Determining complementary properties with quantum clones
DDetermining complementary properties with quantum clones
G.S. Thekkadath*, R.Y. Saaltink, L. Giner, and J.S. Lundeen
Department of Physics, Centre for Research in Photonics,University of Ottawa, 25 Templeton Street,Ottawa, Ontario K1N 6N5, Canada*[email protected]
In a classical world, simultaneous measurements of complementary properties ( e.g. position andmomentum) give a system’s state. In quantum mechanics, measurement-induced disturbance islargest for complementary properties and, hence, limits the precision with which such propertiescan be determined simultaneously. It is tempting to try to sidestep this disturbance by copying thesystem and measuring each complementary property on a separate copy. However, perfect copyingis physically impossible in quantum mechanics. Here, we investigate using the closest quantumanalog to this copying strategy, optimal cloning. The coherent portion of the generated clones’ statecorresponds to “twins” of the input system. Like perfect copies, both twins faithfully reproducethe properties of the input system. Unlike perfect copies, the twins are entangled. As such, ameasurement on both twins is equivalent to a simultaneous measurement on the input system. Forcomplementary observables, this joint measurement gives the system’s state, just as in the classicalcase. We demonstrate this experimentally using polarized single photons.
At the heart of quantum mechanics is the concept ofcomplementarity: the impossibility of precisely determin-ing complementary properties of a single quantum sys-tem. For example, a precise measurement of the positionof an electron causes a subsequent momentum measure-ment to give a random result. Such joint measurementsare the crux of Heisenberg’s measurement-disturbancerelation [1, 2], as highlighted by his famous microscopethought-experiment in 1927 [3]. Since then, methods forperforming joint measurements of complementary prop-erties have been steadily theoretically investigated [4–8],leading to seminal inventions such as heterodyne quan-tum state tomography [9, 10]. More recently, advancesin the ability to control measurement-induced distur-bance have led to ultra-precise measurements that sur-pass standard quantum limits [11], and also simultaneousdetermination of complementary properties with a pre-cision that saturates Heisenberg’s bound [12]. In sum,joint complementary measurements continue to proveuseful for characterizing quantum systems [13–16] andfor understanding foundational issues in quantum me-chanics [11, 12, 17, 18].In this Letter, we address the main challenge in per-forming a joint measurement, which is to circumventthe mutual disturbance caused by measuring two gen-eral non-commuting observables, X and Y . Classically,such joint measurements ( e.g. momentum and position)are sufficient to determine the state of the system, even ofstatistical ensembles. In quantum mechanics, these jointmeasurements have mainly been realized by carefully de-signing them to minimize their disturbance, such as inweak [12–16, 18] or non-demolition [7, 11, 17] measure-ments. In order to avoid these technically complicatedmeasurements, one might instead consider manipulatingthe system, and in particular, copying it. Subsequently,one would perform a standard measurement separately on each copy of the system. Since the measurements areno longer sequential, or potentially not in the same lo-cation, one would not expect them to physically disturbone another. Crucially, as we explain below, the copiesbeing measured must be correlated for this strategy towork. Hofmann recently proposed an experimental pro-cedure that achieves this [19]. Following his proposal, weexperimentally demonstrate that a partial-SWAP two-photon quantum logic gate [20] can isolate the measure-ment results of two photonic “twins”. These twins arequantum-correlated ( i.e. entangled) copies of a photon’spolarization state that are ideal for performing joint mea-surements.We begin by considering a physically impossible, butinformative, strategy. Given a quantum system in a state ρ , consider making two perfect copies ρ ⊗ ρ and then mea-suring observable X on copy one and Y on copy two. Inthis case, the joint probability of measuring outcomes X = x and Y = y is Prob( x, y ) = Prob( x )Prob( y ) [21].Since it is factorable into functions of x and y , this jointprobability cannot reveal correlations between the twoproperties. Even classically, this procedure would gen-erally fail to give the system’s state, since such correla-tions can occur in e.g. statistical ensembles. Less ob-viously, these correlations can occur in a single quan-tum system due to quantum coherence [4]. In turn, thelack of sensitivity to this coherence makes this joint mea-surement informationally incomplete [6], and thus thissimplistic strategy is insufficient for determining quan-tum states [22]. Further confounding this strategy, theno-cloning theorem prohibits any operation that can cre-ate a perfect copy of an arbitrary quantum state, ρ (cid:57) ρ ⊗ ρ [23]. In summary, even if this strategy were allowedin quantum physics, it would not function well as a jointmeasurement.Although perfect quantum copying is impossible, there a r X i v : . [ qu a n t - ph ] J u l has been extensive work investigating “cloners” that pro-duce imperfect copies [24]. Throughout this paper, weconsider a general “1 → ρ a along with a blank ancilla I b / I is the identity operator), and attempts to outputtwo copies of ρ into separate modes, a and b .We now consider a second strategy, one that utilizesa trivial version of this cloner by merely shuffling themodes of the two input states. This can be achieved byswapping their modes half of the time, and for the otherhalf, leaving them unchanged. That is, one applies withequal likelihood the SWAP operation ( S ab : ρ a I b / → I a ρ b / I ab = I a ⊗ I b ): ρ a I b / → ( ρ a I b + I a ρ b ) / ≡ t ab . (1)Each output mode of the trivial cloner t ab contains animperfect copy of the input state ρ . Jointly measuring X and Y , one on each trivial clone, yields the resultProb( x, y ) = (Prob( x ) + Prob( y )) /
