Not all entangled states are created equal (continued)

Transcription

1 Not all entangled states are created equal (continued) Compass states, et cetera: another caveat to the uncertainty principle N00N states revisited, along with their competition 29 Mar 2012 (part I)

2 Background (A) I ve already mentioned N00N states as the most sensitive probe of phase shifts, because of their narrow features in phase space.

3 Background (A) I ve already mentioned N00N states as the most sensitive probe of phase shifts, because of their narrow features in phase space. But these features are narrow in the azimuthal direction: these states are optimizing for measuring rotations about z. Are they also suited to measuring arbitrary SU(2) rotations, or is it better to stick with coherent states, or perhaps use a completely different entangled state?

4 Background (B) Full process tomography can be done on the em field by preparing only coherent states and doing homodyne measurements [Lobino et al., Science 322, 563 (2008).] Easy to imagine that this would be a good way to characterize translation, squeezing, et cetera. But how sensitive is it to generic Fock-space transformations? And could it possibly detect small dephasing as well as, say, a N00N or compass state? The x- and p-displacements δx and δp required to significantly lower the overlap with the initial state may have δxδp << h! [Zurek, Nature 412, 712 (2001)]

5 Full vs partial characterization Complete Characterization Process Tomography VS Detection of a Single Parameter e.g. degree of decoherence due to SU(2) blurring Send a complete set of states through the process, fully characterize each state at the output. Sample the process with a single state, perform a measurement on the state at the output General, requires no prior information and yields a complete description of the process Hard (many (~d 4 ) measurements) but general Requires assumptions about the process, and then yields a only a single parameter Easy (1 measurement) but requires assumptions

6 Process Tomography - Procedure Perform process tomography using different sets of input states Each set generated from SU(2) rotations of one fiducial state O R O R...?????? Experimentally test process tomography result Use the result to predict the output for different states Experimentally send those states through the process

7 The experiment: Prepares desired biphoton states with >95% fidelity see Hofmann & Ono, Phys. Rev. A 76, (07) and Afek, Ambar, &Silberberg, Science 328, (10)

8 Process Tomography - Results Results for several sets of input states and 2 decoherence strengths When x 0.15 the SU(2) orbit generates a 2-design N00N states x=0.5 and SU(2) operations cannot generate a complete set (Lines = simulations)

9 Using a single state as a detector of decoherence or SU(2)* rotations Prepare some initial state and add some decoherence by small random polarisation rotations; project onto original state OR OR... What is the best input state to use? Coherent states? N00N states? Compass states? 2-design promotive states?...? * not U(1) [phase] rotations, which is where N00N shines???????? OR OR... * not arbitrary SU(2 n ) rotations, which would be process tomography 2 2

10 Detecting Decoherence X=0 X=0.15 X=0.5 (solid from expt fits) (dash=theory) theory: exp t (so far):

11 Summary of Tomography/Metrology using Entangled States The sensitivities δx and δp of a state ( how much must I disturb it to see a difference ) obey no uncertainty principle such as δx δp h/2... Although coherent states alone can be used to do process tomography, they are less sensitive than 2-design promotive squeezed states. N00N states actually do not even form a tomographically complete set N00N states are more sensitive to small, random SU(2) rotations than either coherent states or 2-design promotive states (even though the latter are better for full tomography). N00N states are the most sensitive to decoherence, yet they form the worst set of states for process tomography on decoherence

12 Generalized measurement... POVMs, non-orthogonal state discrimination, and a few final remarks The moral of the course (what is measurement?) POVMs (generalized quantum measurements) Discrimination of non-orthogonal states "Best guess" approach Unambiguous discrimination POVMs versus projective measurements A linear-optical experiment State-filtering (discrimination of mixed states) 29 Mar 2012 [part II}

13 The moral of the course Measurement can be almost anything! If two systems interact, such that depending on the initial state of system 1, system 2 may evolve differently, then by looking at system 2, I can learn about the initial state of system 1. The question of how I measure system 2 remains, but if we understand that first step, the measurement interaction, we already know what has happened to system 1, and how much information is available in principle. THIS IS NOT AN APPROXIMATION TO THE REAL MEASUREMENT DESCRIBED BY PROJECTION OPERATORS! Rather, the theory of projective measurement is an approximate treatment of one idealized class of such interactions.

14 Inaccurate measurement All real measurements have finite accuracy. Either this means we never perform true (projective) measurements, or it means that the theory of projective measurements is never an exact description of true (experimental) measurements. There are exactly 2 orthogonal projectors in a 2D space, so 2 A Stern-Gerlach setup: measurement outcomes. spin-1/2

15 Inaccurate measurement All real measurements have finite accuracy. Either this means we never perform true (projective) measurements, or it means that the theory of projective measurements is never an exact description of true (experimental) measurements. There are exactly 2 orthogonal projectors in a 2D space, so 2 A Stern-Gerlach setup: measurement outcomes. spin-1/2 Except really, the screen (or particle position) is continuous; there are an infinite number of possible outcomes each giving different information about the initial state. No point means definitely +, but some mean 99% chance of +, 1% of - ; while others mean it s really a toss-up.

