SIC-POVMs and Compatibility among Quantum States
نویسنده
چکیده
An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.
منابع مشابه
On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States
We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to con...
متن کاملSIC-POVMs exist in all dimensions
The most general type of measurement in a quantum system is a Positive Operator Valued Measure (POVM). In order for the POVM to completely determine the quantum state being measured it must be Informationally Complete. Maximal independence of outcomes is critical in Cryptographic applications [7], thus the pairwise inner products of the POVM elements need to be uniform. Thus we are interested i...
متن کاملSymmetric Informationally Complete Quantum Measurements
We consider the existence in arbitrary finite dimensions d of a POVM comprised of d rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC-POVM) and is equivalent to a set of d equiangular lines in C . SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum me...
متن کاملOn SIC-POVMs in Prime Dimensions
The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a “canonical” unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the...
متن کاملConstruction of all general symmetric informationally complete measurements
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs), and thereby show that SIC-POVMs that are not necessarily rank 1 exist in any finite dimension d. In particular, we show that any orthonormal basis of a real vector space of dimension d 2 − 1 corresponds to some general SIC POVM and vice versa. Our...
متن کامل