4. In contrast to ajoint measurement on perfect copies, this result exhibitscorrelations between x and y . These appear because inany given trial, only one of the observables is measuredon ρ , while the other is measured on the blank ancilla.Hence, the apparent correlations are an artifact causedby randomly switching the observable being measured,and are not due to genuine correlations that could bepresent in ρ . While now physically allowed, this jointmeasurement strategy is still insufficient to determine thequantum state ρ .In order to access correlations in the quantum state,we must take advantage of quantum coherence. Insteadof randomly applying S ab or I ab as in trivial cloning, werequire the superposition of these two processes, i.e. thecoherent sum: Π jab = 12 ( I ab + j S ab ) , (2)where now we are free to choose the phase j . Π j is ageneralized symmetry operation that can implement apartial-SWAP gate [20]. For j = +1 ( − ρ a I b /
2. The symmetricsubspace only contains states that are unchanged by aSWAP operation. A projection onto this subspace in-creases the relative probability that ρ a and the blankancilla are identical. In fact, it has been proven that asymmetric projection on the trivial cloner input is theoptimal cloning process, since it maximizes the fidelityof the clones ( i.e. their similarity to ρ ) [25–27].This brings us to our third and final strategy. Opti-mal cloning achieves more than just producing imperfectcopies: the clones are quantum-correlated, i.e. entan-gled [25]. This can be seen by examining the outputstate of the optimal cloner (i.e. with j = 1): o jab = 23 ( Π jab ρ a I b Π j † ab ) = 23 t ab + 13 Re [ j c ab ] , (3) where c ab = S ab ρ a I b and Re [ s ] = ( s + s † ) /
2. While thefirst term is two trivial clones, the second term is the co-herent portion of the optimal clones, and is the source oftheir entanglement. Considered alone, c ab corresponds totwo “twins” of ρ . Like perfect copies, any measurementon either twin gives results identical to what would be ob-tained with ρ [19]. However, the twins are entangled. Assuch, it is important to realize that they are very differentfrom the uncorrelated perfect copies we considered in thefirst strategy. Relative to these ( i.e. ρ ⊗ ρ ), performingthe same joint measurement as before, but on the twins c ab , provides more information about ρ . Measuring X onone twin and Y on the other yields the expectation value (cid:104) xy (cid:105) ρ = Tr( xyρ ), where x = | x (cid:105) (cid:104) x | and y = | y (cid:105) (cid:104) y | areprojectors onto the eigenstates of observables X and Y ,respectively. Classically, this result would be interpretedas a joint probability Prob( x, y ). However, due to Heisen-berg’s uncertainty principle, (cid:104) xy (cid:105) ρ has non-classical fea-tures that shield precise determination of both X and Y .In fact, (cid:104) xy (cid:105) ρ is a “quasiprobability” distribution muchlike the Wigner distribution [4], and has similar prop-erties such as being rigorously equivalent to the state ρ [15]. Unlike the Wigner distribution, it is generallycomplex since xy is not an observable ( i.e. it is non-Hermitian). Although the measurements of X and Y are performed independently on each twin, because thetwins are entangled, it is equivalent to simultaneouslymeasuring the same two observables on a single copy of ρ . This approach is complementary to other joint mea-surement strategies for state determination in which themeasurement itself is entangling, while the copies beingmeasured are separable [28, 29].Performing a joint measurement directly on twins can-not be achieved in a physical process. This is likely partof the reason why previous theoretical investigations con-cluded that optimal cloners were not ideal for joint mea-surements [25, 30, 31]. However, in a joint measurementon optimal clones, Hofmann showed that the contribu-tion from the twins can be isolated from that of the trivialclones [19]. This is because changing the phase j affectsonly the coherent part of the cloning process. Thus, byadding joint measurement results obtained from the op-timal cloner with different phases j , we can isolate thecontribution from the twins and measure (cid:104) xy (cid:105) ρ [32].The experiment is shown schematically in Fig. 1. Aphotonic system lends itself to optimal cloning becausethe symmetry operation Π j in Eq. 2 can be imple-mented with a beam splitter (BS). If two indistinguish-able photons impinge onto different ports of BS1, Hong-Ou-Mandel interference occurs and the photons always“bunch” by exiting BS1 from a single port. By select-ing cases where photons bunch (anti-bunch), one imple-ments the symmetry projector Π +1 ( Π − ) [33]. Thisenabled previous experimental demonstrations of opti-mal cloners for both polarization [34] and orbital an-gular momentum [35, 36] states. However, we must FIG. 1.
