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van Enk SJ. Pooling probability distributions and partial information decomposition. Phys Rev E 2023; 107:054133. [PMID: 37329048 DOI: 10.1103/physreve.107.054133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2023] [Accepted: 05/09/2023] [Indexed: 06/18/2023]
Abstract
Notwithstanding various attempts to construct a partial information decomposition (PID) for multiple variables by defining synergistic, redundant, and unique information, there is no consensus on how one ought to precisely define either of these quantities. One aim here is to illustrate how that ambiguity-or, more positively, freedom of choice-may arise. Using the basic idea that information equals the average reduction in uncertainty when going from an initial to a final probability distribution, synergistic information will likewise be defined as a difference between two entropies. One term is uncontroversial and characterizes "the whole" information that source variables carry jointly about a target variable T. The other term then is meant to characterize the information carried by the "sum of its parts." Here we interpret that concept as needing a suitable probability distribution aggregated ("pooled") from multiple marginal distributions (the parts). Ambiguity arises in the definition of the optimum way to pool two (or more) probability distributions. Independent of the exact definition of optimum pooling, the concept of pooling leads to a lattice that differs from the often-used redundancy-based lattice. One can associate not just a number (an average entropy) with each node of the lattice, but (pooled) probability distributions. As an example, one simple and reasonable approach to pooling is presented, which naturally gives rise to the overlap between different probability distributions as being a crucial quantity that characterizes both synergistic and unique information.
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Affiliation(s)
- S J van Enk
- Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
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Propp TB, van Enk SJ. On nonlinear amplification: improved quantum limits for photon counting. Opt Express 2019; 27:23454-23463. [PMID: 31510621 DOI: 10.1364/oe.27.023454] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2019] [Accepted: 07/24/2019] [Indexed: 06/10/2023]
Abstract
We show that detection of single photons is not subject to the fundamental limitations that accompany quantum linear amplification of bosonic mode amplitudes, even though a photodetector does amplify a few-photon input signal to a macroscopic output signal. Alternative limits are derived for nonlinear photon-number amplification schemes with optimistic implications for single-photon detection. Four commutator-preserving transformations are presented: one idealized (which is optimal) and three more realistic (less than optimal). Our description makes clear that nonlinear amplification takes place, in general, at a different frequency ω' than the frequency ω of the input photons. This can be exploited to suppress thermal noise and dark counts past what is possible with linear amplification up to a fundamental limit imposed by nonlinear amplification into a single bosonic mode.
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Gao T, Yan F, van Enk SJ. Permutationally invariant part of a density matrix and nonseparability of N-qubit states. Phys Rev Lett 2014; 112:180501. [PMID: 24856681 DOI: 10.1103/physrevlett.112.180501] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2013] [Indexed: 06/03/2023]
Abstract
We consider the concept of "the permutationally invariant (PI) part of a density matrix," which has proven very useful for both efficient quantum state estimation and entanglement characterization of N-qubit systems. We show here that the concept is, in fact, basis dependent but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part ρ(PI) of a general (mixed) N-qubit state ρ, we obtain (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multipartite separability criteria (one of which entails a sufficient criterion for genuine N-partite entanglement) that can be experimentally determined by just 2N+1 measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of N qubits, and (iv) an explicit example that shows there are, for increasing N, genuinely N-partite entangled states lying closer and closer to the maximally mixed state. Moreover, we show that if the PI part of a state is k nonseparable, then so is the actual state. We further argue to add as requirement on any multipartite entanglement measure E that it satisfy E(ρ)≥E(ρ(PI)), even though the operation that maps ρ→ρ(PI) is not local.
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Affiliation(s)
- Ting Gao
- College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
| | - Fengli Yan
- College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050024, China and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
| | - S J van Enk
- Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
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Abstract
While it is known that Trρ(n) can be measured directly (i.e., without first reconstructing the density matrix) by performing joint measurements on n copies of the same state ρ, it is shown here that random measurements on single copies suffice, too. Averaging over the random measurements directly yields estimates of Trρ(n), even when it is not known what measurements were actually performed (so that ρ cannot be reconstructed).
