High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments
Amin Babazadeh, Manuel Erhard, Feiran Wang, Mehul Malik, Rahman Nouroozi, Mario Krenn, Anton Zeilinger
HHigh-Dimensional Single-Photon Quantum Gates: Concepts and Experiments
Amin Babazadeh,
1, 2, 3
Manuel Erhard,
1, 2, ∗ Feiran Wang,
1, 2, 4
MehulMalik,
1, 2
Rahman Nouroozi, Mario Krenn,
1, 2 and Anton Zeilinger
1, 2, † Vienna Center for Quantum Science & Technology (VCQ), Faculty of Physics,University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. Institute for Quantum Optics and Quantum Information (IQOQI),Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. Physics Department, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran. Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,Shaanxi Province, Xian Jiaotong University, Xian 710049, China. (Dated: February 24, 2017)Transformations on quantum states form a basic building block of every quantum informationsystem. From photonic polarization to two-level atoms, complete sets of quantum gates for avariety of qubit systems are well known. For multi-level quantum systems beyond qubits, thesituation is more challenging. The orbital angular momentum modes of photons comprise onesuch high-dimensional system for which generation and measurement techniques are well-studied.However, arbitrary transformations for such quantum states are not known. Here we experimentallydemonstrate a four-dimensional generalization of the Pauli X-gate and all of its integer powers onsingle photons carrying orbital angular momentum. Together with the well-known Z-gate, this formsthe first complete set of high-dimensional quantum gates implemented experimentally. The conceptof the X-gate is based on independent access to quantum states with different parities and can thusbe easily generalized to other photonic degrees-of-freedom, as well as to other quantum systems suchas ions and superconducting circuits.
Introduction – High-dimensional quantum states haverecently attracted increasing attention in both funda-mental and applied research in quantum mechanics [1–5].The possibility of encoding vast amounts of informationon a single photon makes them particularly interestingfor large-alphabet quantum communication protocols [6–9], as well as for investigating fundamental questions con-cerning local realism or quantum contextuality [10, 11].The temporal and spatial structure of a photon providesa natural multi-mode state space in which to encodequantum information. The orbital angular momentum(OAM) modes of light [12] comprise one such basis ofspatial modes that has emerged as a popular choice forexperiments on high-dimensional quantum information[13]. While techniques for the generation and measure-ment of photonic quDits carrying OAM are well known[14–16], efficient methods for their control and transfor-mation remain a challenge. No general recipe is knownso far, and experimentally feasible techniques are knownonly for special cases.Here we experimentally demonstrate a four-dimensional X-gate and all of its integer powerswith the orbital angular momentum modes of singlephotons. The four-dimensional X-gate is a generalizationof the two-dimensional σ x Pauli transformation andacts as a cyclic ladder operator on a four-dimensionalHilbert space. The cyclic transformation required forthis gate was designed through the use of the computeralgorithm M ELVIN [17] and recently demonstrated withclassical states of light [18]. The Z-gate for OAM quDits(the generalization of the two-dimensional σ z Paulitransformation) introduces a mode-dependent phase, which can be implemented simply with a single opticalelement [19, 20]. With all powers of the high-dimensionalX- and Z-gate, we arrive at a complete basis of quDitgates, which in principle allows for the construction ofarbitrary unitary operations in a four-dimensional statespace [21] (see Appendix for details).It is interesting to compare OAM with other high-dimensional degrees of freedom that allow for the en-coding of quantum information. For path-encoding inparticular, it is known how arbitrary single-quDit trans-formations can be performed in a lossless way [22]. Suchtransformations have been implemented recently on in-tegrated photonic chips for the generation and transfor-mation of entanglement [23, 24]. General unitary trans-formations such as these are not known for the photonicOAM degree-of-freedom. In addition to being naturalmodes in optical communication systems with cylindri-cal symmetry, photons carrying OAM offer an importantadvantage over path and time-bin encoding in that quan-tum entanglement can be generated [25] and transmitted[26] without the need for interferometric stability. There-fore, the development of new controlled transformationsfor photonic OAM, as we show here, fills an importantgap.The X-gate demonstrated here uses the ability to sorteven and odd parity modes as a basic building block[19]. This concept can be extended to other photonicdegrees of freedom such as frequency [27, 28], and usedin other quantum systems such as trapped ions [29, 30],cold atoms [31], and superconducting circuits [32] for con-structing similar high-dimensional quantum logic gates.
