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Tomographing Quantum State Tomography

Tomographing Quantum State Tomography

A talk I gave at the Last Frontiers in Quantum Information Science Workshop in Juneau, Alaska.

Released under SAND2016-5734 C

Travis Scholten

June 21, 2016
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  1. Tomographing Quantum State Tomography Travis L Scholten @Travis_Sch Center for

    Quantum Information and Control, UNM Center for Computing Research, Sandia National Labs LFQIS-4 2016 June 21 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. CCR Center for Computing Research
  2. Inference of quantum states near the boundaries of state space

    is complicated and interesting! Inference of rank-deficient states has complications. (This is where we want to do tomography!)
  3. State tomography is an act of statistical inference. ˆ ⇢

    From measurement data, we want to form an estimate ⇢0, ⇢ |↵ih↵| ⇡ ⇢0, {X, Y, Z} (This talk - maximum likelihood estimation (MLE))
  4. Canonical statistical inference (usually) relies on asymptotic normality of MLE.

    Asymptotic normality: The distribution of MLEs is Gaussian (Related idea: local asymptotic normality (LAN))
  5. If we have no boundaries on the parameter manifold, we

    can invoke asymptotic normality of MLE. Distribution is normal
  6. Distribution is normal in a region of interest If we

    are far from boundaries on the parameter manifold, we can invoke asymptotic normality of MLE.
  7. If we are near boundaries on the parameter manifold, we

    cannot invoke asymptotic normality of MLE. Unavoidable if true state is pure! Distribution is not normal in a region of interest
  8. Quantum state tomography lives “on the edge” of statistical inference.

    Want to make and characterize (relatively) pure quantum states! Under C/J isomorphism, we will also want to study pure quantum processes. ⇢0 ⇢0
  9. Inference of quantum states near the boundaries of state space

    is complicated and interesting! We can make it tractable to analysis. Inference of rank-deficient quantum states has complications.
  10. We consider the distribution of MLEs, and assume it is

    an isotropic Gaussian. Some estimates are non-physical (negative matrices) Ignore boundary = asymptotic normality of unconstrained estimates
  11. Truncation = make negative states physical Can be done efficiently

    [1] Piles up estimates on the boundary We consider the distribution of MLEs, and assume it is an isotropic Gaussian, then truncate the estimates.
  12. How does the boundary affect model selection for Hilbert space

    dimension and estimation of low-rank states?
  13. A statistical model is a parameterized family of probability distributions.

    A model is fully described by its parameters: M = {⇢ | some conditions are satisfied } M = {⇢ | dim(⇢) = d} Examples: M = {⇢ | rank(⇢) = r} Pr(k|⇢) = Tr(⇢Ek) Model Parameter (assume POVM fixed)
  14. Suppose we have data [(POVM, counts)] from a true state

    (E1, n1), (E2, n2), · · · We may have different models for the same data… which one should we choose? How do we determine which model to use? We also have two models for the state: M1, M2 Two concepts: Fitting the data the best Being close to the truth Not the same!!
  15. The loglikelihood ratio statistic (LLRS) is a tool for determining

    which model fits the data better. ( M1, M2) = 2 log( L ( M1) /L ( M2)) The LLRS is defined as where the likelihood of a model is L ( M ) = max ⇢2M L ( ⇢ ) > 0 : M2 < 0 : M1 Decision rule:
  16. If one model is nested within the other, we may

    be fooled into choosing the larger model needlessly. Nested models: Have to handicap the larger model! M1 ⇢ M2 > 0 : M2 (?!) M1 : y / ax M2 : y / ax 10
  17. We need to know the behavior when both are equally

    valid to make a good decision! If both contain true state… larger appears to fit data better… …actually just fitting noise Nested models: M1 ⇢ M2 M1 = {⇢ | dim(⇢) = d} M2 = {⇢ | dim(⇢) = d + 1} What’s the behavior of the LLRS when both are equally valid? ⇢0 M1 M2 & h i =) M2
  18. If asymptotic normality holds, the Wilks Theorem gives the distribution

    of the LLRS when two models are equally valid. ⇢0 2 M1 ⇢ M2 =) ⇠ 2 K The Wilks Theorem, in one line: For example: ( M1, M2) = 2 log( L ( M1) /L ( M2)) ⇠ 2 d2 1 ⇢0 M1 M2 M0 = {⇢0 } M1 = {⇢ | dim(⇢) = d}
  19. When positivity is imposed, the distribution of the LLRS changes!

