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Construction and Physical Realization of Povms on a Symmetric Subspace S

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Construction and Physical Realization of Povms on a Symmetric Subspace S Construction and Physical Realization of POVMs on a symmetric subspace S. P. Shilpashree1 3 ∗ Veena Adiga2 † Swarnamala Sirsi3 ‡ 1 K.S. School of Engineering and Management, Bangalore 2 St. Joseph’s College(Autonomous), Bangalore 3 Yuvaraja’s College, University of Mysore, Mysore Abstract. Positive Operator Valued Measures (POVMs) are of great importance in quantum information and quantum computation. Using the well-known spherical tensor operators, we propose a scheme of constructing a set of N−qubit POVMs operating on the symmetric state space of dimension N + 1 which is a subspace of the 2N dimensional Hilbert space. By invoking Neumark’s theorem we show for a 2-qubit case, that POVM’s in the 3 dimensional symmetric space can be physically realized as projection operators in the larger 4-dimensional Hilbert space. Keywords: POVM, quantum information, quantum computation, spherical tensors 1 Introduction 2 Operator representation in terms of ir- Positive-Operator-Valued-Measures(POVMs), are the reducible tensors most general class of quantum measurements, of which The systematic use of tensor operators was first sug- von Neumann measurements are merely a special case. gested by Fano[8]. In this representation, any Hermitian The rapidly developing quantum information theory has matrix ‘H’ in the symmetric space of dimension N + 1 in generated a lot of interest in the construction and ex- terms of spherical tensor parameters can be expressed as perimental implementation of POVMs [1]. Some tasks that cannot be performed using projective measurements 2j +k ~ X X k k† ~ can be completed using POVMs. For example, a set of H(J) = hq τq (J) (1) non-orthogonal states cannot be distinguished using pro- k=0 q=−k jective measurements, but can be discriminated unam- where τ k 0s (with τ 0 = I , the identity operator) are biguously using POVMs [2]. This concept can be used q 0 the irreducible (spherical) tensor operators of rank ‘k’ in to construct POVMs in conclusive teleportation [3]. Re- the (2j + 1) dimensional spin space. The operators τ k 0s cently, POVMs instead of usual projective measurements q of rank ’k’ and projection ’q’ are homogeneous polyno- has been utilized to realize some quantum communica- mials constructed out of angular momentum operators tion protocols [4]. In the context of Remote State Prepa- J , J , J . In particular, following the well known Weyl ration (RSP), POVM has already attracted a lot of at- x y z construction [10] for τ k 0s in terms of angular momentum tention [5]. Recently it has been shown [6] how quantum q operators, we have filtering can be performed in the POVM setting. In this paper we introduce the most general method k ~ ~ ~ k k k τq (J) = Nkj (J · 5) r Yq (ˆr) , (2) of constructing POVMs in terms of detection operators, k on N + 1 dimensional symmetric space which is a sub- where Nkj are the normalization factors and Yq (ˆr) are space of 2N dimensional Hilbert space in which N qubits the spherical harmonics. k 0 reside. A set of N-qubit pure states that respect per- The τq s satisfy the orthogonality relations mutational symmetry are called symmetric states. Sym- 0 metric space can be considered to be spanned by the k† k T r(τq τ 0 ) = (2j + 1) δkk0 δqq0 (3) eigen states |jm >, −j ≤ m ≤ +j of angular momentum q 2 operators J and Jz, where j = N/2. A large number and k† q k of experimentally relevant states [7] possesses symmetry τq = (−1) τ−q , under particle exchange, which significantly reduces the Here the normalization has been chosen so as to be in computational complexity. The detection operators are agreement with Madison convention [11]. The matrix found to be the well known irreducible tensor operators elements of the tensor operator are given by τ k which are used for the standard representation [8] of q √ spin