Fock state interferometry for quantum enhanced phase discrimination
FFock state interferometry for quantum enhanced phase discrimination
Reihaneh Shahrokhshahi, Saikat Guha, and Olivier Pfister ∗ University of Virginia, Dept. of Physics, 382 McCormick Rd., Charlottesville, VA 22904-4714, USA College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA (Dated: February 16, 2021)We study Fock state interferometry, consisting of a Mach-Zehnder Interferometer with two Fockstate inputs and photon-number-resolved detection at the two outputs. We show that it allowsdiscrimination of a discrete number of apriori-known optical phase shifts with an error probabilitylower than what is feasible with classical techniques under a mean photon number constraint. Wecompare its performance with the optimal quantum probe for M -ary phase discrimination, whichunlike our probe, is difficult to prepare. Our technique further allows discriminating a null phaseshift from an increasingly small one at zero probability of error under ideal conditions, a featureimpossible to attain using classical probe light. Finally, we describe one application to quantumreading with binary phase-encoded memory pixels. I. INTRODUCTION
Quantum mechanics dictates the fundamentallimit of phase estimation. A typical interferome-ter, such as the Michelson or Mach-Zehnder inter-ferometer, probes a phase shift by splitting an inputfield into two mutually coherent fields, each prob-ing a different path. The fields are then recombinedand the resulting total intensity yields interferencefringes which inform on the phase-difference betweenthe two paths. The quantum physics of interferom-etry hinges on the initial splitting of the input field:by unitarity, the two split outputs necessarily callfor two inputs, in of which is in a vacuum state. Asshown by Carl Caves [1], it the yields the classical— a.k.a. beam-splitter shot-noise — limit∆ θ cl ∼ (cid:104) N (cid:105) − , (1)where θ is the phase difference between the armsof an interferometer and (cid:104) N (cid:105) is the total averagenumber of photons in the interferometer. The clas-sical limit, however, doesn’t give the ultimate phaseprecision, which is fixed by the Heisenberg number-phase Heisenberg inequality [2] and bounded by theHeisenberg limit (HL)∆ θ Hl ∼ (cid:104) N (cid:105) − . (2)In order to reach the HL, the vacuum input must bereplaced by a nonclassical state such as a squeezedstate [3]. Other possibilities were proposed [4], inparticular the use of correlated Fock states [5] or su-perpositions or statistical mixtures thereof [6]. Manytheoretical studies of nonclassical inputs yielding theHL have been conducted [7], in particular applyinginformation theory to quantum physics [8], whichhas led to the definition of quantum metrology [9].In addition, a recent study has shown that ensur-ing phase estimation at the HL requires, in general,losses to be bounded by (cid:104) N (cid:105) − [10]. ∗ opfi[email protected] In this paper, we investigate the properties of thehighly nonclassical, yet conceptually simple inter-ferometry which we call Fock-state interferometry(FSI), as depicted in Fig. 1: in all of the paper, FSIconsists in a Mach-Zehnder Interferometer (MZI)with Fock state input and photon-number-resolveddetectors (PNRDs). One should note that FSI was | (cid:1) n b in i | (cid:1) n a in i ✓ (cid:1) n b out (cid:1) n a out b eam s plitter s FIG. 1. Phase discrimination by Fock-state interferom-etry. The Fock state | n a (cid:105) a | n b (cid:105) b is input into a Mach-Zehnder interferometer of phase difference θ and the in-terferometer’s output is measured by photon-number-resolving detectors. recently implemented experimentally, and the effectof photon loss was studied [11]. FSI was also studiedtheoretically with multimode inputs [12]. Here, wefocus on FSI not for phase estimation as much as forphase discrimination.It may be useful to give a simple insight on theexpected impact of such a purely corpuscular inter-ferometer input on purely undularory interferomet-ric performance. It is simple to show, as we do be-low, that FSI benefits general phase estimation onlywhen the input state satisfies n a = n b [5]. We outlinea basic derivation, using the Schwinger spin repre-sentation of two boson fields [13], which we’ll use inall of the paper and which is presented, along withits application to quantum interferometry, in Ap-pendix A. The Schwinger spin formalism allows one a r X i v : . [ qu a n t - ph ] F e b to use the convenient formalism of quantum angularmomentum and SO(3) rotation matrices to performquantum optics calculations.To begin with, the expectation value of the outputphoton number difference is (cid:104) N a − N b (cid:105) out = (cid:104) J z (cid:105) out (3)= 2 m cos θ (4)= ( n a − n b ) cos θ (5)which constitutes the quantum expression of an in-terference fringe.Note that, when n a = n b ⇔ m = 0, the outputphoton number difference is zero for all values of θ hence no direct fringe is present and other meth-ods are required to access phase information. How-ever, these methods yield Heisenberg-limited perfor-mance [5, 6].When n a (cid:54) = n b ⇔ m (cid:54) = 0, a direct interferencesignal is present but we show that direct measure-ment performance of FSI is, in fact, at the classicallimit. First, we evaluate the