Quantum conical designs

Graydon, Matthew A.; Appleby, D. M.

doi:10.1088/1751-8113/49/8/085301

Cited by 31 publications

(31 citation statements)

References 87 publications

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“…While these inequalities have been obtained by extensively considering all sets of MUBs and SIC vectors, analytic expressions for the upper and lower bounds can be derived for a quantum 2-design ( ) are proven in [14] and [17], respectively. Lower bounds are shown in [15] and later in [18]. As mentioned earlier, when the full measurement set of a quantum 2-design is used, it is more efficient to exploit the measurements for state tomography, and use theoretical tools to solve the separability problem that is known to be NP-hard.…”

Section: Lower Boundsmentioning

confidence: 93%

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Linking entanglement detection and state tomography via quantum 2-designs

2019

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We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion for bipartite systems with equal dimensions is based on measurements from subsets of a quantum 2-design, e.g. mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for quantum state tomography. Secondly, our method is twice as effective as standard witnesses in the sense that both upper and lower bounds can be derived. Thirdly, the scheme can be readily applied to measurement-device-independent scenarios.For quantum information applications it is often more interesting to learn if multipartite quantum states are entangled than to identify quantum states themselves [1,2]. This is in fact what direct detection of entanglement executes, which utilizes an entanglement witness that works with individual measurements followed be postprocessing of the outcomes [3], to provide an experimentally feasible approach for this task [4]. Entanglement detection under less assumptions, for instance, when detectors are not trusted [5][6][7] or dimensions are unknown [8], is of practical significance for cryptographic applications.For the practical usefulness of entanglement detection, it is worth exploring the experimental resources. If a priori information about a quantum state is given, a set of entanglement witnesses may be constructed accordingly and exploited for entanglement detection. With no a priori information multiple entanglement witnesses may be required. One possible method is quantum state tomography which determines a ddimensional quantum state with O(d 2 ) measurements. Then, theoretical tools such as positive maps [9], e.g. partial transpose, or numerical tests involving semidefinite programming [10] can be applied. For entanglement witnesses, however, little is known about the minimal measurements for their realization. In fact, it may happen that repeating experiments for multiple witnesses may be less cost effective than state tomography [11], and quite possible that no useful information is obtained, neither for entanglement detection nor for quantum state identification. This raises questions on the usefulness of entanglement witnesses, in particular when a priori information about a particular state is not available.A useful experimental setup for entanglement detection may distinguish the largest collection of entangled states with as few measurements as possible. It is noteworthy that a tomographically complete measurement can ultimately identify a quantum state so that theoretical tools may completely determine whether it is entangled or separable. From a practical point of view, it would be therefore highly desirable that measurements for ...

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Section: Lower Boundsmentioning

confidence: 93%

“…The connections between entanglement detection, MUBs, and quantum 2-designs have first been explored in [14,15], and subsequent results were found in, e.g. [16][17][18]. Let us emphasize here that entanglement detection via MUBs can also detect bound entangled states, those mixed entangled states from which no entanglement can be distilled.…”

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confidence: 98%

Linking entanglement detection and state tomography via quantum 2-designs

2019

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“…A cognate notion of quantum conical design was recently proposed [28,29], which concerns operators of an arbitrary rank from the cone of mixed quantum states. However, these designs are not suitable to sample the set Ω d of mixed states according to the flat, Hilbert-Schmidt measure.…”

Section: Pacs Numbersmentioning

confidence: 99%

“…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]. These discrete sets of pure quantum states, and analogous sets of unitary operators called unitary designs [22], proved to be useful for process tomography [23], construction of unitary codes [24], realization of quantum information protocols [25], derandomization of probabilistic constructions [26], and detection of entanglement [27].A cognate notion of quantum conical design was recently proposed [28,29], which concerns operators of an arbitrary rank from the cone of mixed quantum states. However, these designs are not suitable to sample the set Ω d of mixed states according to the flat, Hilbert-Schmidt measure.…”

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confidence: 99%

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

et al. 2020

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Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem whether a complete set of five iso-entangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced matrices of the 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius 3/20 located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design -a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. These peculiar constellations of density matrices constitute a class of generalized measurements with additional symmetries useful for the reconstruction of unknown quantum states. Furthermore, we show that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. PACS numbers:Introduction.-Recent progress of the theory of quantum information triggered renewed interest in foundations of quantum mechanics. Problems related to measurements of an unknown quantum state attract particular interest. The powerful technique of state tomography [1,2], allowing one to recover all entries of a density matrix, can be considered as a generalized quantum measurement, determined by a suitable set of pure quantum states of a fixed size d. Notable examples include symmetric informationally complete measurements (SIC) [3,4] consisting of d 2 pure states, which form a regular simplex inscribed inside the set Ω d of all density matrices of size d, and complete sets of (d + 1) mutually unbiased bases (MUBs) [5] such that the overlap of any two vectors belonging to different bases is fixed.The above schemes are distinguished by the fact that they allow to maximize the information obtained from a measurement and minimize the uncertainty of the results obtained under the presence of errors in both state preparation and measurement stages [4,6]. Interestingly, it is still unknown, whether these configurations exist for an arbitrary dimension. In the case of SIC measurements analytical results were known in some dimensions up to d = 48, see [7] and references therein. More recently, a putative infinite family of SICs starting with dimensions d = 4, 8, 19, 48, 124, 323 has been constructed [8], while the general problem remains open. Nonetheless, numerical results suggest [7] that such configurations might exist in every finite dimension d. For MUBs explicit constructions are known in every prime power dimension d [5], and it is uncertain whether such a solution exists otherwise, in particular [9, 10] in dimension d = 6.When the dimension is a ...

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“…Another path from the sporadic SICs to E 6 starts with the qubit SICs, i.e., regular tetrahedra inscribed in the Bloch sphere. Shrinking a tetrahedron, pulling its vertices closer to the origin, yields a type of quantum measurement (sometimes designated a SIM [32]) that has more intrinsic noise. Apparently, E 6 is part of the story of what happens when the noise level becomes maximal and the four outcomes of the measurement merge into a single degenerate case.…”

Section: Ementioning

confidence: 99%