density matrices. Further, by invoking Neumark’s hjm0|τ k(J~)|jmi = 2k + 1 C(jkj; mqm0) (4) theorem [9], we demonstrate the physical implementa- q tion of our symmetric two-qubit POVMs as projection where C(jkj; mqm0) are the Clebsch-Gordan coefficients. operators in the two qubit tensor product space. ∗[email protected] 3 Positive Operator valued Measure † [email protected] The spherical tensor operators can be used as detection ‡[email protected] operators in constructing a set of POVMs for a symmetric space as is the unitary matrix which transforms computational ba- † τ kτ k sis to the angular momentum basis |11i, |10i, |1−1i, |00i. Ek = q q , (5) q N 2 Thus satisfying the properties of partition of unity, hermiticity 1 0 0 0 a a 1 1 1 1 b+c and positivity. 1 0 2 2 0 b 2 1|ψi = 1 1 = b+c 3 0 2 2 0 c 3 2 3.1 Partition of unity 0 0 0 0 d 0 k P k (13) Since Eq ’s add up to unity i.e., kq Eq = 1, we have where † k k 2 2 2 2 X τq τq 0 |ψi = a| ↑↑i+b| ↑↓i+c| ↓↑i+d| ↓↓i; |a| +|b| +|c| +|d| = 1 hjm| |jm i = δmm0 (6) N 2 (14) kq is the most general pure state in the 2-qubit state space. with Observe that the resultant state is a symmetric state X 2 0 2 0 0 0 2 and is given by 1 a|11i + √1 (b + c)|10i. Thus Ek’s N = [k ] C(jk j; m − q q m) (7) 3 3 2 q k0q0 project vectors in the 4-dimensional Hilbert space onto 1 3-dimensional symmetric space. Note that 1 can be ex- 3.2 Hermiticity pressed in terms of the well-known Pauli spin matrices, Since given by k k† † k† † k† k k† (τq τq ) = (τq ) τq = τq τq , (8) 1 1 1 1 † 1 = (I1 ⊗ I2) + (σz(1) ⊗ I2) + (I1 ⊗ σz(2)) Ek = Ek for all k, q 6 2 2 q q 1 1 + (σ (1) ⊗ σ (2)) + (σ (1) ⊗ σ (2)). (15) 3.3 Positivity 2 x x 2 y y k We can also show that hψ|Eq |ψi ≥ 0 for all k, q and References k all hψ| ∈ H, i.e., Eq are positive operators. [1] S. E. Ahnert and M. C. Payne. Phys. Rev. A 71: It can be shown that the above POVMs do not 012330,2005; M. Ziman and V. Buzek. Phys. Rev. form an orthogonal set. A 72: 022343, 2005; S. E. Ahnert and M. C. Payne. If we perform the measurement and do not record the Phys. Rev. A 73: 022333, 2006. results then the post measurement state is described by [2] I.D. Ivanovic. Phys. Lett. A 123: 257, 1987; D. Dieks. the density operator, Phys. Lett. A 126: 303, 1988. f k i k ρ = Eq ρ Eq (9) [3] T. Mor. arXiv:quant-th/9608005v1,1996; T. Mor arXiv:quant-ph/9906039, 1999. 4 Neumark’s Theorem [4] F.L. Yan and H.W. Ding, Chin. Phys. Lett. 23: 17, k Next, we address the question, can a POVM Eq acting 2006; J. Wu. Int. J. Theor. Phys. 49: 324, 2010; Z.Y. on a symmetric space, be interpreted as resulting from a Wang. Int. J. Theor. Phys. 49: 1357, 2010; J.F. Song measurement on a larger space? The answer is yes. For and Z.Y. Wang. Int. J. Theor. Phys. 50: 2410, 2011; eg., Consider one of the POVM’s, namely P. Zhou. J. Phys. A. 45: 215305, 2012. 1 0 0 [5] Siendong Huang. Physics Letters A 377: 448, 2013 1 E1 = 0 1 0 (10) and the references therein. 1 3 0 0 0 [6] Ram A. Somaraju, Alain Sarlette and Hugo Thien- pont. arXiv:1303.2631v2 [quant-ph] 2013. in the symmetric |1mi basis, m = 1, 0, −1. The relation- ship between |1m > basis and the computational basis is [7] R. Prevedel et al. Phys. Rev. Lett. 103: 020503, 2009. |↑↓i+|↓↑i such that |11i = | ↑↑i, |10i = √ and |1−1i = | ↓↓i. 2 [8] U. Fano. Phys. Rev. 90: 577, 1953. 1 The Matrix representation of E1 in the 2-qubit state space of dimension 4, in the computational basis [9] A. Peres. Quantum Theory: Concepts and Meth- | ↑↑i, | ↑↓i, | ↓↑i, | ↓↓i is given by ods (Kluwer Academic Publishers, Dordrecht, The Netherlands,1993). 1 = U †(E1 ⊕ 0)U, (11) 1 1 [10] Racah, G. Group theory and spectroscopy. CERN, where ⊕ represents the direct sum and report 61-8, 1961; Rose, M. E. Elementary theory of angular momentum. John Wiley, New York, 1957. 1 0 0 0 1 1 [11] Satchler et al. Proceedings of the International Con- 0 √ √ 0 U = 2 2 , (12) ference on Polarization Phenomena in Nuclear Reac- 0 0 0 1 tions. University of Wisconsin Press, Madison, 1971. 0 √1 − √1 0 2 2.
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