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## Determination of all pure quantum states from a minimal number of observables (2014)

Citations: | 13 - 0 self |

### Citations

300 |
Introduction to Smooth Manifolds.
- Lee
- 2012
(Show Context)
Citation Context ...is a submanifold, π1 restricts to a smooth map on N and thus, since π1|N = g ◦ π2, where g : N/G′ 7→M/G sends an element of N/G′ to its G-orbit in M/G, we have that g is smooth by Proposition 5.19 in =-=[16]-=-. By construction, N/G = π1(N) = g ◦ π2(N) = g(N/G′) ⊆M/G. Thus, N/G is the image of a smooth map, in a manifold of dimension dim(M) − dim(G), from a manifold of dimension dim(N) − dim(G′) < dim(M) − ... |

185 | Recovering low-rank matrices from few coefficients in any basis
- Gross
(Show Context)
Citation Context ...hat arise from measuring it. Due to the exponential growth in the state space dimension of many-body systems, work in this field aims to recover low-rank mixed states from few measurements [17,18,23] =-=[12]-=-. A natural question is: what is the minimal number of observables needed to determine any state x? Indeed, a conjecture was attributed to Ron Wright in 1978 that there exist 3 unitary n×n matrices U1... |

132 |
Phaselift : exact and stable signal recovery from magnitude measurements via convex programming
- Candes, Strohmer, et al.
(Show Context)
Citation Context ...esponding to a quadratic embedding of CPn−1 into R4(n−1). The study of phase retrieval, which in essence is quantum tomography of pure states, has been a topic of much recent activity. The authors of =-=[9, 10, 24]-=- formulated phase retrieval as a matrix recovery problem. In [24], this framework, called PhaseLift, is analyzed and the authors prove that PhaseLift recovers a fixed state with high probability from ... |

73 | Phase retrieval via matrix completion
- Candes, Eldar, et al.
- 2013
(Show Context)
Citation Context ...esponding to a quadratic embedding of CPn−1 into R4(n−1). The study of phase retrieval, which in essence is quantum tomography of pure states, has been a topic of much recent activity. The authors of =-=[9, 10, 24]-=- formulated phase retrieval as a matrix recovery problem. In [24], this framework, called PhaseLift, is analyzed and the authors prove that PhaseLift recovers a fixed state with high probability from ... |

62 |
Algebraic Geometry
- Real
- 1998
(Show Context)
Citation Context ...arts Uij, we have φij(P ∩ Uij) = π/ij(Wij)×W 3 Lemma 2.7 W and Wij are semialgebraic sets in R 4n, with dim(W ) ≤ 3n − 3 and dim(Wij) ≤ 3n− 5. By the Nash stratification theorem, Proposition 9.1.8 in =-=[14]-=-, any semialgebraic set is a union of Nash submanifolds. Therefore, we can express any P ∩Uij as a union of Nash submanifolds of Uij and therefore P is a union of submanifolds of Z. Now, since B is a ... |

53 |
Phase recovery, MaxCut and complex semidefinite programming.
- Waldspurger, d’Aspremont, et al.
- 2015
(Show Context)
Citation Context ...n approach that exploits a polarization identity and expander graphs to yield a computationally efficient approach to recovering states from specifically structured rank-1 observables. The authors of =-=[13]-=- formulate phase retrieval as class of instances of MAXCUT, and along with [27], show that PhaseLift is equivalent to the Goemans-Williamson relaxation on the corresponding instance of MAXCUT. 2 2 Mai... |

49 | Painless reconstruction from magnitudes of frame coefficients - Balan, Bodmann, et al. - 2009 |

49 |
Solving quadratic equations via phaselift when there are about as many equations as unknowns, August 2012, ArXiv e-prints
- Candes, Li
(Show Context)
Citation Context ...y from O(n log n) observables ziz ∗ i , provided the zi are iid gaussian on the unit sphere. Moreover, they proved that PhaseLift is stable with respect to measurement noise. A further improvement by =-=[11]-=- showed that PhaseLift recovers all states with O(n) gaussian observables and moreover the authors of [15], show that trace minimization is unnecessary since PhaseLift is actually a feasibility proble... |