quantum standard devi-ation of the output photon-number difference, using j = ( n a + n b ) / m = ( n a − n b ) / | j m (cid:105) z = | n a (cid:105) a | n b (cid:105) b ,∆ J out z = | sin θ | { j ( j + 1) − m ] } (6)from which we obtain the theoretical phase error(whose expression is invalid for n a = n b )∆ θ = ∆ J out z (cid:12)(cid:12)(cid:12) ∂ (cid:104) J z (cid:105) out ∂θ (cid:12)(cid:12)(cid:12) (7)∆ θ = (cid:20) j ( j + 1)2 m − (cid:21) , (8)which is only valid for m (cid:54) = 0, i.e., n a (cid:54) = n b . Sincethe total photon number j is a constant here, it iseasy to see that the minimum error can only be ob-tained by maximizing m , i.e., for m = ± j ( n b,a = 0respectively), in which case we get∆ θ min = (2 j ) − (9)which is the classical limit, Eq. (1), as was firstshown by Caves [1].We show in this paper that FSI can yield en-hanced interferometry performance and break theclassical limit in cases less general than the estima-tion of an unknown phase, namely the discrimina-tion of two or more predetermined phase shifts. Therationales for this study are multiple: (i) , the theo-retical simplicity of the experimental setup; (ii) , itstranslation into sophisticated experimental conceptsthat are nonetheless coming of age, i.e., on-demandphoton sources and photon-number-resolving detec-tors; (iii) , the availability of concrete applications ofphase discrimination, such as the quantum readingof classical optical memories. This paper is organized as follows, in Section II westudy the problem of discriminating the finite num-ber M (cid:62) II. PHASE DISCRIMINATIONII.1. Introduction
In optical communication terms, the ability to dis-criminate between M optical phase shifts can bebeneficial to MPSK, a digital modulation schemethat conveys M messages by modulating the op-tical phase of a probe signal. Here we show thatusing FSI we can accurately discriminate between M = 2 , M = 2 phases and then generalize it to higher val-ues of M . We consider an unknown fixed phase θ ofMZI, which can take one of M = 2 values, denoted θ and θ and we define the estimated phase ˆ θ . Fourdifferent scenarios can occur during the phase dis-crimination process. If the initial phase θ = θ , andthe estimated phase ˆ θ = θ , , respectively, then wehave success. Else ˆ θ = θ , , and we have an error.A natural criterion to measure interferometer per-formance in the phase discrimination problem willthen be the error probability, P e , which we’ll definelater.We consider a MZI with a | j µ (cid:105) input and whosephase θ can be either of two predetermined values θ , . We then perform a single J z measurement ofthe photon number difference at the output ports, ofresult µ (cid:48) , and make a decision about the phase shiftbased on maximum likelihood algorithm: knowingthe probability distribution P ( µ (cid:48) , µ | θ ) of the inter-ferometer (Table I), we compare both cases θ = θ and θ = θ for a given measurement outcome and as-sign the estimated phase shift ˆ θ to the phase whichis more likely to result in this specific outcome µ (cid:48) .The algorithm is thusif P ( µ (cid:48) , µ | θ ) (cid:62) P ( µ (cid:48) , µ | θ )then (cid:26) P (ˆ θ = θ | θ = θ ) = P ( µ (cid:48) , µ | θ ) — success P (ˆ θ = θ | θ = θ ) = P ( µ (cid:48) , µ | θ ) — failureelse (cid:26) P (ˆ θ = θ | θ = θ ) = P ( µ (cid:48) , µ | θ ) — success P (ˆ θ = θ | θ = θ ) = P ( µ (cid:48) , µ | θ ) — failure TABLE I. Probability distribution P ( µ (cid:48) , µ | θ ). The possi-ble measurement outcomes are denoted by µ (cid:48) (columns)and possible phases by θ , (rows). Each element of thisarray is the probability of measuring µ (cid:48) ∈ [ − j, j ], givenphase θ . θ µ (cid:48) − j ... m ... jθ P ( − j, µ | θ ) ... P ( m, µ | θ ) ... P ( j, µ | θ ) θ P ( − j, µ | θ ) ... P ( m, µ | θ ) ... P ( j, µ | θ ) For this procedure to be error free, one would need: P (ˆ θ = θ , | θ = θ , ) = 0 . (10)Of course, this is not the case in general, and theaverage error probability is given by: P e = (cid:88) i,j (cid:54) = i P ( θ i ) P (ˆ θ = θ j | θ = θ i ) . (11)However, we now show that, for a judicious choice ofphases θ , and input µ , phase discrimination withan FSI may perform better than classical methodssuch as a coherent state probe and homodyne orheterodyne detection. II.2. Binary phase discrimination
The analytic expressions of probabilities are givenby rotation matrix elements in the Schwinger repre-sentation, Eq. (A23), Appendix A. Without loss ofgenerality, we may elect to set θ = 0 as this entails P ( µ (cid:48) , µ |
0) = d jµ (cid:48) ,µ (0) = δ µ (cid:48) ,µ (12)and simplifies the situation. The problem will thenreduce to discriminating θ = θ against θ = 0. Notethat this is still different from general phase estima-tion — again classically-limited for a FSI — as we’llrestrict θ to the values that will allow optimized per-formance.Using information theory — see Appendix B fora brief review — we will evaluate two important fig-ures of merit of phase discrimination: first, the errorprobability, Eq. (11), and also the mutual informa-tion I ( θ ; ˆ θ ) between ˆ θ and θ , Eq. (B16). II.2.1. Influence of the total photon number