High-dimensional Pauli gates – The Pauli matrix a r X i v : . [ qu a n t - ph ] F e b group has applications in quantum computation, quan-tum teleportation and other quantum protocols. Thisgroup is defined for a single quDit (a single photon withd-dimensional modes) in the following manner [33, 34]: X = d − (cid:88) (cid:96) =0 | (cid:96) ⊕ (cid:105) (cid:104) (cid:96) | (1a) Z = d − (cid:88) (cid:96) =0 | (cid:96) (cid:105) ω (cid:96) (cid:104) (cid:96) | (1b)where (cid:96) ∈ { , ..., d − } refers to the different modesin the d-dimensional Hilbert space and (cid:96) ⊕ ≡ ( (cid:96) + 1)mod d . The Z-gate introduces a mode-dependent phasein the form of ω = exp( πid ). Furthermore, the Y-gatecan be written Y = X · Z . While the two-dimensionalX-gate swaps two modes with one another, in high di-mensional Hilbert spaces ( d >
2) it takes the form of acyclic operation: X | (cid:96) (cid:105) = | (cid:96) + 1 (cid:105) . (2)This results in each state being transformed to its near-est neighbor in a clockwise direction, with the last state | d − (cid:105) being transformed back to the first one | (cid:105) . TheY-gate can be expressed as a combination of Z andX gates. While powers of Z lead to different mode-dependent phases, integer powers of X shift the modesby a larger number: X n = d − (cid:88) (cid:96) =0 | (cid:96) ⊕ n (cid:105) (cid:104) (cid:96) | (3)The X -gate, for example, transforms each mode to thesecond nearest mode. Likewise, the conjugate of X leadsto a cyclic operation in the counter-clockwise direction, X † = d − (cid:88) (cid:96) =0 | (cid:96) (cid:9) (cid:105) (cid:104) (cid:96) | . (4) Experimental implementation – A Z-gate for photonscarrying OAM can simply be achieved by using a Doveprism, which has been shown recently [4, 35–37]. Sincethe Y-gate can be achieved by a combination of Z and Xgates, it is sufficient to focus on the X-gate and its pow-ers. Fig.1a shows the schematic of the X-gate. It consistsof two parity sorters (PS1 and PS2) and a Mach-Zehnderinterferometer (MZI) that is implemented between them.The input photon is first incident on a spiral phase platethat adds one quantum of OAM quantum (SPP (cid:96) +1 ) ontothe photon before it enters PS1. The parity sorter is aninterferometric device which then sorts the photon ac-cording to its mode parity [19]. For the first-order cyclictransformation, the sign of the odd photon needs to beflipped after PS1. This is achieved by reflecting the oddoutput photon at a mirror placed in one MZI path, while FIG. 1. The conceptual diagrams for the three types of quan-tum logic gates. The input states for each case is (-2,-1,0,1).a) A spiral phase plate (SPP) adds +1 to the mode, leading to(-1,0,1,2). Afterwards, the first parity sorter separates evenand odd modes, while the second one combines them again –which forms a large interferometer. Within the interferome-ter, the even mode have an odd number of reflections, whichleads to the correct output modes (-1,0,1,-2). b) For the X gate, the input modes are directly separated into even andodd modes. After a reflection in each arm, the even modesare increased by 2. The two arms are recombined at the PS2and all of the modes are reflected for changing the signs of themodes. That leads to (0,1,-2,-1). c) In the X † transformation,the different parity modes are separated again, and the evenpart gets reflected twice, while the odd modes are reflectedonce. After recombination, the (cid:96) of the modes is decreased byone, which leads to (1,-2,-1,0). Note that in the experiment,it can be adjusted whether even or odd modes are reflected atthe parity sorter. In this conceptual diagrams, for simplicitywe have chosen PS1 to reflect even modes and PS2 to reflectodd modes. even photons undergo two reflections that preserve thesign of their OAM mode (see Fig.1a). The modes arethen input into PS2, which coherently recombines theminto the same path.Interestingly, the X-gate can be converted into the X -gate and X † -gate with only minor changes to the exper-imental setup (for d=4, X † =X ). For constructing theX -gate, the SPP (cid:96) +1 is removed and an SPP (cid:96) +2 replacesthe extra reflection in the even MZI path (Fig.1b). TheX † -gate is achieved by simply moving the SPP (cid:96) +1 fromthe input of PS1 and replacing it with an SPP (cid:96) − at theoutput of PS2 (Fig.1c). One should note that in princi-ple, these changes can be implemented rapidly and with-out physically moving optical components via the use ofdevices such as a spatial light modulator or a digital mi-cromirror device. FIG. 2. Experimental setup for the four-dimensional X-gate (additional experimental details are partially transpar-ent). A 405nm CW laser pumps a Type-II ppKTP crystal(not shown), creating photon pairs entangled in orbital angu-lar momentum (OAM). The idler photon is used for heraldingthe signal photon in a particular OAM mode. After passingthrough a (cid:96) = +1 spiral phase plate, the signal photon is inputinto a parity sorting interferometer, which separates the oddand even OAM components of the photon. After traversinga series of mirrors, the odd and even components are coher-ently recombined in a Mach-Zehnder interferometer throughthe use of a half-wave plate (HWP) and polarizing beam split-ter (PBS). A spatial light modulator (SLM) and single modefiber are used to perform projective measurements of OAMmodes and their superpositions.
