Laying the Quantum and Classical Embedding Problems

to RestarXiv:0908.2128

Toby Cubitt1, Jens Eisert2, Michael Wolf 3

1University of Bristol, 2Potsdam University, 3Niels Bohr Institute, Copenhagen

Talk Outline

• The Quantum and Classical Embedding Problems

• Motivation: Measurements and Experiments

• The Quantum Embedding Problem (2)

• Laying the Embedding Problems to Rest

• Conclusions

Open Quantum SystemsTwo complementary descriptions of evolution of open quantum systems:

• Master equations:– differential equation– continuous time– describes underlying physics

• Quantum channels (a.k.a. CPT maps):– input-output transformation– discrete time– “black-box” model

Yes iff is of Lindblad form:

[Lindblad; Kossakowski, Gorini 1978]

…

Open Quantum SystemsTwo questions naturally arise about the relationship between these descriptions:

“Quantum embedding problem”(a.k.a. “Markovianity problem”)

• Given , does it generate CPT ?

• Conversely, given CPT , is there a Lindblad master equation that generates it?

Quantum ↔ Classical Systems

Quantum

density matrix

quantum channel

Lindblad generator

master equation

Classical

probability vector

stochastic matrix

“Q” matrix

continuous-timeMarkov process

• Classical channels (a.k.a. stochastic maps):– input-output transformation– discrete time– “black-box” model

Open Classical SystemsTwo complementary descriptions of evolution of open classical systems:

• Continuous-time Markov chains:– differential equation– continuous time– describes underlying physics

Open Classical SystemsTwo questions naturally arise about the relationship between these descriptions:

• Given , does it generate stochastic ?

…

Yes iff matrix satisfies:

[Any good textbook on Markov chains, e.g. Norris]

• Conversely, given a stochastic map , is there a continuous-time Markov process that generates it?

• i.e. can be embedded in a continuous-time Markov process?

“Embedding problem”

A Potted History of the(Classical) Embedding Problem

Em

be

dding

prob

lem rem

ains u

nso

lved

• 1937: first(?) posed in a paper by Elfving.

• 1962: Kingman publishes paper on the embedding problem during his PhD(attributing soln of 2x2 case to Kendall).

• 1973: Kingman+Williams

• 1980: Frydman makes progress on 3x3 case.

• 1985: Kingman → Sir John Kingman

• 1990: Mukherjea

• 1988: Denisov; Fuglede

Talk Outline

• The Quantum and Classical Embedding Problems

• Motivation: Measurements and Experiments

• The Quantum Embedding Problem (2)

• Laying the Embedding Problems to Rest

• Conclusions

Motivation:Measurements and Experiments

State tomography

Motivation: Measurements and Experiments

Process tomography

State tomography

Motivation: Measurements and Experiments

…

…

Scales polynomially in the relevantparameter (system dimension)

Motivation: Measurements and Experiments

Physics!differential equations:

=

…

• Given a quantum channel, does there exist a master equation that generates it?

• Can we extract the underlying physics (i.e. dynamical equations) from experimental data?

• Given a family of quantum channels, can we find a Lindblad master equation that is consistent with them?

• Given a single quantum channel, can we find a Lindblad master equation that generates it?

…

Motivation: Measurements and Experiments

does there exist

Recover the embedding problem.

• Given a quantum channel, does there exist a master equation that generates it?

The Quantum Embedding Problem(a.k.a. the Markovianity Question)

Any quantum channel describes aphysically realisable evolution

All channels are generated bysome master equation

[cf. Wolf, Cirac, CMP 279, 147 (2008)]

If the environment is the dominant effect on the evolution…

…trying to describe the evolution in terms of the system alone is doomed to failure!

• Given a quantum channel, does there exist a master equation that generates it?

The Quantum Embedding Problem(a.k.a. the Markovianity Question)

Talk Outline

• The Quantum and Classical Embedding Problems

• Motivation: Measurements and Experiments

• The Quantum Embedding Problem (2)

• Laying the Embedding Problems to Rest

• Conclusions

• Given a quantum channel, does there exist a master equation that generates it?

The Quantum Embedding Problem(a.k.a. the Markovianity Problem)

• Given a CPT map , does there exist an such that , and is CPT?

The Quantum Embedding Problem(a.k.a. the Markovianity Problem)

• What is “E” ?→ matrix representation of the channel as a linear operator on the vector space of density matrices.

• Closely related to Choi-Jamiołkowski state representation:

“ involution”(not partial transpose)

matrixmultiplication

• Given a CPT map , does there exist an such that , and is CPT?

• Given a CPT map , does there exist an such that , and is CPT?

• Given a stochastic map , does there exist a such that , and is stochastic?