We study the behavior of FSI for input states withdifferent total photon number n a + n b = 2 j . We calculate the error probability, and also the mutualInformation, for all θ . We then determine optimalphases for each input state, based on the perfor-mance.We first consider binary ( M = 2) phase dis-crimination for the two-photon inputs | (cid:105) a | (cid:105) b and | (cid:105) a | (cid:105) b , which are the respective Schwinger spinstates | (cid:105) z and | (cid:105) z . Figure 2 displays the er-ror probability versus the phase. As can be seen ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ θ Pe | i a | i b = | , i z | i a | i b ‡ = ‡ | , i z FIG. 2. Error probability P e ( θ ) for binary phase dis-crimination with a total photon number 2 j = 2. on the figure, both inputs yield zero error prob-ability. However, the smallest phase θ such that P e ( θ ) = 0 is π for a “classical” FSI (vacuum in-put, i.e., m = ± j ) and π/ m = 0 input. Animmediate question is therefore whether this trendcontinues and larger photon numbers yield discrim-ination of smaller phases.The 6-photon ( j = 3) and 8-photon ( j = 4)cases are displayed in Fig. 3(a) and Fig. 3(b), re-spectively. These situations are richer because morephoton number partitions are possible at the inter-ferometer’s input. Two trends are clearly visible: (i) , from Fig. 3(a) and Fig. 3(b) individually, i.e.,for a given j , the first zero of P e ( θ ), or “smallest op-timum phase,” decreases as | m | decreases and, (ii) ,from Fig. 3(a) and Fig. 3(b) together, the smallestoptimum phase decreases as j increases.In order to confirm the latter, we computed thesmallest optimum phase θ sop for larger photon num-bers in the m = 0 and m = j cases, results areplotted in Fig. 4. For the values up to 2 j = 10 com-puted in Fig. 4, a fit for the balanced Fock state with m = 0 (red plot) yields θ sop = 2 . × (2 j ) − . , (13)where the parenthetic numbers (6) and (2) representfit uncertainty. While we do not yet have any cer-tainty about the theoretical form of the dependenceof θ sop on j , these results clearly hint at the abilityfor “twin” Fock states | n (cid:105) a | n (cid:105) b = | j = n m = 0 (cid:105) z to allow the discrimination of phase shifts < j − / , ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚ θ | , i z = | i a | i b | , i z = | i a | i b | , i z = | i a | i b | , i z = | i a | i b a ) ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ θ | , i z = | i a | i b | , i z = | i a | i b | , i z = | i a | i b b ) FIG. 3. Error probability P e ( θ ) for binary phase discrimination with a total photon number of a) 2 j = 4, b) 2 j = 6. which beats the classical limit j − / . This is, indeed,proven by our calculations up to 2 j = 10.This result is reminiscent of Holland-Burnett in-terferometry, [5] which employs the same input state.However, the measurement processing is entirely dif-ferent as it uses maximum likelihood, as opposed toBayesian reconstruction.We know turn to another form of comparison ofFSI phase discrimination with conventional meth-ods, by comparing it with the coherent-state-basedhomodyne detection and Dolinar receivers, as wellas with the use of phase eigenstates. II.2.2. Comparison with coherent state input methodsand with phase eigenstates
Various other states can be used as interferometerinput. Coherent states, in particular, present the ad- P HBI P Single ported
MI Pe1 b ) Ê Ê Ê Ê Ê‡ ‡ ‡ ‡ ‡ j | j, j i z = | n i a | i b | j, i z = | n i a | n i b Smallest o ptimump hase (rad) a ) N avg FIG. 4. Smallest optimum phase vs j , The optimum θ is smallest for twin-Fock state input | n (cid:105) a | n (cid:105) b , ∀ n . vantage of being readily accessible from the outputof a well stabilized laser.In binary phase-shift-keyed encoding using coher-ent states, the phase-modulated coherent states aregiven as | α (cid:105) and | − α (cid:105) , i.e., two fields of oppositephase. In the low photon number regime, thesestates have small amplitudes ( | α | = √ j ) and arelargely overlapping (due to nonorthogonality of co-herent states), so the ability to successfully distin-guish these states at the receiver is limited, ulti-mately by the Helstrom bound [8]. Various detec-tion schemes can approach this bound for coherentstate discrimination. We consider two of them: thehomodyne receiver and the Dolinar receiver.The former is the simplest possible receiver relyingon Gaussian (Wigner function) operations [15]. Theerror probability for coherent state discriminationusing a homodyne receiver is given by [16]: P e ( α ) = 12 (cid:20) − erf (cid:18) | α | (cid:19)(cid:21) . (14)The latter is an adaptive measurement scheme,proposed by Dolinar in 1973, and which reaches theHelstrom bound for discriminating between two purecoherent states [17]. The Dolinar receiver is basedon a combination of photon counting and real-timefeedback control. The minimum error probabilityusing a Dolinar receiver is P min e = 12 (1 − (cid:112) − e − | α | ) , (15)which is the lowest possible error in distinguishingbetween two pure coherent states. Therefore, onereaches zero error probability only asymptotically,using high-amplitude coherent states.Finally, we mention a theoretical proposal, whichdoesn’t rely on coherent states, by Nair et al. [18]for the quantum state inside (between the two beam-splitters of) the MZI, rather than at its input. When j (cid:62) ( m − /
