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A resource theory of superposition Thomas Theurer in cooperation - - PowerPoint PPT Presentation

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Institute of Theoretical Physics Prof. Dr. Martin B. Plenio A resource theory of superposition Thomas Theurer in cooperation with Nathan Killoran, Dario Egloff and Martin B. Plenio Logos ERC Synergy grant BioQ Page 2 A resource theory of

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Institute of Theoretical Physics

  • Prof. Dr. Martin B. Plenio

A resource theory of superposition

Thomas Theurer in cooperation with Nathan Killoran, Dario Egloff and Martin B. Plenio

Logos

ERC Synergy grant BioQ

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A resource theory of superposition| Thomas Theurer| 2017 Page 2

Quantum resource theories

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A resource theory of superposition| Thomas Theurer| 2017 Page 3

Entanglement as a resource theory

Restriction: Local operations and classical communication – physically motivated.

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A resource theory of superposition| Thomas Theurer| 2017 Page 4

Entanglement as a resource theory

Restriction: Local operations and classical communication – physically motivated. Entanglement cannot be created but allows for tasks otherwise forbidden.

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A resource theory of superposition| Thomas Theurer| 2017 Page 5

Entanglement as a resource theory

Restriction: Local operations and classical communication – physically motivated. Entanglement cannot be created but allows for tasks otherwise forbidden. Teleport the state of your system – allows to overcome the restriction

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A resource theory of superposition| Thomas Theurer| 2017 Page 6

Quantum resource theories

  • Main ingredients
  • 1. Free states resource states
  • 2. Free operations
  • Entanglement
  • 1. Separable states

entangled states

  • 2. LOCC
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A resource theory of superposition| Thomas Theurer| 2017 Page 7

Quantum resource theories

  • Main ingredients
  • 1. Free states resource states
  • 2. Free operations
  • Entanglement
  • 1. Separable states

entangled states

  • 2. LOCC
  • Questions to answer
  • Manipulation
  • Detection
  • Quantification
  • Operational advantage
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A resource theory of superposition| Thomas Theurer| 2017 Page 8

Quantum resource theories

  • Main ingredients
  • 1. Free states resource states
  • 2. Free operations
  • Entanglement
  • 1. Separable states

entangled states

  • 2. LOCC
  • Questions to answer
  • Manipulation
  • Detection
  • Quantification
  • Operational advantage
  • Systematic investigation leads to a better usage in applications.
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A resource theory of superposition| Thomas Theurer| 2017 Page 9

Relevance

Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 10

Relevance

In principle, all aspects of quantum mechanics not present in classical physics can lead to operational advantages.

  • non-classicality is a resource

Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 11

Relevance

In principle, all aspects of quantum mechanics not present in classical physics can lead to operational advantages.

  • non-classicality is a resource

Superposition is underlying important types of non-classicality including

  • coherence,
  • entanglement,
  • optical non-classicality.

Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 12

Relevance

In principle, all aspects of quantum mechanics not present in classical physics can lead to operational advantages.

  • non-classicality is a resource

Superposition is underlying important types of non-classicality including

  • coherence,
  • entanglement,
  • optical non-classicality.

Investigate non-classicality in terms of superpositions[1].

Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 13

Relevance

In principle, all aspects of quantum mechanics not present in classical physics can lead to operational advantages.

  • non-classicality is a resource

Superposition is underlying important types of non-classicality including

  • coherence,
  • entanglement,
  • optical non-classicality.

Investigate non-classicality in terms of superpositions[1]. Agreement on the definition of entanglement – but how to define non- classicality?

  • convert it to entanglement

Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 14

Basic framework

  • State space with finite dimension .
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A resource theory of superposition| Thomas Theurer| 2017 Page 15

Basic framework

  • State space with finite dimension .
  • Pure free states: (a linearly independent but not necessarily
  • rthogonal basis of the state space under consideration).
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A resource theory of superposition| Thomas Theurer| 2017 Page 16

Basic framework

  • State space with finite dimension .
  • Pure free states: (a linearly independent but not necessarily
  • rthogonal basis of the state space under consideration).
  • Free states:
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A resource theory of superposition| Thomas Theurer| 2017 Page 17

Basic framework

  • State space with finite dimension .
  • Pure free states: (a linearly independent but not necessarily
  • rthogonal basis of the state space under consideration).
  • Free states:
  • Free operations:
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A resource theory of superposition| Thomas Theurer| 2017 Page 18

Basic framework

  • State space with finite dimension .
  • Pure free states: (a linearly independent but not necessarily
  • rthogonal basis of the state space under consideration).
  • Free states:
  • Free operations:

Alternative definitions possible!