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Affiliation(s)
- S J van Enk
- Department of Physics and Oregon Center for Optics University of Oregon, Eugene, Oregon 97403, USA
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Abstract
We propose and analyze a method to detect and characterize the drift of a nonstationary quantum source. It generalizes a standard measurement for detecting phase diffusion of laser fields to quantum systems of arbitrary Hilbert space dimension, qubits in particular. We distinguish diffusive and systematic drifts, and examine how quickly one can determine that a source is drifting. We show that for single-photon wave packets our measurement is implemented by the Hong-Ou-Mandel effect.
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Affiliation(s)
- Lucia Schwarz
- Department of Physics, Oregon Center for Optics, University of Oregon, Eugene, 97403, USA
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Choi KS, Goban A, Papp SB, van Enk SJ, Kimble HJ. Entanglement of spin waves among four quantum memories. Nature 2010; 468:412-6. [DOI: 10.1038/nature09568] [Citation(s) in RCA: 100] [Impact Index Per Article: 7.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2010] [Accepted: 10/05/2010] [Indexed: 11/09/2022]
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Blume-Kohout R, Yin JOS, van Enk SJ. Entanglement verification with finite data. Phys Rev Lett 2010; 105:170501. [PMID: 21231029 DOI: 10.1103/physrevlett.105.170501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2010] [Indexed: 05/30/2023]
Abstract
Suppose an experimentalist wishes to verify that his apparatus produces entangled quantum states. A finite amount of data cannot conclusively demonstrate entanglement, so drawing conclusions from real-world data requires statistical reasoning. We propose a reliable method to quantify the weight of evidence for (or against) entanglement, based on a likelihood ratio test. Our method is universal in that it can be applied to any sort of measurements. We demonstrate the method by applying it to two simulated experiments on two qubits. The first measures a single entanglement witness, while the second performs a tomographically complete measurement.
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Abstract
So-called direct measurements of entanglement are collective measurements on multiple copies of a (bipartite or multipartite) quantum system that directly provide one with a value for some entanglement measure, such as the concurrence for bipartite states. Multiple copies are needed since the entanglement of a mixed state is not a linear function of the density matrix. The procedures proposed and implemented so far make certain assumptions about the states generated. This feature distinguishes direct measurements from standard entanglement verification tests such as Bell inequalities, entanglement witnesses, and quantum-state tomography, which make no such assumptions. I discuss how a direct measurement can be turned into a quantitative entanglement verification test without such assumptions by exploiting a recent theorem by Renner [Nature Phys. 3, 645 (2007)10.1038/nphys684].
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Affiliation(s)
- S J van Enk
- Department of Physics, University of Oregon Oregon Center for Optics and Institute for Theoretical Science, Eugene, Oregon 97403, USA
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Papp SB, Choi KS, Deng H, Lougovski P, van Enk SJ, Kimble HJ. Characterization of Multipartite Entanglement for One Photon Shared Among Four Optical Modes. Science 2009; 324:764-8. [DOI: 10.1126/science.1172260] [Citation(s) in RCA: 101] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
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McKinstrie CJ, van Enk SJ, Raymer MG, Radic S. Multicolor multipartite entanglement produced by vector four-wave mixing in a fiber. Opt Express 2008; 16:2720-2739. [PMID: 18542357 DOI: 10.1364/oe.16.002720] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
Multipartite entanglement is a resource for quantum communication and computation. Vector four-wave mixing (FWM) in a fiber, driven by two strong optical pumps, couples the evolution of four weak optical sidebands (modes). Depending on the fiber dispersion and pump frequencies, the mode frequencies can be similar (separated by less than 1 THz) or dissimilar (separated by more than 10 THz). In this report, the discrete- and continuous-variable entanglement produced by vector FWM is studied in detail. Formulas are derived for the variances of, and correlations between, the mode quadratures and photon numbers. These formulas and related results show that the modes are four-partite entangled.
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Affiliation(s)
- C J McKinstrie
- Bell Laboratories, Alcatel-Lucent, Holmdel, New Jersey 07733, USA.