Schematic of experimental setup.
A photon in apolarization state ρ a and a photon in a blank state I b / x and y are jointlymeasured by counting coincidences at detectors D1 and D2.When the red (blue) path is blocked, we post-select on thecase where the photons exit the first beam splitter BS1 fromthe same (opposite) port and perform a symmetric projec-tor Π +1 (anti-symmetric projector Π − ), thus making twooptimal clones of ρ . With no path blocked and a phase dif-ference of ϕ = ± π/ Π ± i , respectively. also implement Π ± i . Following a similar strategy asRefs. [20, 37], we use an interferometer to coherentlycombine the symmetric and anti-symmetric projectors,since Π ± i = ( e ± iπ/ Π +1 + e ∓ iπ/ Π − ) / √
2. This isachieved by interfering at BS2 the cases where the pho-tons bunched at BS1 with cases where they anti-bunchedat BS1. In summary, this provides an experimental pro-cedure to vary the phase j and thereby isolate the jointmeasurement contribution of the twins from that of thetrivial clones.We experimentally verify that this procedure works byperforming a joint measurement on trivial clones t ab andshowing that its outcome does not contribute to (cid:104) xy (cid:105) ρ .In particular, we scan the delay between ρ a and I b / Π j . When the delay is larger than thecoherence time of the photons, the BS does not dis-criminate the symmetry of the two-qubit input state.Thus, it simply shuffles the modes of both qubits andproduces trivial clones t ab . We test the procedure bymeasuring (cid:104) xy (cid:105) ρ = (cid:104) dh (cid:105) ρ , where d and h are diago-nal and horizontal polarization projectors, respectively.We use an input state ρ a = h , for which one expects (cid:104) dh (cid:105) ρ = Tr( dhh ) = 0 .
5. In Fig. 2, we show that forlarge delays (cid:104) dh (cid:105) ρ = 0, whereas for zero delay, it ob-tains its full value. This shows that the procedure haseffectively removed the contribution of the trivial clonesto the optimal clone state in Eq. 3, and so the jointmeasurement result is solely due to the twins.A joint measurement on twins of ρ can reveal cor-relations between complementary properties in ρ . We FIG. 2.
Transition from trivial to optimal cloning.
Ahorizontal photon ρ a = h is sent into the cloner. We jointlymeasure complementary observables d and h , one on eachclone, and plot the real ( A ) and imaginary ( B ) parts of (cid:104) dh (cid:105) ρ .For large delays, only trivial clones are produced. Since theycontain no information about (cid:104) dh (cid:105) ρ , our procedure cancelstheir contribution to the joint measurement result. At zerodelay, optimal clones are produced. We isolate the contribu-tion of the twins to the joint measurement, yielding the de-sired value of (cid:104) dh (cid:105) ρ = 0 .