The experimental setup is depicted in Fig.2. Weuse heralded single photons produced via the process ofType-II spontaneous parametric down conversion process(SPDC) in a 5mm long periodically poled Potassium Ti-tanyl Phosphate (ppKTP) crystal pumped by a 405nmdiode laser. In the SPDC process, conservation of thepump angular momentum leads to the generation of pho-ton pairs with a degenerate wavelength of λ =810nm thatare entangled in OAM. Therefore whenever the idler pho-ton is measured to be in mode | + (cid:96) (cid:105) , the signal photon isfound to be in mode |− (cid:96) (cid:105) . Thus, by heralding the idlerphoton in a particular OAM mode, we can select theOAM quantum number of the signal photon that is inputinto the logic gate. Here, we use the OAM quantum num-bers of -2,-1,0 and 1 for demonstrating our 4-dimensionalquantum logic gates. By changing the mode number be-fore and after the transformation, the X-gate can be usedwith every connected 4-dimensional subspace.The parity sorter was originally proposed as an MZIwith a dove prism in each arm [19]. The relative rotationangle between the two dove prisms is set at 90 ◦ , whichintroduces an (cid:96)π phase difference between the two arms.Depending on the parity of OAM mode ( (cid:96) ) of the input FIG. 3. Data showing the operation of the a) Identity, b)X-gate, c) X -gate, and d) X † -gate on the four-dimensionalset of input states {|− (cid:105) , |− (cid:105) , | (cid:105) , | (cid:105)} . Each row shows themeasured normalised coincidence rate in every output modefor a given input mode. The X-gate implements a clockwisecyclic transformation ( − → − → → → − -gate swaps the odd and even modes ( − ↔ , − ↔ † -gate performs a counter-clockwise cyclic transforma-tion (1 ← − ← − ← ← , and X † gates are 87.3%, 90.4%, and88.4% respectively. photon, constructive or destructive interference results ineven and odd modes exiting different outputs of the MZI.For long-term stability, in our case we implement this in-terferometer in a double-path Sagnac configuration [38].Two adjacent Sagnac loops allow for the positioning of adove prism in each loop. The outputs of this Sagnac in-terferometer are then directly input into the second MZI(denoted as OAM manipulating in Fig.2). In the sec-ond interferometer the sign of odd modes is flipped byreflection on an extra mirror. A trombone system in theodd arm is used to adjust the relative path difference toachieve a coherent combination of even and odd modes.The concept of the quantum gates (discussed in Fig. 1)allows in principal for a lossless operation. For simplicity,we replace the second parity sorter with a polarizationbeam splitter (PBS). This allows the odd and even modesin the MZI to be recombined in a stable manner, albeitwith an additional loss of 50%.Now we explain the experimental details of the X-Gate
TABLE I. Transformation efficiency E (cid:96) i for each input state | (cid:96) i (cid:105) . The efficiency is calculated by dividing the number ofphotons in the correct output state by the total number ofcounts measured in all four states.Inputmode |− (cid:105) |− (cid:105) | (cid:105) | (cid:105) X-Gate 88 . ± .
2% 90 . ± .
6% 90 . ± .
3% 80 . ± . -Gate 90 . ± .
1% 87 . ± .
2% 90 . ± .
7% 93 . ± . † -Gate 85 . ± .
2% 87 . ± .
8% 92 . ± .