2, there always exists an optimumquantum state which can resolve m optical phases.This quantum state is Pegg and Barnett’s eigenstateof the optical phase operator [19] | ϕ (cid:105) = 1 √ m m − (cid:88) n =0 e − inϕ | n (cid:105) (16)taken at ϕ = 0 | ϕ = 0 (cid:105) = 1 √ m m − (cid:88) n =0 | n (cid:105) (17)This state is intended to probe the phase shift inthe “signal” arm of the MZI, which it can do at theHeisenberg limit. For m = 2 ( j (cid:54) P e = 12 − (cid:112) j (1 − j ) . (18)We now turn to the comparison of the error prob-abilities for binary phase discrimination (one of thephases being set to zero) of Fock-state interferome-try, coherent-state homodyne detection, the Dolinarreceiver, and phase-eigenstate probe. The respec-tive error probabilities for discriminating between 0and π radians are plotted in Fig. 5. As can be seen Ê Ê Ê Ê Ê (cid:75) - - - - Coherent -s tate input with h omodyne r eceiverCoherent -s tate input with Dolinar r eceiver Phase-eigensta te probe | (cid:19)(cid:75)(cid:1) i a | i b Fock-state interferometry . } (cid:1) | ↵ | (cid:1) = (cid:1) (cid:75) . FIG. 5. Error probability for discriminating between 0and π radians, versus j (where 2 j is the total photonnumber). on the figure, FSI’s performance is remarkable as itmatches that of the optimal phase-eigenstate probein this (0, π ) case, and, like the phase-eigenstateprobe, yields a clear advantage over the Dolinar re-ceiver for small photon numbers. This is also of in-terest as FSI, unlike the phase-eigenstate probe, hasa clear experimental implementation.In addition, we recall that FSI offers the addi-tional capability of resolving decreasing phase shiftsas the total photon number increases (Fig. 4), as perEq. (13). II.3. Ternary phase discrimination
We now turn to the extension of the previous prob-lem to discriminating three phases (0 , θ , θ ) — oneof them being, again, set to zero for convenience andwithout loss of generality. Again, the error proba-bility ( P e ), is a natural criterion to assess the per-formance of the phase discrimination. For all phasesequiprobable, the error probability is, from Eq. (11), P e = 13 [ P (0 | θ ) + P (0 | θ ) + P ( θ | P ( θ | θ ) + P ( θ |
0) + P ( θ | θ )] (19)Figure 6 displays the error probability for all possiblecombinations of (0 , θ , θ ), for a twin-photon input. θ P e θ FIG. 6. Error probability vs phase shifts θ and θ (inradians) for optical phase discrimination between threephase shifts (0 , θ , θ ). MZI input is | (cid:105) a | (cid:105) b = | , (cid:105) z . From the figure, we can spot the optimum phases forthe problem, which label the minimums of the errorprobability.In Fig. 7, we plot the error probability for theother two possible Fock-state inputs with j = 2.As for the binary case, the discrimination perfor-mance depends very much on the input state. Also,a shared trend with the binary case is that the errorprobability displays more oscillations, and thereforemore local minima, versus the phase angle(s) as m decreases, with the most favorable situation occur-ring for m = 0. This case also allows discriminationof smaller phase shifts, as for the binary case.We further determined optimum phase values fordifferent input states up to j = 6. These are listed,along with the error probability and mutual infor-mation [Eq. (B16)], in Table II. Ternary discrimi-nation does differ from binary discrimination: whilethe trend that balanced inputs allow the discrimina-tion of smaller phases as j increases is confirmed byexamination of all results for m = 0, one can notice θ θ P e a ) θ θ P e b ) FIG. 7. error probability ( P e ) vs phase shifts θ and θ , for optical phase discrimination between three phase shifts(0 , θ , θ ). MZI input is a), | (cid:105) a | (cid:105) b = | , (cid:105) z and b), | (cid:105) a | (cid:105) b = | , (cid:105) z TABLE II. Optimal phases θ opt1 , θ opt2 that attain the min-imum error probability P min e , and corresponding mu-tual information I ( { θ } ; { ˆ θ } ) for all input states with2 (cid:54) j (cid:54)
6. (The maximum value of I ( { θ } ; { ˆ θ } ) hereis log M = 1 . θ opt1 θ opt2 − P min e I ( { θ } ; { ˆ θ } ) | (cid:105) z = | (cid:105) a | (cid:105) b π π
160 0 . | (cid:105) z = | (cid:105) a | (cid:105) b π π
160 0 . | (cid:105) z = | (cid:105) a | (cid:105) b π π
40 1 . | (cid:105) z = | (cid:105) a | (cid:105) b . π
140 1 . | (cid:105) z = | (cid:105) a | (cid:105) b π π . | (cid:105) z = | (cid:105) a | (cid:105) b .
32 3 .
16 5 1 . | (cid:105) z = | (cid:105) a | (cid:105) b π π
48 1 . | (cid:105) z = | (cid:105) a | (cid:105) b .
55 1 . . | (cid:105) z = | (cid:105) a | (cid:105) b . π . | (cid:105) z = | (cid:105) a | (cid:105) b . π . | (cid:105) z = | (cid:105) a | (cid:105) b . π . | (cid:105) z = | (cid:105) a | (cid:105) b π π . | (cid:105) z = | (cid:105) a | (cid:105) b . π
96 1 . | (cid:105) z = | (cid:105) a | (cid:105) b π . | (cid:105) z = | (cid:105) a | (cid:105) b . π . | (cid:105) z = | (cid:105) a | (cid:105) b . π
24 1 . | (cid:105) z = | (cid:105) a | (cid:105) b . π .
04 1 . | (cid:105) z = | (cid:105) a | (cid:105) b π π .
68 1 . | (cid:105) z = | (cid:105) a | (cid:105) b .
32 2 . . | (cid:105) z = | (cid:105) a | (cid:105) b .
32 3 .
16 4 1 . | (cid:105) z = | (cid:105) a | (cid:105) b . π . | (cid:105) z = | (cid:105) a | (cid:105) b . π .
07 1 . | (cid:105) z = | (cid:105) a | (cid:105) b . π .
23 1 . | (cid:105) z = | (cid:105) a | (cid:105) b . π .