• EMBEDDABLE CHANNELInstance: quantum channel E; precision ¸ 0Question: assert either

• 9 embeddable (=Markovian) E’ s.t. || E – E’ || ·

• 9 non-embeddable E’ s.t. || E – E’ || ·

The (Classical) Embedding ProblemThe Quantum Embedding Problem(a.k.a. the Markovianity Problem)

Embeddable Non-embeddable

²E2

²E1 ²E1 ²E2

(“Weak-membership” formulation, cf. separability [Gurvits ’03]).

Talk Outline

• The Quantum and Classical Embedding Problems

• Motivation: Measurements and Experiments

• The Quantum Embedding Problem (2)

• Laying the Embedding Problems to Rest

• Conclusions

• Lemma [Lindblad ‘76]: generates a CPT evolution iff it is of Lindblad form:

• Lemma [Lindblad ‘76]: L generates a CPT evolution iff

i. is Hermitian (Hermiticity)

ii. (normalisation)

iii. (ccp)

Solving the Quantum Embedding Problem

• E is embeddable iff there exists an L such that , and is CPTP 8 t ¸ 0 (up to ’s etc.).

)Not unique! (phasesof eigs. modulo 2i)

• E is embeddable iff there exists an L such that , and is CPTP 8 t ¸ 0 (up to ’s etc.).

• Branches parameterised by integers mc:

(some branches ruled out by Hermiticity condition).

• E is embeddable iff some branch of the logarithm has Lindblad form.

Solving the Quantum Embedding Problem

Solving the Quantum Embedding Problem

• LINDBLAD GENERATOR:instance: map L0; precision ¸ 0

promise: 9 map L’ s.t. || L – L’ || · f ( and eL’ is CPTPquestion: assert either

• 9 map L’0 and integers mc s.t. || L0 – L’ || · and

is of Lindblad form;

• 9 map L’0 s.t. || L0 – L’0 || · , eL’0 is CPTP, and

no is of Lindblad form.• Theorem:

LINDBLAD GENERATOR =K EMBEDDABLE CHANNEL(dealing with ‘s and ‘s is non-trivial → need some functional analysis results)

The Journey so Far

EMBEDDABLECHANNEL

EMBEDDABLEMAPLINDBLAD

GENERATOR

1-in-3SAT

Quantum Embedding Problem

NP - Hardness

• Linear integer program:Boolean variables → integer variablesTrue/False →Clause for vars. i, j, k →

• 1-in-3SAT:instance: boolean variables; clauses = sets of 3 vars.question: values s.t. clauses contain exactly 1 true var?

Recall…

• Lemma [Linblad ’76]:

is of Lindblad form iff

i. is Hermitian (Hermiticity)

ii. (normalisation)

iii. (ccp)

NP - Hardness

Encode 1-in-3SAT…

…whilst ensuring i. and ii. are always satisfied.

NP - Hardness• Encoding is very involved!

eigvals & eigvects ofEncoding:

eigvals & eigvects ofConstraints:

–involution is basis-dependent, and doesn’t preserve eigenvalues, eigenvectors, rank…anything useful!

NP - Hardness

EMBEDDABLECHANNEL

EMBEDDABLEMAPLINDBLAD

GENERATOR

1-in-3SAT

Quantum Embedding Problem

What about the 70-year-old(Classical) Embedding Problem?

EMBEDDABLESTOCHASTIC MAP

EMBEDDABLEMAP

Q-MATRIXGENERATOR

1-in-3SAT

(Classical) Embedding Problem≠ Quantum embedding problemThere’s an encoding of 1-in-3SAT into a

Q-matrix “hiding” inside the quantum construction

• For fixed dimension, integer semi-definite programming can be solved in poly-time (scaling with precision).[Khachiyan & Porkolab ’00, generalising Lenstra’s integer prog. result]

• Gives efficient algorithm for finding master equations from quantum channels for fixed dimension.[Wolf, Eisert, Cubitt, Cirac, arXiv:0711.3172, PRL 101, 150402 (2008)]

• NP is a decision class.

• Weak-membership problems are not decision problems.

Cannot technically be in NP, hence not NP-complete.

However…

• P=NP integer semi-definite programming P embedding problems can be solved in poly-time.

• Solving embedding problems ≡ solving P = NP.

“Moral” NP-Completeness

Proof: NP-hardness + algorithm for solving embedding problems using integer semi-definite programming.

Conclusions• Both the quantum and classical embedding problems

are NP-hard (and “morally” NP-complete: solving themis equivalent to solving P = NP).

• This finally lays to rest the 70-year-old (classical) embedding problem for stochastic matrices.

• Raises interesting questions about how we can deduce the underlying physics (dynamical equations) from experimental observations when doing so is NP-hard…

• …but also provides the first provably correct algorithm for extracting master equations from experimental data (efficient for fixed dimension, and optimal unless P = NP).

• Open problem: what about time-inhomogenous case?