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Chou CW, de Riedmatten H, Felinto D, Polyakov SV, van Enk SJ, Kimble HJ. Measurement-induced entanglement for excitation stored in remote atomic ensembles. Nature 2005; 438:828-32. [PMID: 16341008 DOI: 10.1038/nature04353] [Citation(s) in RCA: 133] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2005] [Accepted: 10/19/2005] [Indexed: 11/09/2022]
Abstract
A critical requirement for diverse applications in quantum information science is the capability to disseminate quantum resources over complex quantum networks. For example, the coherent distribution of entangled quantum states together with quantum memory (for storing the states) can enable scalable architectures for quantum computation, communication and metrology. Here we report observations of entanglement between two atomic ensembles located in distinct, spatially separated set-ups. Quantum interference in the detection of a photon emitted by one of the samples projects the otherwise independent ensembles into an entangled state with one joint excitation stored remotely in 10(5) atoms at each site. After a programmable delay, we confirm entanglement by mapping the state of the atoms to optical fields and measuring mutual coherences and photon statistics for these fields. We thereby determine a quantitative lower bound for the entanglement of the joint state of the ensembles. Our observations represent significant progress in the ability to distribute and store entangled quantum states.
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Affiliation(s)
- C W Chou
- Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125, USA
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Affiliation(s)
- S. J. van Enk
- Bell Labs, Lucent Technologies, Room 1D-428, 600-700 Mountain Avenue, Murray Hill, NJ 07974, USA
| | - H. J. Kimble
- Norman Bridge Laboratory 12-33, California Institute of Technology, Pasadena, CA 91125, USA
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van Enk SJ, Kimble HJ. Corrections and Clarifications. Science 2005; 309:1187; author reply 1187. [PMID: 16109867 DOI: 10.1126/science.309.5738.1187a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Affiliation(s)
- S J van Enk
- Bell Labs, Lucent Technologies, Room 1D-428, 600-700 Mountain Avenue, Murray Hill, NJ 07974, USA
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van Enk SJ. Relations between cloning and the universal NOT derived from conservation laws. Phys Rev Lett 2005; 95:010502. [PMID: 16090596 DOI: 10.1103/physrevlett.95.010502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2005] [Indexed: 05/03/2023]
Abstract
We discuss certain relations between cloning and the not operation that can be derived from conservation laws alone. Those relations link the limitations on cloning and the not operation possibly imposed by other laws of nature. Our result is quite general and holds both in classical and quantum-mechanical worlds, for both optimal and suboptimal operations, and for bosons as well as fermions.
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Affiliation(s)
- S J van Enk
- Bell Labs, Lucent Technologies, 600-700 Mountain Ave, Murray Hill, New Jersey 07974, USA
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van Enk SJ. Entanglement capabilities in infinite dimensions: multidimensional entangled coherent states. Phys Rev Lett 2003; 91:017902. [PMID: 12906576 DOI: 10.1103/physrevlett.91.017902] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2003] [Indexed: 05/24/2023]
Abstract
An example is given of an interaction that produces an infinite amount of entanglement in an infinitely short time, but only a finite amount in longer times. The interaction arises from a standard Kerr nonlinearity and a 50/50 beam splitter, and the initial state is a coherent state. For certain finite interaction times multidimensional generalizations of entangled coherent states are generated, for which we construct a teleportation protocol. Similarities between probabilistic teleportation and unambiguous state discrimination are pointed out.
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Affiliation(s)
- S J van Enk
- Bell Labs, Lucent Technologies, 600-700 Mountain Avenue, Murray Hill, New Jersey 07974, USA
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Abstract
We give a quantum information-theoretic description of an ideal propagating cw laser field and reinterpret typical quantum-optical experiments in light of this. In particular, we show that, contrary to recent claims [T. Rudolph and B. C. Sanders, Phys. Rev. Lett. 87, 077903 (2001)], a conventional laser can be used for quantum teleportation with continuous variables and for generating continuous-variable entanglement. Optical coherence is not required, but phase coherence is. We also show that coherent states play a privileged role in the description of laser light.
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Affiliation(s)
- S J van Enk
- Bell Labs, Lucent Technologies, 600-700 Mountain Avenue, Murray Hill, New Jersey 07974, USA
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