Why Quasi-probability?

"Resources"
Negativity Contextuality Bell non-locality Entanglement Interference Quantum speed-up

Tools
Quasi-probability Covariance matrix Stabilizer formalism Matrix product state Monte Carlo Computational complexity


Contextuality Bell non-locality Entanglement Interference Quantum speed-up


Covariance matrix Stabilizer formalism Matrix product state Monte Carlo Computational complexity

Negativity

Quasi-probability

"Resources"

Tools

Why Quasi-probability?

Positive quasi-probabilities

Monte Carlo

Classical simulation

Why Quasi-probability?

Monte Carlo

Classical simulation

Negative quasi-probabilities

No classical simulation

Why Quasi-probability?

Efficiently representable

positive quasi-probabilities

Negativity is a necessary but not sufficient quantum resource

Veitch et al., New J. Phys. 14 113011 (2012)

Free states/ Operations	Wigner function	Discrete Wigner function (Odd dimensions only)
Positive pure states	Gaussians (Hudson thm)	Stabilizer states (Discrete Hudson thm)
(Non-convex-Gaussian) positive mixed states	e.g. Single-photon-added thermal states	"Bounded universal" states
Operations	Quadratic bosonic Hamiltonians	Cliffords
Measurements	Gaussians	Paulis

Veitch et al., New J. Phys. 15 013037 (2013)

Gross, J. Math. Phys. 47 122107 (2006)

Bartlett et al., Phys. Rev. Lett. 88 097904 (2002)

Q functions are always positive

Q(\alpha) = \langle \alpha|\rho|\alpha \rangle

Q(\alpha) = \langle \alpha|\rho|\alpha \rangle

Q function of a photon number state

"In evaluating the possibility of a classical explanation of an experiment, one must consider the negativity of not just the representation of [states] but of measurements as well, and one must look at representations other than that of Wigner."

Spekkens, Phys. Rev. Lett. 101 020401 (2008)

This Talk

\rho

\rho

E

E

State

\mu_{\rho}(\Omega) = \text{Tr}[\rho F(\Omega)]

\mu_{\rho}(\Omega) = \text{Tr}[\rho F(\Omega)]

Measurement

\zeta_{E_k}(\Omega) = \text{Tr}[E_k D(\Omega)]

\zeta_{E_k}(\Omega) = \text{Tr}[E_k D(\Omega)]

Quasi-probability Representations

Outcome k

Hermitian

Quantum experiment

\text{Tr}[A

\text{Tr}[A

A = \int d\Omega \, \text{Tr}[AF(\Omega)]

A = \int d\Omega \, \text{Tr}[AF(\Omega)]

A

A

A

A

F(\Omega)

F(\Omega)

]

]

Frame

D(\Omega)

D(\Omega)

Dual frame

F(\Omega)

F(\Omega)

D(\Omega)

D(\Omega)

Frames and Their Duals

\text{Tr}[A

\text{Tr}[A

A = \int d\Omega \, \text{Tr}[A|\alpha \rangle \langle \alpha|]

A = \int d\Omega \, \text{Tr}[A|\alpha \rangle \langle \alpha|]

A

A

A

A

|\alpha \rangle \langle \alpha|

|\alpha \rangle \langle \alpha|

]

]

Frame

D(\Omega)

D(\Omega)

Dual frame

F(\Omega)

F(\Omega)

D(\Omega)

D(\Omega)

Frames and Their Duals

Frame for the

Q function

Frame for the P function

\rho

\rho

E

E

State

\mu_{\rho}(\Omega) = \text{Tr}[\rho F(\Omega)]

\mu_{\rho}(\Omega) = \text{Tr}[\rho F(\Omega)]

Measurement

\zeta_{E_k}(\Omega) = \text{Tr}[E_k D(\Omega)]

\zeta_{E_k}(\Omega) = \text{Tr}[E_k D(\Omega)]

P(E_k|\rho) = \text{Tr}[E_k \rho] = \int d\Omega \, \mu_{\rho} (\Omega) \zeta_{E_k} (\Omega)

P(E_k|\rho) = \text{Tr}[E_k \rho] = \int d\Omega \, \mu_{\rho} (\Omega) \zeta_{E_k} (\Omega)