    ⇠ 2 d2 1 M0 = {⇢0 } M1 = {⇢ | dim(⇢) = d}
  20. The Wilks Theorem relies on asymptotic normality of MLEs… Loglikelihood

    differentiable Pr(ˆ ⇢) log( L ( ⇢ )) h (M0 , M1)i ⇡ Tr(hH|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(Ih|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(II 1) = d2 1
  21. Split expectation values The Wilks Theorem relies on asymptotic normality

    of MLEs… Pr(ˆ ⇢) log( L ( ⇢ )) h (M0 , M1)i ⇡ Tr(hH|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(Ih|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(II 1) = d2 1
  22. Asymptotic normality/ efficiency The Wilks Theorem relies on asymptotic normality

    of MLEs… Pr(ˆ ⇢) log( L ( ⇢ )) h (M0 , M1)i ⇡ Tr(hH|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(Ih|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(II 1) = d2 1
  23. …consequently, the Wilks Theorem breaks down for inference on boundaries.

    No asymptotic normality/efficiency Loglikelihood differentiable (?) Pr(ˆ ⇢) log( L ( ⇢ )) How do we proceed? h (M0 , M1)i ⇡ Tr(hH|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(Ih|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ⇡ Tr(II 1) = d2 1
  24. To model the expected value of the LLRS when boundaries

    are present, we make several assumptions. We can do a Taylor series: The Fisher information is isotropic (in Hilbert-Schmidt basis): Unconstrained estimates are normally distributed and asymptotically efficient: Non-physical estimates are truncated back to positive states. h (M0 , M1)i ⇡ Tr (hH|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) ˆ ⇢ ⇠ N(⇢0, I 1) I = ✏2I
  25. Under these assumptions, the LLRS takes a very simple form.

    Simple expression What do we do with it?? Can we even compute the distribution?? Truncated!! h (M0, M1)i ⇡ 1 ✏2 Tr (h|ˆ ⇢ ⇢0)(ˆ ⇢ ⇢0 |i) = 1 ✏2 X jk h||(ˆ ⇢ ⇢0)jk ||2i
  26. To understand the LLRS under the positivity constraint, we study

    individual terms in the sum. What are the individual terms? Depending on the support of the true state, how does imposing positivity change the terms? How do they compare to the Wilks Theorem? Time for some numerics! h (M0, M1)i ⇡ 1 ✏2 X jk h||(ˆ ⇢ ⇢0)jk ||2i
  27. Numerical results show dramatically different, but understandable, behavior of the

    terms. Where’s the support of the truth? (fixed basis)
  28. What are terms in the sum (from our assumptions)? 1

    ✏2 h||(ˆ ⇢ ⇢0)jk ||2i Numerical results show dramatically different, but understandable, behavior of the terms.
  29. What are terms in the sum (from Wilks)? 1 ✏2

    h||(ˆ ⇢ ⇢0)jk ||2i Numerical results show dramatically different, but understandable, behavior of the terms.
  30. How do we make sense of this?? For different true

    states, we have different behavior.
  31. We partition the terms into 2 groups, depending on how

    they relate to the support of the true state. “L” “Kite” “L” “L” h i = L + kite
  32. Elements in the “L” are simple to model, and have

    a nice geometric interpretation. “L” Arises from unitaries Elements do not “see” boundary Positivity unimportant
  33. Elements in the “L” are simple to model, and have

