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Experimental Realization of Quantum Tomography of Photonic Qudits Via Symmetric Informationally Complete Positive Operator-Valued Measures

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Experimental Realization of Quantum Tomography of Photonic Qudits Via Symmetric Informationally Complete Positive Operator-Valued Measures PHYSICAL REVIEW X 5, 041006 (2015) Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures N. Bent,1 H. Qassim,1 A. A. Tahir,1 D. Sych,2 G. Leuchs,1,2 L. L. Sánchez-Soto,2,3 E. Karimi,1,* and R. W. Boyd1,4 1Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, K1N 6N5 Canada 2Max-Planck-Institut für die Physik des Lichts, Günther-Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany 3Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain 4Institute of Optics, University of Rochester, Rochester, New York 14627, USA (Received 3 May 2015; published 12 October 2015) Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension. In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime. Fidelities of reconstruction within the range of 0.81–0.96 are obtained for both pure and mixed states. These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown. DOI: 10.1103/PhysRevX.5.041006 Subject Areas: Optics, Quantum Physics, Quantum Information I. INTRODUCTION generalizes the concept of a quantum measurement [10,11]. In the class of POVMs, symmetric informationally Quantum state tomography (QST) is an experimental complete (SIC) POVMs are optimal [12]. These POVMs procedure that allows the reconstruction of a quantum state have the significant advantage that they have been conjec- via direct measurements on identical copies of the state. An tured to exist in arbitrary dimensions [4,12,13]. Furthermore, important issue in QST is optimality, a term that has been they have been calculated numerically for dimensions up to definedinavarietyofways[1–5]. In general, optimality 67 [14], which facilitates their use in configurations that refers to maximizing the amount of information extracted per utilize higher-dimensional spaces, such as the orbital angular single measurement, or equivalently to maximizing the momentum (OAM) degree of freedom of light. In addition, accuracy of the estimation of the quantum state while their optimality means that minimum information is sacri- simultaneously minimizing the sample size. It has been ficed in the QST process, which is of paramount importance shown that QST using mutually unbiased bases (MUBs) in applications such as quantum key distribution (QKD), achieves optimality in the class of projective measurements where the two parties need to minimize the informational loss [1], in which a state is projected onto elements of sets of that comes with verifying the security of the source, as well as orthogonal bases. They have also been shown to minimize the that which comes from the actual key generation [15].For amount of measurements required. However, the complete set instance, the Singapore protocol for performing QKD relies of MUBs cannot in general be constructed for a Hilbert space on QST using SIC POVMs [4,15]. Although, this SIC- of arbitrary dimension, even for a dimension as low as six POVM-based QST has been experimentally realized for [6,7]. This can make QST more complicated for higher qubits [16,17] and qutrits [18], the most interesting cases, i.e., dimensions, although recently a QST method using MUBs higher-dimensional spaces, are yet to be demonstrated. has been examined for dimensions ranging from two to OAM of optical beams provides an unbounded Hilbert five [8]. space in the single-photon regime [19], as they carry a well- The widely believed impossibility of finding complete set defined quantized OAM per photon along the direction of of MUBs for Hilbert spaces whose dimension is not power of propagation. There has been great interest in innovating and aprime[9] calls for the use of positive operator-valued – measures (POVMs), a mathematical construction that verifying quantum protocols using this variable [20 23].In this work, we experimentally demonstrate SIC POVMs for photonic qudit spaces with dimensions up to ten. We then *[email protected] implement this concept to perform optimal QSTof both pure and mixed states in dimensions of 6 and 10, where complete Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- sets of MUBs are unknown. The results show that this bution of this work must maintain attribution to the author(s) and technique can yield QST with high fidelities of estimation, the published article’s title, journal citation, and DOI. and can therefore be utilized in quantum cryptography and 2160-3308=15=5(4)=041006(12) 