Quasi-probability Representations

Born rule as phase space average

Outcome k

Hermitian

Quantum experiment

The frame or the dual frame must contain a non-positive operator

Spekkens, Phys. Rev. Lett. 101 020401 (2008)

Ferrie and Emerson, J. Phys. A: Math. Theor. 41 352001 (2008)

Quasi-probability Representations

\rho

\rho

E

E

State

Measurement

Quantum experiment

Fermions

Veitch et al., New J. Phys. 14 113011 (2012)

	Wigner function	Discrete Wigner function (Odd dimensions only)
Positive pure states	Gaussians (Hudson thm)	Stabilizer states (Discrete Hudson thm)
(Non-convex-Gaussian) positive mixed states	e.g. Single-photon-added thermal states	"Bounded universal" states
Operations		Cliffords
Measurements	Gaussians	Paulis

Veitch et al., New J. Phys. 15 013037 (2013)

Gross, J. Math. Phys. 47 122107 (2006)

Bartlett et al., Phys. Rev. Lett. 88 097904 (2002)

Free states/ Operations

Quadratic bosonic Hamiltonians

Reversible free operations simply permute points of the phase space

U(g) F(\Omega) U^{\dagger}(g) = F(g \cdot \Omega)

U(g) F(\Omega) U^{\dagger}(g) = F(g \cdot \Omega)

G-covariant frame

Free operations should not increase "resource" (negativity)

The Wigner frame is self-dual and covariant w.r.t. all quadratic bosonic Hamiltonian evolutions

\{ a_j, a^{\dagger}_k \} = \delta_{jk}

\{ a_j, a^{\dagger}_k \} = \delta_{jk}

H = \sum_{jk}

H = \sum_{jk}

Quadratic Fermionic Hamiltonians

Free Fermionic Evolution

\{ a_j, a_k \} = 0

\{ a_j, a_k \} = 0

\alpha_{jk} a_j^{\dagger}a_k

\alpha_{jk} a_j^{\dagger}a_k

+

+

\beta_{jk} a_j^{\dagger} a_k^{\dagger}

\beta_{jk} a_j^{\dagger} a_k^{\dagger}

+ \text{H.C.}

+ \text{H.C.}

Number-preserving

Squeezing

\{ c_{\alpha}, c_{\beta} \} = 2 \delta_{\alpha \beta}

\{ c_{\alpha}, c_{\beta} \} = 2 \delta_{\alpha \beta}

Majorana operators

a_j = \displaystyle{\frac{c_{2j-1} + ic_{2j}}{2}}

a_j = \displaystyle{\frac{c_{2j-1} + ic_{2j}}{2}}

Quadratic Fermionic Hamiltonians

Free Fermionic Evolution

H = \frac{i}{4} \sum_{\alpha\beta,\alpha < \beta}

H = \frac{i}{4} \sum_{\alpha\beta,\alpha < \beta}

h_{\alpha\beta}

h_{\alpha\beta}

[c_{\alpha},c_{\beta}]

[c_{\alpha},c_{\beta}]

Antisymmetric

\{ a_j, a^{\dagger}_k \} = \delta_{jk}

\{ a_j, a^{\dagger}_k \} = \delta_{jk}

\{ a_j, a_k \} = 0

\{ a_j, a_k \} = 0

Majorana operators

\{ c_{\alpha}, c_{\beta} \} = 2 \delta_{\alpha \beta}

\{ c_{\alpha}, c_{\beta} \} = 2 \delta_{\alpha \beta}

a_j = \displaystyle{\frac{c_{2j-1} + ic_{2j}}{2}}

a_j = \displaystyle{\frac{c_{2j-1} + ic_{2j}}{2}}

Quadratic Fermionic Hamiltonians

Free Fermionic Evolution

H = \frac{i}{4} \sum_{\alpha\beta,\alpha < \beta}

H = \frac{i}{4} \sum_{\alpha\beta,\alpha < \beta}

h_{\alpha\beta}

h_{\alpha\beta}

[c_{\alpha},c_{\beta}]

[c_{\alpha},c_{\beta}]

Antisymmetric

\vec{c}(t) = e^{\bold{h}t} \vec{c}(0)

\vec{c}(t) = e^{\bold{h}t} \vec{c}(0)