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A resource theory of superposition| Thomas Theurer| 2017 Page 19

Relevance

  • Definition: Classical rank[1,2] (for an arbitrary set of free states):

Sperling, J., & Vogel, W. (2015). Convex ordering and quantification of quantumness. Physica Scripta, 90(7), 074024. Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1] [2]

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Relevance

  • Definition: Classical rank[1,2] (for an arbitrary set of free states):
  • Faithful conversion operation (to and from entanglement):

Sperling, J., & Vogel, W. (2015). Convex ordering and quantification of quantumness. Physica Scripta, 90(7), 074024. Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1] [2]

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A resource theory of superposition| Thomas Theurer| 2017 Page 21

Relevance

  • Definition: Classical rank[1,2] (for an arbitrary set of free states):
  • Faithful conversion operation (to and from entanglement):
  • Faithful conversions allow for the definition of the controversial notion
  • f non-classicality based on the well-founded principles of

entanglement.

Sperling, J., & Vogel, W. (2015). Convex ordering and quantification of quantumness. Physica Scripta, 90(7), 074024. Killoran, Nathan, Frank ES Steinhoff, and Martin B. Plenio. "Converting Nonclassicality into Entanglement." Physical review letters 116.8 (2016): 080402. [1] [2]

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Relevance

Theorem: If the free states in a finite dimensional Hilbert space form a countable set, then linear independence of the free states is a necessary and sufficient condition for the existence of a faithful conversion operation.

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Relevance

  • Generalization of coherence theory[1,2,3].
  • Linear independence versus orthogonality.

Aberg, J. (2006). Quantifying superposition. arXiv preprint quant-ph/0612146. Baumgratz, T., Cramer, M., & Plenio, M. B. (2014). Quantifying coherence. Physical review letters, 113(14), 140401. Streltsov, A., Adesso, G., & Plenio, M. B. (2016). Quantum coherence as a resource. arXiv preprint arXiv:1609.02439. Vogel, W., & Sperling, J. (2014). Unified quantification of nonclassicality and entanglement. Physical Review A, 89(5), 052302. [1] [2] [3] [4]

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A resource theory of superposition| Thomas Theurer| 2017 Page 24

Relevance

  • Generalization of coherence theory[1,2,3].
  • Linear independence versus orthogonality.
  • Example: Quantify non-classicality in the superposition of a finite

number of optical coherent states.

  • Faithful conversion can be done using a beam splitter[4].

Aberg, J. (2006). Quantifying superposition. arXiv preprint quant-ph/0612146. Baumgratz, T., Cramer, M., & Plenio, M. B. (2014). Quantifying coherence. Physical review letters, 113(14), 140401. Streltsov, A., Adesso, G., & Plenio, M. B. (2016). Quantum coherence as a resource. arXiv preprint arXiv:1609.02439. Vogel, W., & Sperling, J. (2014). Unified quantification of nonclassicality and entanglement. Physical Review A, 89(5), 052302. [1] [2] [3] [4]

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A resource theory of superposition| Thomas Theurer| 2017 Page 25

Relevance

  • Generalization of coherence theory[1,2,3].
  • Linear independence versus orthogonality.
  • Example: Quantify non-classicality in the superposition of a finite

number of optical coherent states.

  • Faithful conversion can be done using a beam splitter[4].
  • Starting point for more general resource theories.

Aberg, J. (2006). Quantifying superposition. arXiv preprint quant-ph/0612146. Baumgratz, T., Cramer, M., & Plenio, M. B. (2014). Quantifying coherence. Physical review letters, 113(14), 140401. Streltsov, A., Adesso, G., & Plenio, M. B. (2016). Quantum coherence as a resource. arXiv preprint arXiv:1609.02439. Vogel, W., & Sperling, J. (2014). Unified quantification of nonclassicality and entanglement. Physical Review A, 89(5), 052302. [1] [2] [3] [4]

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Free operations

Reciprocal vectors:

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Free operations

Reciprocal vectors: Free Kraus operators:

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Free operations

Reciprocal vectors: Free Kraus operators: Free completion of maps:

  • valid for coherence theory
  • not possible in entanglement theory
  • important for applications
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Superposition measures

  • Compare/Quantify superposition: measures
  • A function mapping quantum states to the

positive real numbers is called a superposition measure if it is

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Superposition measures

  • Compare/Quantify superposition: measures
  • A function mapping quantum states to the

positive real numbers is called a superposition measure if it is

  • 1. Faithful
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A resource theory of superposition| Thomas Theurer| 2017 Page 31

