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Byron Smith December 11, 2013 What is Quantum State Tomography? What is Bayesian Statistics? 1. 2. 1. 2. 3. Example: Schrodinger’s Cat 3. 1. 2. 3. 4. 5. 6. 7. 8. 9. Conditional Probabilities Bayes’ Rule Frequentist vs. Bayesian Interpretation Analysis with a Non-informative Prior Analysis with an Informative Prior Sources of Error in Tomography Error Reduction via Bayesian Analysis Adaptive Tomography Conclusion References Supplementary Information Tomography comes from tomos meaning section. Classically, tomography refers to analyzing a 3dimensional trajectory using 2-dimensional slices. Quantum State Tomography refers to identifying a particular wave function using a series of measurements. P ( A) P ( A B ) P ( A C ) P(A | B) P(A B) P(B) P(A | B) P(A | C ) P ( B | A) P( A | B)P(B) ( x | y ) is the Likelihood P ( A) ( y ) is the Prior Probability ( y | x) ( x | y ) ( y ) ( x | y ) ( y ) dy (or just prior) ( y | x ) is the Posterior Probability (or just posterior) Inference on probability arises from the frequency that some outcome is measured and that measurement is random. Inference on probability arises from a prior probability assumption weighted by empirical evidence. In other words, a measurement is random only due to our ignorance. Measurements are inherently random. Frequentist Bayesian Frequentist: Given N cats, there are some that are alive and some that are dead. The probability that the cat is alive is associated with the fact that we sample randomly and a*N of the cats are alive. cat Palive alive a alive alive Pdead dead alive (1 a ) dead dead dead Bayesian: The probability a is a random variable (unknown) and therefore there is an inherent probability associated with each particular cat. p ( x | a ) a (1 a ) x N ( x | a) a i (1 a ) x 1 x i 1 x a (1 a ) Y i 1 N cat Palive alive a alive alive Pdead dead alive (1 a ) dead Y dead x i i 1 dead a ( x | a ) 0 aˆ Y N N Y N ( x | a) a xi (1 a ) 1 x i a (1 a ) Y N Y i 1 (a ) (a | x) cat Palive alive a alive alive Pdead dead alive (1 a ) dead ( ) ( ) ( ) a 1 (1 a ) ( N ) ( Y ) ( N Y ) a 1 Y 1 (1 a ) dead dead a | x ~ Beta ( Y , N Y ) aˆ Y N N Y 1 True value of a = 0.3 N Frequentist Bayesian 3 0.67 0.6 25 0.2 0.22 75 0.24 0.25 250 0.336 0.337 Flourine-18, halflife=1.8295 Hours Cat has been in for 3 hours N Frequentist Bayesian 3 0.67 0.39 25 0.28 0.29 75 0.267 0.272 250 0.316 0.316 Error in the measurement basis. Error in the counting statistics. Error associated with stability. ◦ Detector efficiency ◦ Source intensity Model Error aˆ Y aˆ N var( aˆ ) var( Y ) Y (N Y ) N N 3 var( aˆ ) Y N N ( Y )( N Y ) ( N ) N Var(a) N Var(a) 3 0.027 3 0.013 25 0.0074 25 0.0057 75 0.0029 75 0.0026 250 0.00079 250 0.00076 Frequentist 3 Bayesian (α=3.1, β=6.9) There are several measures of “lack of fit.” Likelihood Infidelity Shannon Entropy N ( nˆ n ) 2 exp 2 ˆ 2 1 F ( ˆ , ) 1 Tr ˆ H p i log( p i ) i 2 The number of measurements required to sufficiently identify ρ can be reduced when using a basis which diagonalizes ρ. To do so, measure N0 particles first, change the measurement basis, then finish the total measurement. Naturally this can be improved with Bayesian by using the first measurements as a Prior. Quantum State Tomography is a tool used to identify a density matrix. There are several metrics of density identification. Bayesian statistics can improve efficiency while providing a new interpretation of quantum states. 1. 2. 3. 4. J. B. Altepeter, D. F. V. James, and P. G. Kwiat, Qubit Quantum State Tomography, in Quantum State Estimation (Lecture Notes in Physics), M. Paris and J. Rehacek (editors), Springer (2004). D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, Phys. Rev. Lett. 111, 183601 (2013). F. Huszár and N. M. T. Houlsby, Phys. Rev. A 85, 052120 (2012). R. Blume-Kohout, New J. Phys. 12, 043034 (2012). For density operators which are not diagonal, we use a basis of spin matrices: a i i i If choose an instrument orientation such that the first parameter is 1, we are left with the Stoke’s Parameters, Si: 1 0 S i i 2 i The goal is then to identify the Si. Used to visualize a superposition of photon polarizations. The x-axis is 45 degree polarizations. The y-axis is horizontal or vertical polarizations. The z-axis is circular polarizations. Using a matrix basis similar to that for Stoke’s Parameters, one can exactly identify a polarization with three measurements. Note that an orthogonal basis is not necessary. Errors in measurement and instrumentation will manifest themselves as a disc on the Poincare Sphere: (a) Errors in measurement basis. (b) Errors in intensity or detector stability. Because there is a finite sample size, states are not characterized exactly. Each measurement constrains the sample space. Optimization can lead to states which lie outside the Poincare Sphere. Unobserved values a registered as zeros in the density matrix optimization. This can be unrealistic. There is no direct measure of uncertainty within maximum likelihood estimators. Parameter space can be constrained through the prior. Unobserved values can still have some small probability through the prior. Uncertainty analysis can come directly from the variance of the posterior distribution, regardless of an analytical form.