Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Quantum theory is a quasi-stochastic process theory Radboud University John van de Wetering wetering@cs.ru.nl Institute for Computing and Information Sciences Radboud University Nijmegen QPL2017 5th of July 2017 J. van de Wetering QPL2017 quasi-stochastic representations 1 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion “The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative.” – Richard Feynman, 1981 J. van de Wetering QPL2017 quasi-stochastic representations 2 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion A bit of background • Wigner (1932): Representing a quantum state as a distribution over classical phase space allowing negative probabilities. J. van de Wetering QPL2017 quasi-stochastic representations 3 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion A bit of background • Wigner (1932): Representing a quantum state as a distribution over classical phase space allowing negative probabilities. • Negativity in representations is “equivalent” to contextuality (Spekkens 2008). • Quantum speed up requires sufficient negativity in representations (Pashayan, Walman & Bartlett 2015). J. van de Wetering QPL2017 quasi-stochastic representations 3 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Related work • Appleby, Fuchs, Stacey, Zhu 2016 “Introducing the Qplex”. • Hardy 2013 “The duotensor framework” • Ferrie & Emerson 2008 “Frame representations of quantum mechanics” J. van de Wetering QPL2017 quasi-stochastic representations 4 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Informationally complete POVMs Definition • Let M n be the set of n × n complex matrices. • An effect is an E ∈ M n such that 0 ≤ E ≤ 1. • A POVM is a set of effects { E i } such that � i E i = I n . J. van de Wetering QPL2017 quasi-stochastic representations 5 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Informationally complete POVMs Definition • Let M n be the set of n × n complex matrices. • An effect is an E ∈ M n such that 0 ≤ E ≤ 1. • A POVM is a set of effects { E i } such that � i E i = I n . • A POVM is called informationally complete if it spans M n and minimal informationally complete (MIC) if it is a basis. A MIC-POVM always has n 2 elements. J. van de Wetering QPL2017 quasi-stochastic representations 5 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Informationally complete POVMs Definition • Let M n be the set of n × n complex matrices. • An effect is an E ∈ M n such that 0 ≤ E ≤ 1. • A POVM is a set of effects { E i } such that � i E i = I n . • A POVM is called informationally complete if it spans M n and minimal informationally complete (MIC) if it is a basis. A MIC-POVM always has n 2 elements. Definition A quantum state is ρ ∈ M n such that ρ ≥ 0 and tr( ρ ) = 1. J. van de Wetering QPL2017 quasi-stochastic representations 5 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Quantum states as probability distributions Let ρ ∈ M n be a quantum state and { E i } a POVM. → p ( i ) = tr( ρ E i ) forms a probability distribution. J. van de Wetering QPL2017 quasi-stochastic representations 6 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Quantum states as probability distributions Let ρ ∈ M n be a quantum state and { E i } a POVM. → p ( i ) = tr( ρ E i ) forms a probability distribution. Now suppose { E i } is MIC, then it is a basis so there are coefficients α such that E j � ρ = α j tr( E j ) j J. van de Wetering QPL2017 quasi-stochastic representations 6 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Quantum states as probability distributions Let ρ ∈ M n be a quantum state and { E i } a POVM. → p ( i ) = tr( ρ E i ) forms a probability distribution. Now suppose { E i } is MIC, then it is a basis so there are coefficients α such that E j � ρ = α j tr( E j ) j Now: � E j � � p ( i ) = tr( ρ E i ) = α j tr tr( E j ) E i j J. van de Wetering QPL2017 quasi-stochastic representations 6 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Quantum states as probability distributions - cont. � E j � � p ( i ) = tr( ρ E i ) = α j tr tr( E j ) E i j � � E j Define the transition matrix T ij = tr tr ( E j ) E i . J. van de Wetering QPL2017 quasi-stochastic representations 7 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Quantum states as probability distributions - cont. � E j � � p ( i ) = tr( ρ E i ) = α j tr tr( E j ) E i j � � E j Define the transition matrix T ij = tr tr ( E j ) E i . then we can succinctly write α = T − 1 p p = T α or equivalently Which allows us to reconstruct the original state: E i � ( T − 1 p ) i ρ = tr( E i ) i J. van de Wetering QPL2017 quasi-stochastic representations 7 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion Quantum states as probability distributions - cont. � E j � � p ( i ) = tr( ρ E i ) = α j tr tr( E j ) E i j � � E j Define the transition matrix T ij = tr tr ( E j ) E i . then we can succinctly write α = T − 1 p p = T α or equivalently Which allows us to reconstruct the original state: E i � ( T − 1 p ) i ρ = tr( E i ) i NOTE: T − 1 can contain negative components! J. van de Wetering QPL2017 quasi-stochastic representations 7 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion (quasi-)stochasticity The transition matrix T is an example of a stochastic matrix . J. van de Wetering QPL2017 quasi-stochastic representations 8 / 18
Introduction Quantum states as probability distributions Quantum channels as quasi-stochastic matrices Radboud University Nijmegen Quantum theory as a quasi-stochastic process theory Conclusion and Discussion (quasi-)stochasticity The transition matrix T is an example of a stochastic matrix . Definition • A real-valued matrix S is called stochastic when S ij ∈ R ≥ 0 for all i , j and all the columns sum up to 1. • It is quasi-stochastic when the positivity requirement is dropped. • S is doubly (quasi-)stochastic when its transpose is also (quasi-)stochastic. Stochastic matrices are precisely those matrices that map the space of probability distributions to itself. J. van de Wetering QPL2017 quasi-stochastic representations 8 / 18
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