Superposition measures

  • Compare/Quantify superposition: measures
  • A function mapping quantum states to the

positive real numbers is called a superposition measure if it is

  • 1. Faithful
  • 2. Monotonic under
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A resource theory of superposition| Thomas Theurer| 2017 Page 32

Superposition measures

  • Compare/Quantify superposition: measures
  • A function mapping quantum states to the

positive real numbers is called a superposition measure if it is

  • 1. Faithful
  • 2. Monotonic under
  • 3. Monotonic under selective free

measurements on average

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A resource theory of superposition| Thomas Theurer| 2017 Page 33

Superposition measures

  • Compare/Quantify superposition: measures
  • A function mapping quantum states to the

positive real numbers is called a superposition measure if it is

  • 1. Faithful
  • 2. Monotonic under
  • 3. Monotonic under selective free

measurements on average

  • 4. Convex
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The 𝒎𝟐-measure of superposition

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State transformations

Question:

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State transformations

Question: Answer:

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State transformations

Question: Answer: Highest probability is the solution of a semidefinite program.

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Qubit transformations

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States with maximal superposition

Golden unit – exists for qubits x z

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States with maximal superposition

  • Not existent in general.
  • Exist in the limit of coherence theory.
  • How to proof? Counter example for qutrits.
  • 1. Classical rank can never increase.
  • 2. Maximize the 𝒎𝟐-measure of superposition.
  • 3. Consider transformation to max-rank states and formulate

semidefinite program.

  • 4. Bound solution by duality .
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Unitary qubit operations

Theorem: For every qubit-unitary there exists a fixed independent of acting on two qubits such that

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Superposition as a resource in decision tasks

Napoli, C., Bromley, T. R., Cianciaruso, M., Piani, M., Johnston, N., & Adesso, G. (2016). Robustness of coherence: an operational and observable measure of quantum coherence. Physical review letters, 116(15), 150502. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 43

ask for new turn

Superposition as a resource in decision tasks

Napoli, C., Bromley, T. R., Cianciaruso, M., Piani, M., Johnston, N., & Adesso, G. (2016). Robustness of coherence: an operational and observable measure of quantum coherence. Physical review letters, 116(15), 150502. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 44

ask for new turn

Superposition as a resource in decision tasks

return

Napoli, C., Bromley, T. R., Cianciaruso, M., Piani, M., Johnston, N., & Adesso, G. (2016). Robustness of coherence: an operational and observable measure of quantum coherence. Physical review letters, 116(15), 150502. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 45

ask for new turn

Superposition as a resource in decision tasks

return Arbitrary quantum

  • peration

Napoli, C., Bromley, T. R., Cianciaruso, M., Piani, M., Johnston, N., & Adesso, G. (2016). Robustness of coherence: an operational and observable measure of quantum coherence. Physical review letters, 116(15), 150502. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 46

ask for new turn

Superposition as a resource in decision tasks

return try again Arbitrary quantum

  • peration

Napoli, C., Bromley, T. R., Cianciaruso, M., Piani, M., Johnston, N., & Adesso, G. (2016). Robustness of coherence: an operational and observable measure of quantum coherence. Physical review letters, 116(15), 150502. [1]

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A resource theory of superposition| Thomas Theurer| 2017 Page 47

ask for new turn

Superposition as a resource in decision tasks

return try again guess Arbitrary quantum

  • peration
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A resource theory of superposition| Thomas Theurer| 2017 Page 48

ask for new turn

Superposition as a resource in decision tasks

return try again guess Arbitrary quantum

  • peration

correct: win wrong: loss

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A resource theory of superposition| Thomas Theurer| 2017 Page 49

ask for new turn

Superposition as a resource in decision tasks

return try again guess Arbitrary quantum

  • peration

correct: win wrong: loss

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A resource theory of superposition| Thomas Theurer| 2017 Page 50

Summary

  • Relevance
  • Definition of non-classicality using

entanglement

  • Generalization of coherence theory
  • Step toward optical non-classicality
  • Mathematical structure
  • Free maps and free completion of maps
  • Superposition measures
  • Superposition manipulation
  • Semidefinite program
  • States with maximal superposition
  • Advantages by superposition
  • Unitary qubit operations
  • Decision task

Outlook

  • Manipulation
  • Mixed state, catalytic and approximate

transformations

  • Transformations in the asymptotic limit
  • Counterpart to Nielsen’s theorem
  • Combine with further restrictions
  • Distributed scenarios
  • Energy conservation
  • Generalizations
  • Drop linear independence
  • Infinite dimensional systems
  • Continuous settings

Conclusions