\bold{h}

\bold{h}

\text{SO(2n)}

\text{SO(2n)}

\in

\in

Group of rotations

in 2n dimensions

\begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_{2n} \end{pmatrix}

\begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_{2n} \end{pmatrix}

| \Omega \rangle = U(g) |0 \rangle

| \Omega \rangle = U(g) |0 \rangle

The group action on the vacuum define an SO(2n)-covariant Q Function

Orbit

Coset space

Number-preserving

Perelomov, Generalized Coherent States... (1986)
Zhang, Feng and Gilmore, Rev. Mod. Phys. 62 867 (1990)

\{\Omega \} = \text{SO}(2n)/\text{U(n)}

\{\Omega \} = \text{SO}(2n)/\text{U(n)}

Fermionic Gaussian states

Phase space

=

=

Q(\alpha) = \langle \Omega|\rho|\Omega \rangle

Q(\alpha) = \langle \Omega|\rho|\Omega \rangle

Q function

S^2 \simeq \text{SO}(3)/\text{SO}(2)

S^2 \simeq \text{SO}(3)/\text{SO}(2)

|n\rangle

|n\rangle

\theta

\theta

\phi

\phi

|0\rangle

|0\rangle

Guiding Example

Q functions on a sphere for spin-j systems

Spherical Q Function

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

Quasi-probability representation for spin-j systems

|n\rangle

|n\rangle

\theta

\theta

\phi

\phi

|0\rangle

|0\rangle

n

n

m

m

Spherical Q Function

R_{n \to m}

R_{n \to m}

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

Quasi-probability representation for spin-j systems

|n\rangle

|n\rangle

\phi

\phi

|0\rangle

|0\rangle

\theta

\theta

0

0

0

0

Spherical Q Function

R_Y

R_Y

R_Z

R_Z

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

Quasi-probability representation for spin-j systems

|n\rangle

|n\rangle

\phi

\phi

|0\rangle

|0\rangle

R_Z

R_Z

0

0

0

0

\theta

\theta

Spherical Q Function

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

R(\eta,\theta,\phi) = R_Z(\phi) R_Y(\theta) R_Z(\eta)

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

|\langle n|m \rangle|^2 = \frac{(1+\cos\theta)}{2}^{2j}

Quasi-probability representation for spin-j systems

|n\rangle

|n\rangle

\phi

\phi

|0\rangle

|0\rangle

\theta

\theta

R_Y

R_Y

0

0

0

0

|0\rangle

|0\rangle

|0\rangle

|0\rangle

|0\rangle

|0\rangle

R_{Y^+}

R_{Y^+}

|0\rangle

|0\rangle

R_{Z^+}

R_{Z^+}

R_{Y^-}

R_{Y^-}

R_{Z^-}

R_{Z^-}

Fermionic Q Function

\text{U}(4)

\text{U}(4)

2 commuting

two-mode squeezings

|\langle \Omega|\Omega' \rangle|^2 = (\frac{1+\cos \theta^+_{12}}{2}) (\frac{1+\cos \theta^+_{34}}{2})

|\langle \Omega|\Omega' \rangle|^2 = (\frac{1+\cos \theta^+_{12}}{2}) (\frac{1+\cos \theta^+_{34}}{2})

Number-preserving

R_{Y^+}

R_{Y^+}

R_{Z^+}

R_{Z^+}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

Jordan-Wigner transform to 4-qubit quantum circuit

|0\rangle

|0\rangle

|0\rangle

|0\rangle

|0\rangle

|0\rangle

R_{Y^+}

R_{Y^+}

|0\rangle

|0\rangle

R_{Z^+}

R_{Z^+}

R_{Y^-}

R_{Y^-}

R_{Z^-}

R_{Z^-}

Fermionic Q Function

\text{U}(4)

\text{U}(4)

commuting

two-mode squeezings

|\langle \Omega|\Omega' \rangle|^2 = (\frac{1+\cos \theta^+_{12}}{2}) (\frac{1+\cos \theta^+_{34}}{2})

|\langle \Omega|\Omega' \rangle|^2 = (\frac{1+\cos \theta^+_{12}}{2}) (\frac{1+\cos \theta^+_{34}}{2})