16 A real theory of measurement? The real theory of measurement should recognize that (a) there s no reason the measurement device should have exactly the same dimensionality as the system (b) the measurements may be uncertain (c) the system may change state when being measured (recall the decaying atoms) POVMs (positive operator-valued measures), or generalized measurements -- a set of operators E i defining probabilites P i of outcome i: They need to be positive and sum to I, but they are not necessarily orthogonal projectors; therefore, there may be an arbitrary number of them.

17 Note that the probability rule is the same as the familiar one. And the update rule is that upon finding result i, you know: The example of projectors: A destructive measurement: If you detect a photon in mode i, you get result i ; but you end up in the vacuum. The Ei POVM elements are the same as for projective measurements, but the Mi s also show the different action of this measurement. Another simple example an imperfect spin measurement: But remember: there is no limitation on the number of operators in this sum.

18 (Continuous measurement) We never had a chance to discuss continuous measurement, beyond the case of the decaying atom, but it becomes trivial once you use POVMs. Each instant of time Δt yields possible outcomes, one of which is always no new information (no photon emitted, for instance). The limit of Δt -> 0 is well behaved, and one can now ask about constructing different measurement series, even measuring X for a while, P for a while, et cetera. (See the references on Jessen & Deutsch s work on continuous measurement of cold atoms, for example, or Mabuchi s work on adaptive phase estimation)

19 Can one distinguish between nonorthogonal states? H-polarized photon 45 o -polarized photon Single instances of non-orthogonal quantum states cannot be distinguished with certainty. Obviously, ensembles can. This is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications. Crucial element: we must learn how to distinguish quantum states as well as possible -- and we must know how well a potential eavesdropper could do.

20 What's the best way to tell these apart? "A" a θ b "B" (if they occur with equal a priori probability) Error rate = (1- sin θ)/2 0 if <a b> = 0 (ideal measurement) 1/2 if <a b> = 1 (pure guessing) BUT: can we ever tell for sure? 15% for 45 o Some interaction would take input states a> and b> to "meter states" "A"> and "B">, which we could distinguish perfectly. But unitary interactions preserve overlap:

21 Non-unitarity just means a success rate < 100%... b "A" a θ b "B" a The only way to be sure "A" means a is to be sure it doesn't mean b... Assuming, as always, equal probability of a> or b>, we choose in which basis to measure randomly. The success probability is then: (For 45 o, you can succeed up to 25% of the time)

22 Theory: how to distinguish nonorthogonal states optimally Step 1: Repeat the letters "POVM" over and over. Step 2: Ask your theorist friends for help. [or see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys. Rev. A 66, (2002).] The view from the laboratory: A projective measurement of a two-state system can only yield two possible results. If the measurement isn't guaranteed to succeed, there are three possible results: (1), (2), and ("I don't know"). Therefore, to discriminate between two non-orth. states, we need three measurement outcomes a POVM: or practically speaking, some interaction with a higher-dimensional system.

23 General bounds on the information... We need a nonunitary transformation to take non-orthogonal a and b to orthogonal "A" and "B". This can be accomplished by post-selection i.e., by throwing out events. But wait = 29.3% > 25%... is this upper bound unattainable?

24 The geometric picture 1 Θ Θ 2 90 o 1 Two non-orthogonal vectors The same vectors rotated so their projections onto x-y are orthogonal (The z-axis is inconclusive )

25 The POVM picture We want one measurement operator to be a><a and one to be b><b. If a & b are not orthogonal, these don t sum to I, and can t form part of a projective-measurement basis. But we can have three measurement operators α a><a β b><b I - α a><a - β b><b which sum to the identity. The third one is the inconclusive result. We require α & β < 1 to maintain the positivity of the third operator.

26 a θ b attenuate the 45 o component if you choose the right attenuation, a -> V and b -> H. Really, the attenuation is a third output port indicating DK. a = V + e DK b = H + e DK see also:

27 How do they compare? POVM von Neumann measurement At 0, the von Neumann strategy has a discontinuity-- only then can you succeed regardless of measurement choice. At <a b> = 0.707, the von Neumann strategy succeeds 25% of the time, while the optimum is 29.3%.