066 1 . | (cid:105) z = | (cid:105) a | (cid:105) b . π . . also that the error probability tends to be higherthan for other cases, even though it does decreasewith increasing j . Mutual information follows thesame trend, being lowest for m = 0 and increasingwith j . In fact, the m = j − j , as plotted in Fig. 8. This is better performance if the magnitude of the phasesto be resolved isn’t important. | j i a | j i b = | j, i z Ê Ê Ê Ê Ê‡ ‡ ‡ ‡ ‡Ï Ï Ï Ï Ï j ¥ -4 ¥ -4 Pe | j i a | i b = | j, j i z | j i a | i b = | j, j i z FIG. 8. Minimum error probability ( P e ) vs N for M=3and for all forms of input states. The effective spin | j, j − (cid:105) z shows the best performance. The problem of discriminating between more thanthree phases grows exponentially in size, and lowerror probabilities appear elusive for FSI with
M > III. QUANTUM READING
We now examine an application of binary phasediscrimination, which is the reading of digital clas-sical memory using quantum light [20, 21].We already know that FSI enables binary dis-crimination of smaller phase shifts than any othermethod, and also yields zero error probability at thefew photon level, beating the best classical protocols.Here, we compare the channel capacity and pho-ton information efficiency (PIE) of quantum opticalreading to that of classical reading and show that,whereas standard optical drives using a laser probeand direct detection are limited to a PIE of 0.5 bit ofinformation per transmitted photon, FSI can reach 1bit of information per transmitted photon for binaryphase discrimination, and log (3) (cid:39) . θ and the measured phaseshifts ˆ θ for each pixel (see Appendix B), C ( n s ) = max I ( θ ; ˆ θ ) , (20)where n s is the average number of signal photonsin the reading probe, here the interferometer armthat contains the phase shift. It is straightforwardto show that n s = (cid:104) n a | (cid:104) n b | U † BS a † a U † BS | n a (cid:105) | n b (cid:105) = j. (21)The PIE is then the number of bits read per signalphotons: P IE = C ( n s ) n s . (22)Here we consider a binary phase shift keyed(BPSK) phase encoding (different from the AMscheme used in optical disks) and apply binary phasediscrimination by FSI to optical reading.The mutual information (MI) for FSI with n s =1 , m = 0 statesyield maximum MI for the smallest minimum phase.In the following, we use the MI at these optimumphases to calculate the channel capacity and PIE.The classical capacity of a quantum channel is lim-ited by the Holevo bound, which imposes an upperlimit on the continued reliable rate of reading clas-sical information from a quantum channel [22, 23].For the lossless phase only encoding the Holevo ca-pacity can be written as [24] C ( n s ) = (1 + n s ) log (1 + n s ) − n s log ( n s ) . (23)In order to reach the Holevo capacity bound, oneneeds to use an optimum probe state as well as anoptimum receiver design. However, no feasible ex-perimental scheme is known to achieve this limit.As for binary phase discrimination, we compare FSIperformance with coherent-state encoding with ho-modyne, heterodyne, or Dolinar receivers. For thesebinary channels the capacity is given by C ( n s ) = 1 + P e log ( P e ) + (1 − P e ) log (1 − P e )(24) where P e for homodyne and Dolinar is given byEqs. (14-15). In Fig. 10, we plot the PIE versus n s for these different approaches. As can be seenfrom the figure, the PIE of FSI slightly outperformsthe Dolinar receiver for n s = 1.In Fig. 11, we plot the PIE versus the chan-nel capacity for the same BPSK receivers and alsoadd the coherent-state on-off encoding, which is thecurrently used (AM) technology for reading opticaldisks. Again, FSI outperforms all other receivers.Finally, we examine the use of ternary phaseshifted keyed (TPSK) modulation, where infor-mation is encoded as three distinct phases shifts(0 , θ , θ ) in each memory pixel for optical reading,which could result in denser encoding. From ourstudy of ternary phase discrimination in this paper,we already know the following: while an m = 0 in-put will resolve smaller phases, it also does that athigher probability of error than other, “less quan-tum” inputs such as m = j . The latter, however,gives lower error probability for larger phase shifts,with m = j − (3) (cid:39) . m = j, j − m = 0. Based on this wecan expect a PIE of 0.8 bit/photon for n s = 2 forthe | j j (cid:105) z input, for a choice of phases such as (0, π/ π ). IV. CONCLUSION
We have studied Fock-state interferometry, whoseimplementation is now within reach with the com-ing of age of on-demand single-photon sources andphoton-number-resolving detectors. We showed thatFSI is well suited to the particular task of phasediscrimination, rather than that of phase estima-tion. To begin with, FSI with twin inputs ( m = 0)can discriminate smaller phases than other methods;the phase magnitude also decreases with the pho-ton number more rapidly than j − / in the domainwhere our study was conducted (where 2 j is the to-tal photon number). For binary discrimination, FSIcan reach zero error probability; it outperforms allpreviously known coherent-state-based approaches,including the Dolinar receiver, at low photon lev-els, and matches the performance of the specificallyoptimized phase-eigenstate interferometry. In themore exotic case of ternary phase distribution, FSIstill provides a smaller phase shift discrimination(for m = 0 inputs), as well as very low error prob-ability and near maximal mutual information (for m = j − , j ). We finally examined the applicationof phase discrimination to phase-shift-keyed opticalreading and confirmed that the photon informationefficiency of FSI exceeds all previously known exper-imentally feasible approaches, as well as the industrystandard on-off AM reading. ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ (cid:81)
ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ (cid:81) | (cid:75) i a | (cid:75) i b (cid:1) = (cid:1) | j (cid:1) i z | (cid:75) i a | i b (cid:1) = (cid:1) | j (cid:1) j i z | i a | i b (cid:1) = (cid:1) | i z (cid:1) , b ) a ) b) FIG. 9. Mutual information of optical reading. The binary information is encoded in optical phase shifts (0 , θ ); a), n s = j = 1; b), n s = j = 2. Holevo b oundCoherent -s tate BPSK encoding with h omodyne r eceiverCoherent -s tate BPSK encoding with Dolinar r eceiver FSI . Ê Ê Ê ÊÊ - (cid:21) ns C H ns Lê ns FIG. 10. Photon information efficiency versus n s forvarious interferometry schemes. ACKNOWLEDGMENTS
This work was funded by Raytheon-BBN, un-der the U.S. Defense Advanced Research ProjectsAgency “Information in a Photon” program, andby U.S. National Science Foundation grants PHY-0855632, PHY-1206029, and PHY-1708023.
