Indirect measurements of a harmonic oscillator Gian Michele Graf ETH Zurich Quantissima in the Serenissima III Palazzo Pesaro Papafava, Venezia August 19-23, 2019
Indirect measurements of a harmonic oscillator Gian Michele Graf ETH Zurich Quantissima in the Serenissima III Palazzo Pesaro Papafava, Venezia August 19-23, 2019 based on joint work with M. Fraas, L. H¨ anggli discussions with J. Fr¨ ohlich, K. Hepp
Outline Setting the stage The model defined at last Results
Setting the stage The model defined at last Results
Born rule (BR) If an ideal measurement of the observable � A = A ∗ = λ P λ λ ∈ spec ( A ) (spectral decomposition) is done in the state ψ , then the outcome λ is found with probability � ψ | P λ | ψ � .
Born rule (BR) If an ideal measurement of the observable � A = A ∗ = λ P λ λ ∈ spec ( A ) (spectral decomposition) is done in the state ψ , then the outcome λ is found with probability � ψ | P λ | ψ � . Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation ( A = − i d / dx ) is premised (e.g. correspondence principle).
Born rule (BR) If an ideal measurement of the observable � A = A ∗ = λ P λ λ ∈ spec ( A ) (spectral decomposition) is done in the state ψ , then the outcome λ is found with probability � ψ | P λ | ψ � . Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation ( A = − i d / dx ) is premised (e.g. correspondence principle). ◮ The relation between the physical and the operational meaning of A is not even addressed. The apparatus remains exophysical; the observable dry.
Born rule (BR) If an ideal measurement of the observable � A = A ∗ = λ P λ λ ∈ spec ( A ) (spectral decomposition) is done in the state ψ , then the outcome λ is found with probability � ψ | P λ | ψ � . Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation ( A = − i d / dx ) is premised (e.g. correspondence principle). ◮ The relation between the physical and the operational meaning of A is not even addressed. The apparatus remains exophysical; the observable dry. Cf. Heisenberg 1927: If one wants to be clear about what is meant by ”position of an object,” for example of an electron (relatively to a given reference frame), then one has to specify definite experiments by which the ”position of an electron” can be measured; otherwise this term has no meaning at all.
Born rule (BR) If an ideal measurement of the observable � A = A ∗ = λ P λ λ ∈ spec ( A ) (spectral decomposition) is done in the state ψ , then the outcome λ is found with probability � ψ | P λ | ψ � . Remarks on what goes with BR, though untold: ◮ The relation between physical meaning (e.g. momentum) and mathematical operation ( A = − i d / dx ) is premised (e.g. correspondence principle). ◮ The relation between the physical and the operational meaning of A is not even addressed. The apparatus remains exophysical; the observable dry. ◮ State ψ changes with time. Hence the ideal measurement is instantaneous. (But any real measurement takes time.)
Remarks on duration of measurement ◮ If the measurement of the intended observable A takes a time T , then the observable actually measured is (at best) the time average � T A T := 1 e i Ht A e − i Ht dt T 0
Remarks on duration of measurement ◮ If the measurement of the intended observable A takes a time T , then the observable actually measured is (at best) the time average � T A T := 1 e i Ht A e − i Ht dt T 0 ◮ General protocol: Measure A T before it deviates a lot from A (von Neumann: Strong coupling to apparatus)
Remarks on duration of measurement ◮ If the measurement of the intended observable A takes a time T , then the observable actually measured is (at best) the time average � T A T := 1 e i Ht A e − i Ht dt T 0 ◮ General protocol: Measure A T before it deviates a lot from A (von Neumann: Strong coupling to apparatus) ◮ Special protocol: If A is a constant of motion, then T does not matter: A T = A . (To the contrary: the larger T , the better, because the initialization of the apparatus matters ever less; e.g. quantum non-demolition experiments (QND) and their theory, T → ∞ .)
Stability of QND A intended observable � T A T := 1 e i Ht A e − i Ht dt actual observable T 0
Stability of QND A intended observable � T A T := 1 e i Ht A e − i Ht dt actual observable T 0 ◮ If A = f ( H ) , then A T = A (as seen).
Stability of QND A intended observable � T A T := 1 e i Ht A e − i Ht dt actual observable T 0 ◮ If A = f ( H ) , then A T = A (as seen). ◮ If A = f ( H 0 ) with H − H 0 small (Hamiltonian error, unavoidable), then � A T − A � ≤ 2 � f ( H ) − f ( H 0 ) �
Stability of QND A intended observable � T A T := 1 e i Ht A e − i Ht dt actual observable T 0 ◮ If A = f ( H ) , then A T = A (as seen). ◮ If A = f ( H 0 ) with H − H 0 small (Hamiltonian error, unavoidable), then � A T − A � ≤ 2 � f ( H ) − f ( H 0 ) � ∴ stability: � A T − A � small uniformly in T > 0 (no apparatus, though).
The dilemma and the way out ◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)?
The dilemma and the way out ◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes.
The dilemma and the way out ◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A , in order to analyze the system proper S (indirect measurement).
The dilemma and the way out ◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A , in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut).
The dilemma and the way out ◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A , in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut). ◮ The shift is real, and not just of perspective, because S and A are now coupled; the test will be a pointer mirroring A
The dilemma and the way out ◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A , in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut). ◮ The shift is real, and not just of perspective, because S and A are now coupled; the test will be a pointer mirroring A ◮ We’ll see: The stability of QND is lost for T → ∞ , because A is feeding back on S .
The dilemma and the way out ◮ BR applies to ideal (hence instantaneous) measurements only ◮ There are no instantaneous measurements ◮ There is no rule for other measurements ◮ Can we nonetheless tell how close real measurements come to ideal ones (and hence test BR)? ◮ Yes. By applying BR at a different levels: Postulate it for the pointer observable of the apparatus A , in order to analyze the system proper S (indirect measurement). ◮ What it takes: Make A endophysical (shift of Heisenberg cut). ◮ The shift is real, and not just of perspective, because S and A are now coupled; the test will be a pointer mirroring A ◮ We’ll see: The stability of QND is lost for T → ∞ , because A is feeding back on S . There is a time window of opportunity ( t ∈ [ 0 , T ] ) for measurement.
Requisites for the apparatus A and their implementation ◮ A is quantum, but establishes a record that is classical.
Requisites for the apparatus A and their implementation ◮ A is quantum, but establishes a record that is classical. → Dyson: “As a general rule, knowledge about the past can only be expressed in classical terms.”
Requisites for the apparatus A and their implementation ◮ A is quantum, but establishes a record that is classical. → Fr¨ ohlich: “ETH approach”
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