35 | Stable optimizationless recovery from phaseless linear measurements. Arxiv preprint arXiv:1208.1803
- Demanet, Hand
- 2012
(Show Context)
Citation Context ...y proved that PhaseLift is stable with respect to measurement noise. A further improvement by [11] showed that PhaseLift recovers all states with O(n) gaussian observables and moreover the authors of =-=[15]-=-, show that trace minimization is unnecessary since PhaseLift is actually a feasibility problem with high probability. In direct relevance to the setting of this paper, the author of [28] proved that ... |

32 | Array imaging using intensity-only measurements
- Chai, Moscoso, et al.
(Show Context)
Citation Context ...esponding to a quadratic embedding of CPn−1 into R4(n−1). The study of phase retrieval, which in essence is quantum tomography of pure states, has been a topic of much recent activity. The authors of =-=[9, 10, 24]-=- formulated phase retrieval as a matrix recovery problem. In [24], this framework, called PhaseLift, is analyzed and the authors prove that PhaseLift recovers a fixed state with high probability from ... |

31 | Saving phase: Injectivity and Stability for phase retrieval, arXiv submission , arXiv: 1302.4618
- Bandeira, Cahill, et al.
(Show Context)
Citation Context ...Wright’s conjecture is false since at least 4n− 2α(n− 1)− 4, where α(n− 1) is the number of 1’s in the binary expansion of n− 1, quadratic measurements of any kind are needed to determine every state =-=[2,20,25]-=-, which essentially follows from general obstructions to embedding CPn into Rm [19, 21, 22]. We shall refer to a set of observables {Ai}mi=1 as informationally complete, if they together determine any... |

26 | Stable phase retrieval with lowredundancy frames,” arXiv:1302.5487
- Bodmann, Hammen
- 2013
(Show Context)
Citation Context ...dy of informational completeness of quantum measurements. For instance, the authors of [4–7] showed that measurements with 4n-2 generic rank1 observables are injective modulo phase and the authors of =-=[8]-=- gave an example of 4n− 4 specific rank 1 observables with the same property. It is conjectured in [20] that 4n − 4 is the minimal number of rank-1 observables required for injectivity modulo phase. N... |

21 |
Quantum tomography via compressed sensing: Error bounds, sample complexity, and efficient estimators,”
- Flammia, Gross, et al.
- 2012
(Show Context)
Citation Context ...ributions that arise from measuring it. Due to the exponential growth in the state space dimension of many-body systems, work in this field aims to recover low-rank mixed states from few measurements =-=[17,18,23]-=- [12]. A natural question is: what is the minimal number of observables needed to determine any state x? Indeed, a conjecture was attributed to Ron Wright in 1978 that there exist 3 unitary n×n matric... |

20 |
Immersing Projective Spaces
- Milgram
- 1967
(Show Context)
Citation Context ... of 1’s in the binary expansion of n− 1, quadratic measurements of any kind are needed to determine every state [2,20,25], which essentially follows from general obstructions to embedding CPn into Rm =-=[19, 21, 22]-=-. We shall refer to a set of observables {Ai}mi=1 as informationally complete, if they together determine any pure state x ∈ CPn−1. A slightly different convention is used in the field of phase retrie... |

18 |
Phase retrieval from power spectra of masked signals
- Bandeira, Chen, et al.
(Show Context)
Citation Context ...(recovery from all states, as opposed to a fixed state). There have also been other recent algorithmic advances in phase retrieval with recovery and stability guarantees. For instance, the authors of =-=[1, 3]-=- propose an approach that exploits a polarization identity and expander graphs to yield a computationally efficient approach to recovering states from specifically structured rank-1 observables. The a... |

16 | On Signal Reconstruction without Noisy Phase.
- Balan, Casazza, et al.
- 2006
(Show Context)
Citation Context ...any x, y ∈ Cn, AA(xx∗) = AA(yy∗) =⇒ xx∗ = yy∗ Note that U4n is a redundant representation of maps A in the sense that for any A ∈ U4n, only the range of Ā determines whether A is injective mod phase =-=[6]-=-. That is, if A ∈ U4n, then A is injective mod phase ⇐⇒ every element of AUn is injective mod phase. We can now state the main theorem: Theorem 2.1 Consider Un, for n ≥ 1, acting on U4n by right multi... |

16 | Equivalence of Reconstruction from the Absolute Value of the Frame Coefficients to a Sparse Representation Problem - Balan, Casazza, et al. - 2007 |

12 | Fast algorithms for signal reconstruction without phase - Balan, Bodmann, et al. - 2007 |