5. The bold lines are theory curvescalculated for intermediate delays [32]. Error bars are calcu-lated using Poissonian counting statistics. measure the entire joint quasiprobability distribution (cid:104) xy (cid:105) ρ for the complementary polarization observables x = { d , a } using diagonal and anti-diagonal projectors,and y = { h , v } using horizontal and vertical projec-tors. This is repeated for a variety of different inputstates ρ . For the input state indicated by the dashedline in Fig. 3A, correlations can be seen in Im (cid:104) xy (cid:105) ρ , asshown in Fig. 3B. With the ability to exhibit correlations, (cid:104) xy (cid:105) ρ is now a complete description of the quantum state ρ [32]. In particular, the wave function of the state (seeFig. 3A) is any cross-section of (cid:104) xy (cid:105) ρ . Moreover, the den-sity matrix (see Fig. 3C) can be obtained with a Fouriertransform of (cid:104) xy (cid:105) ρ . This is the key experimental re-sult. In the classical world, simultaneously measuringcomplementary properties gives the system’s state. Thisresult demonstrates that simultaneously measuring com-plementary observables on twins, similarly, gives the sys-tem’s state.In addition to its fundamental importance, our re-sult has potential practical advantages as a state deter-mination procedure. It is valid for higher dimensionalstates [32] for which standard quantum tomography re-quires prohibitively many measurements. Specifically,a d -dimensional state typically requires O ( d ) measure-ments in O ( d ) bases to be reconstructed tomographically.In contrast, here the wave function is obtained directly( i.e. without a reconstruction algorithm) from 4 d exper-imental measurements of only two observables, X and Y .Our results uncover striking connections with otherjoint measurement techniques, despite the physics of each FIG. 3.
Measuring the quantum state.
Various polarization states | ψ (cid:105) = α | h (cid:105) + β | v (cid:105) are produced by rotating thefast-axis angle θ of a quarter-wave plate with increments of 10 ◦ . We plot the real and imaginary parts of α = (cid:104) h | ψ (cid:105) = (cid:113) cos(4 θ ) + + i sin θ cos θ in A (theory is bold lines, | r (cid:105) = ( | v (cid:105) + i | h (cid:105) ) / √ | l (cid:105) = ( | v (cid:105)− i | h (cid:105) ) / √ B ) and densitymatrix ( C ) are also shown for the input state indicated by the dashed line (color represents amplitude). After processing thecounts with a maximum-likelihood estimation, the average fidelity |(cid:104) ψ | ρ | ψ (cid:105)| of the 18 measured states is 0 . ± . approach being substantially different. For example, thejoint quasiprobability (cid:104) xy (cid:105) ρ is also the average outcomeof another joint measurement strategy: the weak mea-surement of y followed by a measurement of x on a singlesystem ρ [15, 16, 19]. Furthermore, in the continuous-variable analogue of our work, measurements of comple-mentary observables on cloned Gaussian states [38] givea different, but related, quasiprobability distribution forthe quantum state known as the Q-function [10]. Finally,the result of a joint measurement on phase-conjugatedGaussian states can be used in a feedforward to pro-duce optimal clones [39]. These connections emphasizethe central role of optimal cloning in quantum mechan-ics [23, 27] and clarify the intimate relation between jointmeasurements of complementary observables and deter-mining quantum states [4, 6].We anticipate that simultaneous measurements of non-commuting observables can be naturally implemented inquantum computers using our technique, since the oper-ation Π j can be achieved using a controlled-SWAP quan-tum logic gate [19, 40]. 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A detailed figure containing the experimental setup is shown in Fig. S1. A 40 mW continuous-wave diode laser at404 nm pumps a type-II β -barium borate crystal. Through spontaneous parametric down-conversion, pairs of 808nm photons with orthogonal polarization are generated collinearly with the pump laser. The latter is then blockedby a long pass filter. The photon pair splits at a polarizing beam splitter (PBS), and each photon is coupled into apolarization-maintaining single mode fiber. The path length difference between the photon paths is adjusted with adelay stage. A spinning (2 Hz) half-wave plate produces a completely mixed state I / ρ to be cloned at the other fiber