5% 88 . ± . (Fig. 1&2). A 4-dimensional subset of OAM modes (cid:96) ∈{− , − , , } is shifted by one leading to {− , , , } .The parity sorter separates even and odd modes. Thepath for the even modes experience an odd numberof reflections that causes in a sign flip and results in {− , , , − } . The coherent combination at the PBS andsubsequent erasure of polarization information completesthe X-gate: ( − → − → → → − and X † gate work similarly, see Fig. 1. The experimental resultsof the gate operations are depicted in Fig. 3. The proba-bility P i,j to detect a photon in mode j when sending inone in mode i is given by P ( i, j ) = |(cid:104) j out | i in (cid:105)| (cid:80) n |(cid:104) n out | i in (cid:105)| . Theaverage probability of the expected mode for the X, X ,and X † gates are 87.3%, 90.4%, and 88.4% respectively,see Table I.In order to demonstrate a transformation of a coherentsuperposition, we use | ψ in (cid:105) = ( | (cid:105) ± | (cid:105) ) / √ | φ out (cid:105) = ( | (cid:105)±|− (cid:105) ) / √ | ¯ φ out (cid:105) = ( | (cid:105) ∓ |− (cid:105) ) / √ Conclusion – We have shown the experimental gener-ation of the four-dimensional X-gate and all of its uniquehigher orders, including the X and X gates. Togetherwith the well known Z-gate, this forms a complete basis oftransformations on a four-dimensional quantum system.This means that it can in principle be used to constructevery four-dimensional unitary operation. The X-gateis a basic element required for generating large classesof entangled states, such as the set of four-dimensionalBell states [39] or general high-dimensional multi-particlestates [5]. Such states can be used, for example, in testsof quantum contextuality [40] and for Bell-like tests oflocal-realism in a higher-dimensional state space [10, 41].These quantum logic gates can find application in var-ious high-dimensional quantum protocols, such as high- FIG. 4. Action of the X-gate on coherent superpositionsof quantum states. It shows that the operation conservesthe phases. The correlation matrix confirms that the inputquantum states | ψ i (cid:105) = | (cid:105)±| (cid:105)√ are transformed coherently tothe output quantum states ( | φ o (cid:105) = | (cid:105)±|− (cid:105)√ ). The visibilityof the transformation process is calculated to be 90.9%. dimensional quantum key distribution [6, 7, 42, 43] wheretransformations between mutually unbiased bases arenecessary. Other applications could include multi-partysecret sharing [8] or dense coding [44], where transfor-mations between orthogonal sets of entangled states arerequired. In quantum computing where complete setsof quantum gates are necessary, high-dimensional quan-tum states allow for the efficient implementation of gates[45, 46] and offer advantages in quantum error correction[47].Interestingly, a high-dimensional generalization of theCNOT gate consists of a controlled-cyclic transformation[48]. In combination with polarization, one can immedi-ately create a three-, six- and eight-dimensional general-ization of our method [17]. An important next step isthe construction of high-dimensional two-particle gates.This would allow the implementation of complex quan-tum algorithms such as quantum error correction in highdimensions [47]. ACKNOWLEGDEMENTS
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APPENDIXAll unitaries from X and Z
Here we briefly show that it is possible to constructevery unitary transformation in four dimensions in termsof X and Z-gates, and integer powers of them. Here wefollow the construction in [21]. The derivation is gen-eral for arbitrary dimensions d . For the purpose of ourexperiment, in the end we will make d → D ( l, m ) = exp (cid:18) i πlm (cid:19) Z l X m . (5)From these operators, we can write a minimum and com-plete set of Hermitian operators Q l,m = χ D ( l, m ) + χ ∗ D ( l, m ) † , (6)with χ = i . With a linear superposition of the basis el-ements, arbitrary hermitian matrices can be constructedin the form A = d − (cid:88) l =0 d − (cid:88) m =0 c l,m Q l,m , (7)with real coefficients c l,m , and d being the dimension ofthe Hilbert space. Every unitary transformation can bewritten as the exponent of a hermitian generator, suchthat U = exp ( iA ) = ∞ (cid:88) n =0 i n n ! A n . (8)U has infinitely many terms which are combinations ofX and Z gates, and X and Z do not commute. However,because of X · Z = − iZX , one can always write U = d − (cid:88) l =0 d − (cid:88) m =0 g l,m X l Z m , (9)with complex coefficients g l,m . For d = 4, due to X = 1and Z = 1, the sum only goes from 0 ≤ l, m ≤ U = (cid:88) l =0 3 (cid:88) m =0 h l,m X l Z m , (10)with complex coefficients h l,ml,m