11 | Embedding complex projective spaces in Euclidean space - Mukherjee - 1981 |

9 |
Position and momentum distributions do not determine the quantum mechanical state
- Vogt
- 1978
(Show Context)
Citation Context ...on Wright in 1978 that there exist 3 unitary n×n matrices U1, U2, U3, such that for x ∈ Cn, the measurements {|Ui(x)|}3i=1, where the modulus is taken 1 component-wise, determine any state x uniquely =-=[26]-=-. It is now known that Wright’s conjecture is false since at least 4n− 2α(n− 1)− 4, where α(n− 1) is the number of 1’s in the binary expansion of n− 1, quadratic measurements of any kind are needed to... |

7 |
On the embedding of projective spaces in Euclidean space
- Steer
- 1970
(Show Context)
Citation Context ... of 1’s in the binary expansion of n− 1, quadratic measurements of any kind are needed to determine every state [2,20,25], which essentially follows from general obstructions to embedding CPn into Rm =-=[19, 21, 22]-=-. We shall refer to a set of observables {Ai}mi=1 as informationally complete, if they together determine any pure state x ∈ CPn−1. A slightly different convention is used in the field of phase retrie... |

5 |
A comparison between the phaselift and phasecut algorithms. Working paper
- Voroninski
- 2012
(Show Context)
Citation Context ... computationally efficient approach to recovering states from specifically structured rank-1 observables. The authors of [13] formulate phase retrieval as class of instances of MAXCUT, and along with =-=[27]-=-, show that PhaseLift is equivalent to the Goemans-Williamson relaxation on the corresponding instance of MAXCUT. 2 2 Main result Consider U4n, where Un ∈ Cn×n is the group of unitary n × n matrices. ... |

2 |
Phase retrieval with polarization. arXiv:1210.7752
- Alexeev, Bandeira
- 2012
(Show Context)
Citation Context ...(recovery from all states, as opposed to a fixed state). There have also been other recent algorithmic advances in phase retrieval with recovery and stability guarantees. For instance, the authors of =-=[1, 3]-=- propose an approach that exploits a polarization identity and expander graphs to yield a computationally efficient approach to recovering states from specifically structured rank-1 observables. The a... |

1 |
Rolando Somma Olivier Landon-Cardinal Yi-Kai Liu
- Bartlett
(Show Context)
Citation Context ...ributions that arise from measuring it. Due to the exponential growth in the state space dimension of many-body systems, work in this field aims to recover low-rank mixed states from few measurements =-=[17,18,23]-=- [12]. A natural question is: what is the minimal number of observables needed to determine any state x? Indeed, a conjecture was attributed to Ron Wright in 1978 that there exist 3 unitary n×n matric... |

1 |
Yi-Kai Liu Jens Eisert Matthias Ohliger, Vincent Nesme. Continuous-variable quantum compressed sensing
- Gross
(Show Context)
Citation Context ...ributions that arise from measuring it. Due to the exponential growth in the state space dimension of many-body systems, work in this field aims to recover low-rank mixed states from few measurements =-=[17,18,23]-=- [12]. A natural question is: what is the minimal number of observables needed to determine any state x? Indeed, a conjecture was attributed to Ron Wright in 1978 that there exist 3 unitary n×n matric... |

1 |
Teiko Heinosaari, Luca Mazzarella. Quantum tomography under prior information. Arxiv e-print
- Wolf
- 2011
(Show Context)
Citation Context ...Wright’s conjecture is false since at least 4n− 2α(n− 1)− 4, where α(n− 1) is the number of 1’s in the binary expansion of n− 1, quadratic measurements of any kind are needed to determine every state =-=[2,20,25]-=-, which essentially follows from general obstructions to embedding CPn into Rm [19, 21, 22]. We shall refer to a set of observables {Ai}mi=1 as informationally complete, if they together determine any... |

1 |
Phase retrieval from unitary quadratic measurements and implications for wright’s conjecture. http://math.berkeley.edu/ vladv
- Voroninski
(Show Context)
Citation Context ...e authors of [15], show that trace minimization is unnecessary since PhaseLift is actually a feasibility problem with high probability. In direct relevance to the setting of this paper, the author of =-=[28]-=- proved that there is some integer r, such that for n large enough, PhaseLift succeeds in recovering a fixed quantum state from the observables A1, . . . , Ar with high probability, with respect to Ha... |