output. A displaced Sagnacinterferometer composed of two BS is used instead of the interferometer in Fig. 1 (of main text), since it is more robustto air fluctuations and other instabilities. The phase ϕ between red and blue paths is adjusted by slightly rotatingone of the mirrors in the interferometer in order to change the path length difference between both paths. A series ofwave plates and a PBS are used to implement the projectors x and y . Detectors are single photon counting siliconavalanche photodiodes. Using time-correlation electronics, we count coincidence events that occur in a 5 nanosecondwindow and average over 60 seconds for each measurement. JOINT MEASUREMENT ON OPTIMAL CLONES
For qudits, the d -dimensional observables X and Y are complementary if their eigenstates {| x (cid:105)} and {| y (cid:105)} allsatisfy |(cid:104) x | y (cid:105)| = 1 / √ d . We use the notation x = | x (cid:105) (cid:104) x | . The output of an optimal cloner for qudits is: o jab = 2 d + 1 (cid:16) Π jab ρ a I b Π j † ab (cid:17) (S4)with j = +1. Consider measuring X in mode a and Y in mode b . As shown in Ref. [19], the joint probability ofmeasuring outcome X = x and Y = y is:Prob j ( x, y ) = Tr (cid:104) x a y b o jab (cid:105) = 12( d + 1) (cid:16) (cid:104) x (cid:105) ρ + (cid:104) y (cid:105) ρ + 2Re (cid:16) j (cid:104) xy (cid:105) ρ (cid:17)(cid:17) . (S5)The terms (cid:104) x (cid:105) ρ and (cid:104) y (cid:105) ρ could be obtained from a joint measurement on trivial clones. In contrast, the last term (cid:104) xy (cid:105) ρ is obtained from a joint measurement on twins, and is a joint quasiprobability of simultaneously measuringboth x and y on ρ . In order to isolate the latter term, we use the fact that the joint measurement contribution ofthe trivial clones does not depend on the phase j , giving: (cid:104) xy (cid:105) ρ = d + 12 (cid:88) j = ± , ± i j ∗ Prob j ( x, y ) . (S6)When the input state is pure, that is ρ = | ψ (cid:105) (cid:104) ψ | , then (cid:104) xy (cid:105) ψ = ν (cid:104) y | ψ (cid:105) , where ν = (cid:104) ψ | x (cid:105) (cid:104) x | y (cid:105) . For some x = x , thephase of ν is constant for all y and so the wave function | ψ (cid:105) can be expressed in the basis of Y as | ψ (cid:105) = ν (cid:80) y (cid:104) x y (cid:105) ψ | y (cid:105) .As usual, the constant ν is found by normalizing | ψ (cid:105) . Thus, using Eq. S6, any complex amplitude ψ ( y ) = (cid:104) y | ψ (cid:105) ofthe wave function can be found from: ψ ( y ) = d + 12 ν (cid:88) j = ± , ± i j ∗ Prob j ( x , y ) . (S7)The choice of x is equivalent to choosing a phase reference for the wave function. In Fig. 3A of the main text, weuse x = d , which defines the diagonal polarization as | d (cid:105) = ( | h (cid:105) + | v (cid:105) ) / √
2. We choose to make the normalizationconstant ν a real number, i.e. ν = ( |(cid:104) dh (cid:105)| + |(cid:104) dv (cid:105)| ) / .For mixed input states, the joint quasiprobability (cid:104) xy (cid:105) ρ is related to the density matrix ρ via a discrete Fouriertransform (see derivation below). RELATING THE JOINT QUASIPROBABILITY TO THE DENSITY MATRIX
Here we summarize the connection between the joint quasiprobability distribution and the density matrix. Considerthe d -dimensional complementary observables X and Y with eigenstates { x i } and { y i } such that |(cid:104) x i | y j (cid:105)| = 1 / √ d for any i, j . Without loss of generality [41], one can take the { y i } basis to be defined in terms of a discrete Fouriertransform of { x i } : | y j (cid:105) = (cid:80) d − i =0 | x i (cid:105) exp ( i πx i y j /d ) / √ d . This fixes a phase relation for the inner product of all theeigenstates of both bases: (cid:104) x i | y j (cid:105) = exp ( i πx i y j ) / √ d. (S8)A general d -dimensional quantum state ρ can be written in the basis of Y as ρ = (cid:80) k,l p kl | y k (cid:105) (cid:104) y l | . We wish torelate the coefficients p kl to the joint quasiprobability distribution described in the main text, i.e. (cid:104) xy (cid:105) ρ . Recall thatwe use the notation x = | x (cid:105) (cid:104) x | . Thus (cid:104) xy (cid:105) ρ takes the form of D ij ≡ (cid:104)| x i (cid:105) (cid:104) x i | y j (cid:105) (cid:104) y j |(cid:105) ρ = Tr [ | x i (cid:105) (cid:104) x i | y j (cid:105) (cid:104) y j | ρ ] = (cid:104) x i | y j (cid:105) (cid:104) y j | ρ | x i (cid:105) . Inserting the expanded form of ρ : D ij = (cid:104) x