Experimental scheme for qubit and qutrit symmetric informationally complete positive operator-valued measurements using multiport devices

Abstract

It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and by performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather non-trivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been demonstrated by several experimental groups around the world. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.

Conference key agreement with single-photon interference

“…(iii) The central node applies a Bell-multiport beam splitter [43][44][45][46][47][48] with M input and output ports 1 to the incoming pulses and features a threshold detector D i at each output port (i=1, K, M). The action of the multiport beam splitter is defined by the unitary transformation given in figure 1.…”

Isoentangled Mutually Unbiased Bases, Symmetric Quantum Measurements, and Mixed-State Designs

“…Further entanglement properties of SICs were studied in [13,14]. Although entanglement of states forming MUBs in composed dimensions was analyzed [15][16][17], the analogous problem of finding a full set of iso-entangled MUBs remained open till now even for two-qubit system.Collections of states forming a SIC measurement or a set of MUBs found numerous applications in the theory of quantum information [6,12,18,19]. They belong to the class of projective designs: finite sets of evenly distributed pure quantum states in a given dimension d such that the mean value of any function from a certain class is equal to the integral over the of pure states with respect to the unitarily invariant Fubini-Study measure [3,20,21].…”

“…It differs also from the regular dodecahedron of Zimba [36], which describes a basis of five orthogonal anticoherent states in H 5 in the stellar representation Projective and unitary designs.-Let us recall the standard definition of a projective t-design. It is an ensemble of M pure states, |ψ j ∈ H d M j=1 , such that for any polynomial f t of degree at most t in both components of the states and 3 their conjugates its average value is equal to the integral with respect to the unitarily invariant Fubini-Study measure dψ FS over the entire complex projective space of pure states,(2)The notions of pure-state t-designs and unitary t-designs, consisting of matrices evenly distributed over the unitary group [22], found numerous applications in quantum information processing [23][24][25][26] and has been applied in experiments [19,27,37]. They can be considered as a special case of averaging sets, which are known to exists for arbitrary sets endowed with a probability measure [30].…”

Quantum conical designs

“…However, these are not the only examples; in particular, it is known that such designs exist in every finite dimension [11,12]. Among other things they have applications to quantum tomography [4,7,8,[12][13][14][15][16], cryptography [8,[17][18][19][20][21], densecoding [8,22], teleportation [8,23], entanglement detection [24][25][26][27], quantum communication [28][29][30][31] and cloning [4,8,32,33] (references given being representative only). They are important in quantum foundations, sics being a mathematical cornerstone of QBism [34,35].…”