28 The advantage is higher in higher dim. Consider these three non-orthogonal states: Projective measurements can distinguish these states with certainty no more than 1/3 of the time. (No more than one member of an orthonormal basis is orthogonal to two of the above states, so only one pair may be ruled out.) But a unitary transformation in a 4D space produces: and these states can be distinguished with certainty up to 55% of the time

29 Experimental schematic (ancilla)

30 A 14-path interferometer for arbitrary 2-qubit unitaries...

31 Success! "Definitely 3" "Definitely 2" "Definitely 1" "I don't know" The correct state was identified 55% of the time-- Much better than the 33% maximum for standard measurements. Further interesting result: mixed states may also be discriminated, contrary to earlier wisdom.

32 STATE-DISCRIMINATION SUMMARY Non-orthogonal states may be distinguished with certainty ("unambiguously") if a finite rate of "inconclusives" is tolerated. The optimal (lowest) inconclusive rate is the absolute value of the overlap between the states (in 2D), and cannot be achieved by any projective measurement. POVMs, implementable by coupling to a larger Hilbert space, can achieve this optimum. In optics, they may be realized with optical multiports (interferometers). We successfully distinguish among 3 non-orthogonal states 55% of the time, where standard quantum measurements are limited to 33%. More recent observation: "state filtering" or discrimination of mixed states is also possible.

33 If measurement affects things, it can be useful! Examples: Which-path information -- measurement destroys interference Zeno effect -- measurement modified dynamics. Squeezing -- if I measure a previously unknown variable to some accuracy, then I know it to that accuracy; if I measure N to better than sqrt(n), for instance, then I have a squeezed state. Error correction / teleportation / et cetera -- by doing the right measurement, you can project all the possible (continuous) range of errors to a few discrete possibilities which can be compactly described and efficiently corrected. KLM-style LOQC (linear-optical quantum computation) Cluster-state computation Entanglement... - Recall that entanglement of two modes can just be related to squeezing of their sums and differences. Polzik, for instance, has used this to entangle two atomic ensembles. - If I have two independent excited ions, and I detect one photon the right way, I don t know which ion is still excited -- Monroe, for instance, has used this to entangle two separated trapped ions. - If I can measure the sum of the photon number in two beams (by letting the same probe interact with both before observing it), then I entangle that: Munro & Nemoto have shown that this interaction would be enough to build a whole quantum computer.

34 References STATE DISCRIMINATION C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976) I. D. Ivanovic, Phys. Lett. A \23} 257 (1987). A. Chefles and S. M. Barnett, J. Mod. Opt. 45, 1295 (1998) S. M. Barnett and E. Riis, J. Mod. Opt. 44, 1061 (1997) B. Huttner et al., Phys. Rev. A 54, 3783 (1996) R. B. M. Clarke et al., Phys Rev A 63, (2001) R. B. M. Clarke et al., Phys Rev A 64, (2001) T. Rudolph, R. W. Spekkens, and P. S. Turner, Phys. Rev. A 68, (2003) M. Takeoka, M. Ban, and M. Sasaki, Phys. Rev. A 68, (2003). A. Chefles, Phys. Lett. A 239, 339 (1998) D. Dieks, Phys. Lett. A 126, 303 (1998) A. Peres, Phys. Lett. A 128, 19 (1988) A. Chefles and S. M. Barnett, Phys. Lett. A 250, 223 (1998) Y. Sun, M. Hillery, and J. A. Bergou, Phys. Rev. A 64, (2001) J. A. Bergou, M. Hillery, and Y. Sun, J. Mod. Opt. 47, 487 (2000) Y. Sun, J. A. Bergou, and M. Hillery, Phys. Rev. A 66, (2002) J. A. Bergou, U. Herzog, and M. Hillery, Phys. Rev. Lett. 90, (2003) EXCITING TOPICS WE DIDN T REACH (or not fully) Compass states: Zurek, Nature 412, 712 (2001) Coherent-state process toography: Lobino et al, Science 322, 563 (2008) Reference frames & superselection: Bartlett et al, quant-ph/ Phase estimation using entangled states: Krischek et al, PRL 107, (2011) Adaptive homodyne measurement of optical phase: Armen et al, PRL 89, (2002) Quantum State Reconstruction via Continuous Measurement: Silberfarb et al, PRL 95, (2005) Continuous Weak Measurement and Nonlinear Dynamics: Smith et al, PRL 93, (2004) Spin-squeezing of a large-spin system via QND measurement: Sewell et al, quant-ph/ Can the quantum state be interpreted statistically?: Pusey, Barrett, & Rudolph, quant-ph Mladen Pavicic (and Oliver Benson?), to appear: a claim that all 4 Bell states can be efficiently discriminated using linear optics?! Menzel, Puhlmann, Heuer, & Schleich: Wave-particle dualism..., a suggestion that my presentation of complementarity this year has been wrong? To appear in PNAS