i | y j (cid:105) (cid:104) y j | (cid:88) k,l p kl | y k (cid:105) (cid:104) y l | x i (cid:105) = (cid:104) x i | y j (cid:105) (cid:88) l p jl (cid:104) y l | x i (cid:105) = exp ( i πx i y j ) d d − (cid:88) l =0 p jl exp ( − i πx i y l ) . (S9)This shows that the joint quasiprobability is the discrete Fourier transform of the density matrix. The equation canbe inverted by taking the inverse Fourier transform of both sides: p jl = d − (cid:88) i =0 D ij exp ( i πx i ( y l − y j )) . (S10)In the case of polarization qubits, x = { d , a } and y = { h , v } . Then the equation relating the two density matrixand the joint quasiprobability is: ρ = (cid:18) (cid:104) dh (cid:105) ρ + (cid:104) ah (cid:105) ρ (cid:104) dh (cid:105) ρ − (cid:104) ah (cid:105) ρ (cid:104) dv (cid:105) ρ − (cid:104) av (cid:105) ρ (cid:104) dv (cid:105) ρ + (cid:104) av (cid:105) ρ (cid:19) . (S11)Eq. S11 is used to calculate the density matrix in Fig. 3C of the main text. TRIVIAL TO OPTIMAL CLONES
When the two input photons are temporally distinguishable, Hong-Ou-Mandel interference does not occur at thefirst beam splitter, and the cloner produces trivial clones t ab . Conversely, for temporally indistinguishable photons,we produce o jab . Photons that are partially distinguishable can be decomposed into the form σ jab = | α | o jab + (1 − | α | ) t ab , (S12)where α ∈ [0 ,
1] is a temporal distinguishability factor. In particular, the temporal mode of the delayed photon inmode a can be written as | ζ a (cid:105) = (cid:82) dωφ ( ω ) e − iωτ a † ( ω ) | (cid:105) where τ is the delay, while the other photon in mode b isdescribed by | ζ b (cid:105) = (cid:82) dωφ ( ω ) b † ( ω ) | (cid:105) . For a Gaussian spectral amplitude φ ( ω ) = √ π ∆ ω e − ( ω − ω ω where ω is thecentral frequency of the photons and ∆ ω is their spectral width, the distinguishability factor is given by | α | = |(cid:104) ζ a | ζ b (cid:105)| = e − ∆ ω τ . (S13)In the experiment, we adjust the delay τ by moving the delay stage. The parameter ∆ ω is extracted from fitting aGaussian to the Hong-Ou-Mandel dip (see Fig. S2). EXTENDED DATA
FIG. S4.
Experimental setup.
Details of the experimental setup can be found in the Methods section. A simplified schematicof this setup is shown in the main text. Detector D3 is used for alignment purposes, but otherwise is not used in the experiment.CW: continuous-wave, BBO: β -barium borate, LP: long pass, (P)BS: (polarizing) beam splitter, PM: polarization-maintaining, λ/
2: half-wave plate, λ/
4: quarter-wave plate, D: avalanche photodiode detector.
FIG. S5.
Hong-Ou-Mandel interference.
In order to characterize the spectral width of the photons and to ensure that weare performing the symmetry projector Π ± ab , we measure the width and visibility of the Hong-Ou-Mandel dip at BS1. Withboth input photons horizontally polarized and the blue path blocked, we measure the number of coincidences at detectors D1and D2 as a function of the position of the delay stage. The visibility V = ( C max − C min ) / ( C max + C min ) (where C is thenumber of coincidences) of the dip is ∼ Quantum state tomography of Π +1 ab output. In order to determine the fidelity of our clones, we perform two-photon quantum state tomography on the output o +1 ab of the cloner. Here the input state to be cloned is ρ a = h . By tracingover each subsystem of the measured o +1 ab , we can compute the fidelities F a = (cid:12)(cid:12) (cid:104) h | o +1 a | h (cid:105) (cid:12)(cid:12) and F b = (cid:12)(cid:12) (cid:104) h | o +1 b | h (cid:105) (cid:12)(cid:12) . We obtain F a = 0 .
832 and F b = 0 . / ∼ . Quantum state tomography of Π + iab output. In order to achieve the Π + iab operation, both paths in the interfer-ometer are unblocked and the phase between them is ϕ = π/
2. We test our ability to implement Π + iab by performing two-photonquantum state tomography on the state after the Π + iab operation. As an input, we use the state | hv (cid:105) (cid:104) hv | . In this case, thefidelity of the output state o + iab is 0.850. FIG. S8.
Absolute value squared of the measured wave function.
The data in this figure is the same as the data usedin Fig. 3 of the main text. The polarization state of the input photon as a function of the the quarter wave-plate fast-axis canbe written in the form | ψ (cid:105) = α | h (cid:105) + β | v (cid:105) . Here we plot both | α | = cos θ + sin θ and | β | = 2 sin θ cos θθ