Appendix A: The Schwinger Representation
The Schwinger representation [13] introduces amathematical description of passive lossless four-
ÊÊÊÊÊ H n s L H n s Lê n s F SI Coherent - state o n - off encoding with direct detectionHolevo b oundCoherent -s tate BPSK encoding with h omodyne r eceiverCoherent -s tate BPSK encoding with Dolinar r eceiver . FIG. 11. Photon information efficiency (bits per pho-ton) vs the encoding efficiency (bits encoded per pixel)for various input states and receivers. port optical devices based on rotations in an ab-stract 3D space. The application of the Schwingerrepresentation to the analysis of optical interferome-ters was first demonstrated by Yurke et al. [4], afterW´odkiewicz and Eberly presented the relevance tooptics of the SU(2) and SU(1,1) groups [25].Any linear passive lossless optical device with twoinput and two output ports can be described by a2 × U = (cid:18) cos β e i ( α + γ ) / sin β e i ( α − γ ) / − sin β e − i ( α + γ ) / cos β e − i ( α + γ ) / (cid:19) (A1)where α , β and γ are the Euler angles. U operateson the two-dimensional vector ( a, b ) T , whose com-ponents are the annihilation operators for the twoinput fields at each port of the system.The homomorphism from SU(2) to the rotationgroup in three dimensions, SO(3), allows us to vi-sualize the action of two-mode optical devices, suchas beam splitters and phase shifters, as rotationsin 3D space. The general rotation in Eq. (A1) ismathematically equivalent to the rotation of the thefollowing tridimensional vector (cid:126)J in 3D space: J = J x J y J z = 12 a † b + b † a − i ( a † b − b † a ) a † a − b † b . (A2)Components J x , J y and J z follow the canonical com-mutation relations for quantum angular momentumoperators [ J k , J l ] = i (cid:126) ε klm J m (A3)where k, l, m ∈ { x, y, z } , and ε klm is the Levi-Civitasymbol. So (cid:126)J can be deemed a quantum angularmomentum, or effective spin.The magnitude of the angular momentum J is J = J x + J y + J z = a † a + b † b (cid:18) a † a + b † b (cid:19) (A4) J = N (cid:18) N (cid:19) (A5)where N = N a + N b = a † a + b † b (A6)is the total photon number operator.Fock states | n a (cid:105) | n b (cid:105) are therefore also eigen-states of J and J z , | j µ (cid:105) z = | n a (cid:105) a | n b (cid:105) b (A7)with respective eigenvalues j ( j + 1) and µ given bythe total photon number and the photon numberdifference j = n a + n b µ = n a − n b j photons in mode a and vacuum in mode b isidentical to, | j (cid:105) a | (cid:105) b = | j j (cid:105) z and the twin Fockstate input which is required for Holland-Burnett in-terferometry [5] is | j (cid:105) a | j (cid:105) b = | j (cid:105) z .A unitary operation on the quantum fields a and b can be viewed as the SO(3) rotation of the corre-sponding spin (cid:126)J , Eq. (A2). Any rotation of spin (cid:126)J can be described with the 3 Euler rotations: (cid:126)J out = e iαJ z e iβJ y e iγJ z (cid:126)J in e − iγJ z e − iβJ y e − iαJ z (A10) | ψ (cid:105) out = e iαJ z e iβJ y e iγJ z | ψ (cid:105) in (A11)respectively in the Heisenberg and Schrodinger pic-tures. In the Schwinger representation this arbi-trary tridimensional rotation of the effective spin (cid:126)J in is equivalent to the Euler angle parametrization ofthe SU(2) rotation of the two modes a and b basis,Eq. (A1). The SO(3) Euler matrix is c α c β c γ − s α s γ − c γ s α − c α c β s γ c α s β c α s γ + c β c γ s α c α c γ − c β s α s γ s α s β − c γ s β s β s γ c β (A12)where c ( α/β,γ ) =cos ( α/β, γ ), s ( α/β,γ ) =sin ( α/β, γ ).The SO (3) matrix for the beam splitter with Fresnelcoefficients ρ = cos φ/ τ = sin φ/ J out x J out y J out z = φ sin φ − sin φ cos φ J in x J in y J in z (A13)The effect of the beam splitter on modes a and bis a rotation of effective spin (cid:126)J by ( − φ ) around the x axis. The special case of the 50 /
50 beam splitterwith ρ = τ = √ , φ = π/ − π/ x axis: (cid:126)J out = e iπ/ J x (cid:126)J in e − iπ/ J x (A14) | ψ (cid:105) out = e iπ/ J x | ψ (cid:105) out . (A15)The SO(3) matrix for the phase shift θ between thearms of an interferometer is J out x J out y J out z = cos θ − sin θ θ cos θ
00 0 1 J in x J in y J in z (A16)which is a rotation of effective spin around z by θ . (cid:126)J out = e iθ J z (cid:126)J in e − iθJ z (A17) | ψ (cid:105) out = e iθJ z | ψ (cid:105) out . (A18)The Mach-Zehnder Interferometer (MZI) consists oftwo 50 /
