
0; 0 , N + (cid:105) ≡ | N + (cid:105) a | (cid:105) b + | (cid:105) a | N + (cid:105) b and applying a set of coherentoperations to each mode, as shown in Fig. 5 (b), we obtain QOOQ state, N ∏ k = ( ˆ a + α e i π kN )( ˆ b + α e i π kN ) | N + ,
0; 0 , N + (cid:105) = (cid:112) ( N + ) ! | ,
0; 0 , (cid:105) + ( − ) N − α N | N + ,
0; 0 , N + (cid:105) , (15)where | ,
0; 0 , (cid:105) = | (cid:105) a | (cid:105) b + | (cid:105) a | (cid:105) b . Discussion
We have introduced a generic class of path-symmetric entangled states | ϕ (cid:105)| (cid:105) + | (cid:105)| ϕ (cid:105) for phase estimation, which includesthe well-studied NOON state and entangled-coherent states as special cases. The phase-sensitivity is largely determined bythe photon statistics of the single-mode component state | ϕ (cid:105) , and in particular, the QFI is given in terms of the Mandel- Q factor of | ϕ (cid:105) . We have shown that the QCRB can be lowered with the super-Poissonian statistics of | ϕ (cid:105) and that even anarbitrarily small QCRB is possible with a finite energy, e.g. by a state of the form | ϕ (cid:105) = √ q | (cid:105) + √ − q | N (cid:105) . On the otherhand, for specific measurement schemes, we have considered a parity measurement and a full photon-counting method toobtain phase-sensitivity. Without photon loss, the latter scheme employing any path-symmetric states | ϕ (cid:105) + | ϕ (cid:105) achievesthe QCRB over the entire range [ , π ] of unknown phase shift φ whereas the former does so in a restricted range of φ . Thecase of | ϕ (cid:105) = √ q | (cid:105) + √ − q | N (cid:105) turns out to provide the most robust resource against loss among the considered entangledstates over the range of the average input energy N av >
1, particularly under the loss channel with transmittance rate T ∼ . .Furthermore, we have proposed a generation scheme for SOOS state by using polarization entangled states and conditionalphase shift (CPS) operations, and a generation scheme for
QOOQ state by a successive application of a coherent operation a + c . The generation scheme of the SOOS state can also be modified to produce other type entangled states. Replacing one ofthe single-mode components with a coherent state before the CPS operation, we can produce a
SOOA state. Or, by controllingthe phase shift operation and the local operations in the setup, we can produce
OOSS and
OOSA states. Moreover, we mayextend the squeezing component to the superposition of different squeezing operations which can provide more enhancementof QCRB, similar to the phenomenon by multi-headed cat state .In this work we have addressed the estimation of a locally fixed, unknown, phase. A worthwhile direction for future workwill be the estimation of a completely unknown phase over a finite range . Moreover, we may investigate how quantumphase estimation can be improved with other entangled states by employing non-Gaussian operations, e.g., superposition ofphoton operations ( t ˆ a + r ˆ a † and t ˆ a + r ˆ a †2 ) with a controllable parameter t ( | t | + | r | = Remarks : While preparing this work, we became aware of a related work that also considered
SOOS state, differently termedas squeezed entangled state (SES) by Knott et al. in . They showed that, in local quantum phase estimation without photonloss, the SES gives lower bound of phase sensitivity than NOON state and a separable two-mode squeezed state | ξ (cid:105) a | ξ (cid:105) b . Theyalso proposed a separable squeezed cat state (SCS), i.e. a product of two single-mode squeezed cat states, which provideseven lower bound of the phase sensitivity than the SES. For a fixed average input energy N av =
1, the separable SCS exhibitsadvantage against 27% loss in terms of QFI and 10% loss under a full photon counting scheme. In comparison to , we alsoconsidered the SOOS state equivalent to the SES, which actually belongs to a broader class of path-symmetric entangled states | ϕ (cid:105)| (cid:105) + | (cid:105)| ϕ (cid:105) in our work. We showed that QOOQ state among them provided much lower bound of phase sensitivity thanthe
SOOS state. We also proposed experimental schemes to generate those path-entangled states using current technology,though they could be practically demanding. Under the parity measurement scheme, we showed the sub-Poisonianity of thesingle-mode component in the states | ϕ (cid:105)| (cid:105) + | (cid:105)| ϕ (cid:105) exhibits the robustness over photon loss. Under the full photon countingscheme, we showed that the QOOQ state is more robust than any other states | ϕ (cid:105)| (cid:105) + | (cid:105)| ϕ (cid:105) , under 10% loss, over the rangeof the average input energy N av > References Abbott, B. P. et al.
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Authors greatly appreciate M.J.W. Hall and C.C. Gerry for useful comments. This research was supported by the IT R&Dprogram of MOTIE/KEIT [1004346 ( ) ] and by an NPRP grant 7-210-1-032 from Qatar National Research Fund. Author contributions statement
S.-Y.L. developed the scenario with H.N. All authors have contributed to analyzing the results. S.-Y.L. wrote the manuscriptand all authors reviewed.
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