Number-preserving

R_{Y^+}

R_{Y^+}

R_{Z^+}

R_{Z^+}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Y^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

R_{Z^-}

\lfloor \frac{n}{2} \rfloor

\lfloor \frac{n}{2} \rfloor

Jordan-Wigner transform to 4-qubit quantum circuit

|\langle \Omega|\Omega' \rangle|^2 = |\langle 0| U^{\dagger}(\Omega) U(\Omega') |0 \rangle|^2

|\langle \Omega|\Omega' \rangle|^2 = |\langle 0| U^{\dagger}(\Omega) U(\Omega') |0 \rangle|^2

= |\langle 0| U(\Omega^{-1}\Omega') |0 \rangle|^2

= |\langle 0| U(\Omega^{-1}\Omega') |0 \rangle|^2

= |\langle 0| U(Kg^{-1}g'K') |0 \rangle|^2

= |\langle 0| U(Kg^{-1}g'K') |0 \rangle|^2

U(n)-bi-invariant function

Fermionic Q Function

\in \text{Span}\{ \lfloor \frac{n}{2} \rfloor + 1 \, \, \text{zonal spherical functions} \}

\in \text{Span}\{ \lfloor \frac{n}{2} \rfloor + 1 \, \, \text{zonal spherical functions} \}

K = \text{U}(n)

K = \text{U}(n)

From the Q function, we can generate a continuous family of G-covariant frames and dual frames that gives a unique self-dual "Wigner function"

F^{(s)}(\Omega) = \int d\Omega' \,

F^{(s)}(\Omega) = \int d\Omega' \,

\text{Tr} [F^{(s)}(\Omega) F^{(1)}(\Omega')]

\text{Tr} [F^{(s)}(\Omega) F^{(1)}(\Omega')]

|\Omega' \rangle \langle \Omega'|

|\Omega' \rangle \langle \Omega'|

P

Q

Wigner

s

s

-1

0

1

Stratonovich-Weyl axioms

Brif and Mann, Phys. Rev. A 59 971 (1999)

Fermionic Wigner Function

From the Q function, we can generate a continuous family of G-covariant frames and dual frames that gives a unique self-dual "Wigner function"

P

Q

Wigner

s

s

-1

0

1

Stratonovich-Weyl axioms

= \text{Tr} [U(\Omega) F^{(s)}(0) U^{\dagger}(\Omega) U(\Omega') F^{(-s')}(0) U^{\dagger}(\Omega')]

= \text{Tr} [U(\Omega) F^{(s)}(0) U^{\dagger}(\Omega) U(\Omega') F^{(-s')}(0) U^{\dagger}(\Omega')]

= \text{Tr} [F^{(s)}(0) U(\Omega^{-1}\Omega') ^{(-s')}F(0) U^{\dagger}(\Omega^{-1}\Omega')]

= \text{Tr} [F^{(s)}(0) U(\Omega^{-1}\Omega') ^{(-s')}F(0) U^{\dagger}(\Omega^{-1}\Omega')]

U(n)-bi-invariant function!

\text{Tr} [F^{(s)}(\Omega) F^{(s')}(\Omega')]

\text{Tr} [F^{(s)}(\Omega) F^{(s')}(\Omega')]

Fermionic Wigner Function

Brif and Mann, Phys. Rev. A 59 971 (1999)

4-mode fermionic Gaussian state

Fermionic Wigner Function

4-mode fermionic Gaussian state

Closest analogue, but may not be a fair comparison to bosonic quasi-probability representations
- The continuous family of bosonic quasi-probability representations are not covariant under the full quadratic Hamiltonian evolution
- There is no analogue of the Heisenberg group for fermions

Why Negativity?

Scaling of negativity and simulatability
Negativity of non-Gaussian states
Connection to existing quasi-probability representations for fermions

Future Directions

Summary

We define quasi-probability representations in terms of frames and dual frames.
We obtain a continuous family of SO(2n)-covariant quasi-probability representations from fermionic Gaussian states.
We prove the uniqueness of the self-dual "Wigner function".
The construction applies to phase spaces that are symmetric homogeneous spaces and more.

ND, "Quantum Phase Space Representations and Their Negativities", University of New Mexico, PhD Dissertation (2018)

Quasi-probabilities on Fermionic Phase Spaces