    a nice geometric interpretation. ˆ ⇢ ⇠ 0 B B B B @ N(0, 1) · · · · · · N(0, 1) N(0, 1) · · · . . . N(0, 1) . . . . . . 1 C C C C A h||(ˆ ⇢ ⇢0)jk ||2i = 1 “L” L = 2rd r(r + 1)
  34. We split the kite into two pieces - a “body”,

    and a “tail”. “Kite” “Body” “Tail”
  35. “Kite” Can take large-dimensional limit. Can diagonalize estimate within kernel

    of true state! The body of the kite is a Wigner matrix, an instance of the Gaussian Unitary Ensemble (GUE). ˆ ⇢ ⇠ 0 @ N(0, 1) N(0, 1) GUE(d r) 1 A
  36. Large-dimensional limit = convergence of eigenvalues to Wigner semicircle distribution

    ˆ ⇢ ⇠ 0 B B B B B @ .5 .5 e1 e2 ... 1 C C C C C A We study separately the fluctuations within the body and the tail of the kite.
  37. Negative eigenvalues = need to truncate them! We study separately

    the fluctuations within the body and the tail of the kite. ˆ ⇢ ⇠ 0 B B B B B @ .5 .5 e1 e2 ... 1 C C C C C A Large-dimensional limit = convergence of eigenvalues to Wigner semicircle distribution
  38. We then impose the positivity constraint, and integrate over perturbations

    on the tail. Then, integrate over perturbations on support ˆ ⇢ ⇠ 0 B B B B B @ .11 .11 e0 1 e0 2 ... 1 C C C C C A P jk h||(ˆ ⇢ ⇢0)jk ||2i = hf(r)i
  39. After completing the calculation, we find a new formula for

    the expected value. A complicated, but straightforward formula! Dramatically different than Wilks’ result! Valid for rank-deficient states h (M0, M1)i ⇡ 2rd r(r + 1) + g(r) M0 = {⇢0 } M1 = {⇢ | dim(⇢) = d} r = rank(⇢0) “Kite” “L”
  40. Our result is very different than Wilks, in part because

    it depends on the rank of the truth.
  41. For rank one true states, our result does a good

    job of predicting the expected value.
  42. Boundaries affect our ability to judge whether models are fitting

    data well. This can be overcome. has been
  43. What, if anything, does this model have to say about

    compressed sensing? tomography of low-rank states General problem - when true state is low rank, how does tomography behave?
  44. Paradigmatic example: sparse matrix completion/ vector estimation Quantum: sufficient POVMs,

    robustness to noise, and sample complexity [2-4] Recently, the importance of the positivity constraint has been investigated Provides guarantees on recovery! Suggests geometry of quantum state space is important [5] What, if anything, does this model have to say about compressed sensing? tomography of low-rank states
  45. Assuming an isotropic Gaussian distribution of MLEs, what does the

    distribution of ranks of the truncated estimates look like? Is there a difference between the quantum and classical cases?
  46. The quantum case exhibits a higher rank on average, and

    has a lower standard deviation. ???
  47. As the rank of the truth goes up, the quantum

    and classical cases behave similarly.
  48. A reason for the different behavior could be the tails

    of the distribution within the kernel. Classical = bigger tails Making positive = less subtraction = higher rank Quantum = smaller tails Making positive = more subtraction = lower rank
  49. The distribution of loglikelihood ratio statistics is different than that

    of the Wilks Theorem. Necessary for quantum model selection “L” “Kite” “L” “L”
  50. The distribution of ranks of estimates is peaked towards lower

    rank. Useful for low-rank state estimation?
  51. Exact Formula for <LLRS> h (M0, M1)i = 2rd r(r

    + 1) + X2 0 r + n(n + X2 0 ) ⇡ ⇣⇡ 2 sin 1 X02 p n ⌘ X0(X2 0 + 26n) 24⇡ q 4n X2 0 X0 ⇡ 2 p n (240r⇡)2/5 4 n1/10 + (240r⇡)4/5 80 n 3/10 n = d r