041006-1 Published by the American Physical Society N. BENT et al. PHYS. REV. X 5, 041006 (2015) quantum communication in all dimensions, including those one to construct a set of d2 minimally overlapping pro- for which the complete set of MUBs are unknown. jectors onto pure qudit states [compare with the dðdþ1Þ projectors for the case of MUBs]. II. THEORETICAL BACKGROUND In more general terms, a POVM can be defined as a set of ˆ positive operators fEig that achieve a decomposition of the A d-dimensional quantum system is represented by a P Eˆ 1 positive semidefinite d × d density matrix that requires identity: i i ¼ . In order for those operators to reveal d2 − 1 independent real numbers for its specification. A full information about the state, they must be IC. von Neumann measurement fixes at most d − 1 real Furthermore, in analogy to MUBs, the set of rank-1 parameters, so d þ 1 tests have to be performed to recon- projectors that minimizes the informational overlap is struct the state. This means that dðd þ 1Þ histograms have to called symmetric [9] and is defined as be recorded: the approach is, thus, suboptimal because this 1 1 Eˆ Φ Φ ; Φ Φ 2 ;i≠ j: number is higher than the number of parameters in the i ¼ d j iih ij jh ij jij ¼ d 1 ð2Þ density matrix. The von Neumann strategy can be further þ optimized regarding this redundancy when the bases in The structure of a general SIC POVM is quite complex [12]. which the measurements are performed are MUBs: two However, the particular case of group-covariant SIC POVMs orthonormal bases are MUBs if whenever we choose one has been investigated thoroughly and it has been conjectured state in the first basis, and a second state in the second basis, that they exist in arbitrary dimensions [12,14]. In particular, 1=d the modulus squared of their overlap is equal to [24]. Weyl-Heisenberg covariant SIC POVM elements can be When the dimension of the Hilbert space is a prime power, it generated by applying the d2 displacement operators is known that there exists a maximal set of d þ 1 MUBs [25]. Xd−1 Ψ k ˆ jk=2 jm Put in a more formal way, let us denote by j αki the th Djk ¼ ωd ωd jk⊕mihmjð3Þ element (0 ≤ k ≤ d − 1) of the αth orthonormal basis of a m¼0 maximal set of MUBs (0 ≤ α ≤ d). Then we have on some reference (or fiducial) vector jfi. Here, fjiig is an ˆ 1 ω Eαk ¼ jΨαkihΨαkj; orthonormal basis of the space in question, d ¼ d þ 1 expð2πi=dÞ is the dth root of the identity, and ⊕ represents 2 1 addition modulo d [12]. The SIC POVM elements are then jhΨαkjΨαlij ¼ ; α ≠ β; ð1Þ d obtained as the subnormalized rank-1 projectors onto the ˆ resulting states. In this manner, an entire set of SIC POVM where the orthogonal projectors Eαk are normalized so P elements can be generated if a single fiducial vector is Eˆ 1 p αk αk ¼ . The probability αk of obtaining an outcome determined. Numerical solutions for fiducial vectors for Ψ ’ p Eˆ ρˆ ρˆ j αki is given by Born s rule αk ¼ Trð αk Þ, where dimensions up to 67 can be found in Ref. [12]. Once a set stands for density operator of the system. of SIC POVM elements is identified, the probability pi of Of course, more general classes of measurements exist obtaining an outcome jΦii is also given by Born’srule, that generalize the von Neumann measurements. This class ˆ pi ¼ TrðEiρˆÞ. is represented by the POVM measurements. To gain To gain more insights into these matters, let us look at the physical insights, we recall [10] that the most general simple yet relevant example of a single qubit (e.g., a spin POVM can always be obtained by coupling a system A to 1=2 B ). Now, MUBs measurements correspond to three suc- an ancilla and performing a von Neumann measurement cessive Stern-Gerlach measurements performed along on the complete system. When both the system and its orthogonal directions. This allows one to directly infer the ancilla are qudits, the full system lives in a d2-dimensional three Bloch parameters, sx ¼hσˆ xi, sy ¼hσˆ yi, and sz ¼hσˆ zi d2 Hilbert space, which makes it possible to measure (where σ’s denote Pauli matrices), and hence to determine probabilities during a von Neumann measurement, and so, unambiguously the density matrix of the system via this fixes d2 − 1 parameters. When the coupling to the ancilla and the von Neumann measurement are judiciously 1 ρˆ 1 s σˆ s σˆ s σˆ : chosen, we are able in principle to infer the value of the ¼ 2 ð þ x x þ y y þ z zÞ ð4Þ density matrix of the initial qudit system from the knowl- edge of those d2 − 1 parameters, in which case the POVM The three MUBs determine six points, which are the vertices is said to be informationally complete (IC).
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