50 beam splitters, and a phase shifter. So,the effect of MZI is equivalent to a ( − π/
2) rotationaround x axis, a θ rotation around z , and another π/ x , which yields a θ rotationaround y (cid:126)J out = e iθJ y (cid:126)J in e − iθJ z y (A19) | ψ out (cid:105) = e iθJ y | ψ in (cid:105) (A20)So the effect of MZI is equivalent to a single rota-tion of effective spin by θ around the y axis. We areinterested on the effect of MZI on Fock states, the0eigenstates of effective spin (cid:126)J , | j, µ (cid:105) . The proba-bility function P ( µ (cid:48) , µ | θ, j ) for the input spin | j, µ (cid:105) to be measured after the interferometer as | j, µ (cid:48) (cid:105) for fixed θ and J (the total photon number) can bedescribed as a rotation matrix, which is a squarematrix of dimension 2 j + 1 with general element P ( µ (cid:48) , µ | θ, j ) = |(cid:104) j, µ (cid:48) | ψ out (cid:105)| (A21)= |(cid:104) j, µ (cid:48) | e iθJ y | jµ (cid:105) z | (A22)= d jµ (cid:48) ,µ ( θ ) (A23)These rotation matrix elements can be expressed interms of Jacobi polynomials d jµ (cid:48) ,µ ( θ ) = (cid:20) ( j + µ )!( j − µ )!( j + µ (cid:48) )!( j − µ (cid:48) )! (cid:21) / (cid:18) sin β (cid:19) µ − µ (cid:48) × (cid:18) cos β (cid:19) µ + µ (cid:48) P ( µ − µ (cid:48) ,µ + µ (cid:48) ) j − µ (cos β )(A24) Appendix B: Information theory
In this section we briefly recall some basic con-cepts of information theory. We will use these con-cepts to study the behavior of FSI, then compare itwith other schemes.
1. Entropy
For any probability distribution, we recall the def-inition of Shannon entropy. Let X be a discrete ran-dom variable with the probability function, P ( x ), x ∈ X .The Shanon entropy of the random variable X is H ( X ) = − (cid:88) x ∈ X P ( x ) log P ( x ) (B1)and the conditional entropy of random variables X, Y , is the expected value of the entropies of theconditional distributions, averaged over the condi-tioning random variable, H ( Y | X ) = (cid:88) x ∈ X P ( x ) H ( Y | X = x ) (B2)= − (cid:88) x ∈ X P ( x ) (cid:88) y ∈ Y P ( y | x ) log P ( y | x )(B3)where P ( y | x ) is the conditional probability of mea-suring y given that x occurred.
2. Mutual information
The mutual information (MI) is a measure of theamount of information that one random variable X contains about another random variable Y , is equiv-alent to the reduction in the uncertainty of one ran-dom variable due to the knowledge of the other. Forrandom variables X and Y with probability func-tions P ( x ) , x ∈ X and P ( y ) , y ∈ Y and the condi-tional entropies H ( X | Y ) and H ( Y | X ), the mutualinformation can be written as: I ( X ; Y ) = H ( X ) − H ( X | Y ) = H ( Y ) − H ( Y | X )(B4)One way to study the general problem of encodingthe information in M optical phases, or the morespecified problem of optical reading is to look atthe mutual information between the applied opti-cal phases, θ and the measured phases ˆ θ . Ideally, ˆ θ should contain all the information about θ and themutual information I ( θ ; ˆ θ ) should be equal to theamount of information in random variable θ , H ( θ ).Lets assume information is encoded in M opticalphases with equal a priori probabilities so, θ = { θ , . . . , θ i , . . . , θ M } (B5) P ( θ ) = (cid:26) M , . . . , M , . . . , M (cid:27) (B6) H ( θ ) = log M (B7)Then, one seeks to estimate the phases ˆ θ ˆ θ = { ˆ θ , . . . , ˆ θ j , . . . , ˆ θ M } (B8) P (ˆ θ ) = (cid:110) P (ˆ θ ) , . . . , P (ˆ θ j ) , . . . , P (ˆ θ M ) (cid:111) (B9)Note that ˆ θ is not necessarily the same random vari-able as θ but their similarity and the overlap in theirinformation contents, defined as the mutual informa-tion I ( θ ; ˆ θ ), is a good measure of success in the phaseencoding problem. I ( θ ; ˆ θ ) can be calculated as I ( θ ; ˆ θ ) = H ( θ ) − H ( θ | ˆ θ ) (B10)= H ( θ ) − (cid:88) j P (ˆ θ j ) (cid:88) i P ( θ i | ˆ θ j ) log P ( θ j | ˆ θ i )(B11) P (ˆ θ j | θ i ) is the probability that one can directly ex-tract from experiment so, we employ the Bayes the-orem, P ( θ i | ˆ θ j ) = P (ˆ θ j | θ i ) P ( θ i ) P ( ˆ θ j ) (B12) P (ˆ θ j ) = (cid:88) i P (ˆ θ j | θ i ) P ( θ i ) (B13)and substitute P ( θ i | ˆ θ j ) with P (ˆ θ j | θ i ) in I ( θ ; ˆ θ ): I ( θ ; ˆ θ )= H ( θ ) − (cid:88) i,j P ( θ j ) P (ˆ θ j | θ i ) P ( θ i ) P ( ˆ θ i ) log P (ˆ θ j | θ i ) P ( θ i ) P ( ˆ θ i )(B14)1This gives I ( θ ; ˆ θ ) = (cid:88) i P ( θ i ) log P ( θ i ) − (cid:88) i,j P ( θ i ) P (ˆ θ j | θ i ) log P (ˆ θ j | θ i ) (cid:80) k P ( ˆ θ j | θ k ) (B15)= log M − (cid:88) i,j P (ˆ θ j | θ i ) M log P (ˆ θ j | θ i ) (cid:80) k P ( ˆ θ j | θ k ) (B16) Thus I ( θ ; ˆ θ ) max = log M , when P (ˆ θ j | θ i ) i = j = 1and P (ˆ θ j | θ i ) i (cid:54) = j = 0. P (ˆ θ j | θ i ) i (cid:54) = j is the probabilityof having phase θ i and estimating the wrong phaseˆ θ j which results in error. [1] C. M. Caves, Quantum-mechanical radiation-pressure fluctuations in an interferometer, Phys.Rev. Lett. , 75 (1980).[2] J.-M. L´evy-Leblond and F. Balibar, Quantics:Rudiments of Quantum Physics (North-Holland,1990).[3] C. M. Caves, Quantum-mechanical noise in an in-terferometer, Phys. Rev. D , 1693 (1981).[4] B. Yurke, S. L. McCall, and J. R. Klauder, SU(2)and SU(1,1) interferometers, Phys. Rev. A , 4033(1986).[5] M. J. Holland and K. Burnett, Interferometric de-tection of optical phase shifts at the Heisenberglimit, Phys. Rev. Lett. , 1355 (1993).[6] T. Kim, O. Pfister, M. J. Holland, J. Noh, andJ. L. Hall, Influence of decorrelation on Heisenberg-limited interferometry using quantum correlatedphotons, Phys. Rev. A , 4004 (1998).[7] A. Luis and L. S´anchez-Soto, Quantum phase dif-ference, phase measurements and Stokes operators,Prog. Opt. , 421 (2000).[8] C. W. Helstrom, Quantum Detection and Estima-tion Theory (Mathematics in Science and Engineer-ing, 123,(Academic Press, New York), 1976).[9] V. Giovannetti, S. Lloyd, and L. Maccone, Quantummetrology, Phys. Rev. Lett. , 010401 (2006).[10] B. M. Escher, R. L. de Matos Filho, and L. Davi-dovich, General framework for estimating the ul-timate precision limit in noisy quantum-enhancedmetrology, Nat. Phys. , 406 (2011).[11] G. S. Thekkadath, M. E. Mycroft, B. A. Bell, C. G.Wade, A. Eckstein, D. S. Phillips, R. B. Patel,A. Buraczewski, A. E. Lita, T. Gerrits, S. W. Nam,M. Stobi´nska, A. I. Lvovsky, and I. A. Walmsley,Quantum-enhanced interferometry with large her-alded photon-number states, npj Quantum Informa-tion , 89 (2020).[12] M. Perarnau-Llobet, A. Gonz´alez-Tudela, and J. I.Cirac, Multimode fock states with large photonnumber: effective descriptions and applications inquantum metrology, Quantum Science and Technol-ogy , 025003 (2020).[13] J. Schwinger, On angular momentum, U.S. AtomicEnergy Commission Report. No. NYO–3071 (1952), reprinted in Quantum Theory of Angular Momen-tum, edited by L. C. Biedenharn and H. van Dam(Academic Press, New York, 1965), pp. 229–279 .[14] G. P. Agrawal, Fiber-Optic Communication Sys-tems , 4th ed. (Wiley Series in Microwave and Opti-cal Engineering,, Wiley- Interscience, Hoboken, NJ,2010).[15] S. L. Braunstein and P. van Loock, Quantum infor-mation with continuous variables, Rev. Mod. Phys. , 513 (2005).[16] S. Olivares and M. G. A. Paris, Binary optical com-munication in single-mode and entangled quantumnoisy channels, Journal of Optics B: Quantum andSemiclassical Optics Opt. 6 69 (2004).[17] S. Dolinar, Quarterly progress report, Tech. Rep.,RLE at MIT (1973).[18] R. Nair, B. J. Yen, S. Guha, J. H. Shapiro, andS. Pirandola, Symmetric m-ary phase discriminationusing quantum-optical probe states, Phys. Rev. A , 022306 (2012).[19] D. T. Pegg and S. M. Barnett, Phase propertiesof the quantized single-mode electromagnetic field,Phys. Rev. A , 1665 (1989).[20] S. Pirandola, Quantum reading of a classical digitalmemory, Phys. Rev. Lett. , 090504 (2011).[21] R. Nair, Discriminating quantum-optical beam-splitter channels with number-diagonal signalstates: Applications to quantum reading and tar-get detection, Phys. Rev. A , 032312 (2011).[22] P. Hausladen, R. Jozsa, B. Schumacher, M. West-moreland, and W. K. Wootters, Classical informa-tion capacity of a quantum channel, Phys. Rev. A , 1869 (1996).[23] A. S. Holevo, Problemy Peredachi lnformatsii 15, 3[Problems of lnformation Transmission (USSR) ,247 (1979).[24] S. Guha and J. H. Shapiro, Reading boundless error-free bits using a single photon, Phys. Rev. A ,062306 (2013).[25] K. W´odkiewicz and J. Eberly, Coherent states,squeezed fluctuations, and the SU(2) and SU(1,1)groups in quantum-optics applications, J. Opt. Soc.Am. B2