Laying the Quantum and Classical Embedding Problems to Rest

Laying the Quantum and Classical Embedding Problems to Rest arXiv:0908.2128 Toby Cubitt1, Jens Eisert2, Michael Wolf3 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

Quantum channels (a.k.a. CPT maps): • input-output transformation • discrete time • “black-box” model Open Quantum Systems Two complementary descriptions of evolution of open quantum systems: • Master equations: • differential equation • continuous time • describes underlying physics

Given , does it generate CPT ? Yes iff is of Lindblad form:[Lindblad; Kossakowski, Gorini 1978] … • Conversely, given CPT , is there a Lindblad master equation that generates it? Open Quantum Systems Two questions naturally arise about the relationship between these descriptions: “Quantum embedding problem” (a.k.a. “Markovianity problem”) Open Problem

Quantum density matrix quantum channel Lindblad generator master equation Classical probability vector stochastic matrix “Q” matrix continuous-timeMarkov process Quantum ↔ Classical Systems

Classical channels (a.k.a. stochastic maps): • input-output transformation • discrete time • “black-box” model Open Classical Systems Two complementary descriptions of evolution of open classical systems: • Continuous-time Markov chains: • differential equation • continuous time • describes underlying physics

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 embeddedin a continuous-time Markov process? Open Classical Systems Two questions naturally arise about the relationship between these descriptions: Open Problem (since 1937) “Embedding problem”

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 Embedding problem remains unsolved • 1980: Frydman makes progress on 3x3 case. • 1985: Kingman → Sir John Kingman • 1988: Denisov; Fuglede • 1990: Mukherjea A Potted History of the(Classical) Embedding Problem

Motivation:Measurements and Experiments State tomography

State tomography Motivation: Measurements and Experiments Process tomography

… … Motivation: Measurements and Experiments Scales polynomially in the relevantparameter (system dimension)

… = differential equations: Motivation: Measurements and Experiments Physics!

does there exist … Motivation: Measurements and Experiments • 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? Recover the embedding problem.

Any quantum channel describes aphysically realisable evolution All channels are generated bysome master equation The Quantum Embedding Problem(a.k.a. the Markovianity Question) • Given a quantum channel, does there exist a master equation that generates it?  ?  [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! The Quantum Embedding Problem(a.k.a. the Markovianity Question) • Given a quantum channel, does there exist a master equation that generates it?

Given a CPT map, does there exist an such that , and is CPT? The Quantum Embedding Problem(a.k.a. the Markovianity Problem) • Given a quantum channel, does there exist a master equation that generates it?

Given a CPT map, does there exist an such that , and is CPT? • Closely related to Choi-Jamiołkowski state representation: “ involution” (not partial transpose) 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. matrixmultiplication

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? E1 ² ² E2 ² ² E2 E1 Embeddable Non-embeddable The (Classical) Embedding Problem The Quantum Embedding Problem(a.k.a. the Markovianity Problem) • EMBEDDABLE CHANNELInstance: quantum channelE; precision ¸ 0Question: assert either • 9embeddable (=Markovian) E’ s.t. || E – E’ || · ; • 9non-embeddableE’ s.t. || E – E’ || · ; (“Weak-membership” formulation, cf. separability [Gurvits ’03]).

E is embeddable iff there exists an L such that , and is CPTP 8t¸ 0 (up to  ’s etc.). • Lemma [Lindblad ‘76]: generates a CPT evolution iff it is of Lindblad form: • Lemma [Lindblad ‘76]:Lgenerates a CPT evolution iff • is Hermitian (Hermiticity) • (normalisation) • (ccp) ) Solving the Quantum Embedding Problem Not unique! (phasesof eigs. modulo 2i)

E is embeddable iff there exists an L such that , and is CPTP 8t¸ 0 (up to  ’s etc.). • Branches parameterised by integers mc:(some branches ruled out by Hermiticity condition). Solving the Quantum Embedding Problem • E is embeddable iff some branch of the logarithm has Lindblad form.

LINDBLAD GENERATOR:instance: map L0; precision  ¸ 0promise: 9 map L’ s.t. || L – L’ || ·f ( ) and eL’ is CPTPquestion: assert either • 9map L’0 and integers mc s.t. || L0 – L’0|| · and is of Lindblad form; • 9map L’0 s.t. || L0 – L’0|| ·, eL’0 is CPTP, andnois of Lindblad form. • Theorem:LINDBLAD GENERATOR =K EMBEDDABLE CHANNEL (dealing with ‘s and ‘s is non-trivial → need some functional analysis results) Solving the Quantum Embedding Problem

The Journey so Far EMBEDDABLECHANNEL Quantum Embedding Problem 1-in-3SAT EMBEDDABLEMAP LINDBLADGENERATOR

1-in-3SAT:instance: boolean variables; clauses = sets of 3 vars.question: values s.t. clauses contain exactly 1 true var? • Linear integer program:Boolean variables → integer variablesTrue/False →Clause for vars. i, j, k → NP - Hardness

NP - Hardness • Recall… • Lemma [Linblad ’76]: is of Lindblad form iff • is Hermitian (Hermiticity) • (normalisation) • (ccp) Encode 1-in-3SAT… …whilst ensuring i. and ii. are always satisfied.

Constraints: Encoding: eigvals & eigvects of eigvals & eigvects of NP - Hardness • Encoding is very involved! –involution is basis-dependent, and doesn’t preserve eigenvalues, eigenvectors, rank…anything useful!

NP - Hardness EMBEDDABLECHANNEL Quantum Embedding Problem NP-Hard 1-in-3SAT EMBEDDABLEMAP LINDBLADGENERATOR

What about the 70-year-old(Classical) Embedding Problem? EMBEDDABLESTOCHASTIC MAP (Classical) Embedding Problem There’s an encoding of 1-in-3SAT into aQ-matrix “hiding” inside the quantum construction ≠ Quantum embedding problem NP-Hard 1-in-3SAT EMBEDDABLEMAP Q-MATRIXGENERATOR

“Moral” NP-Completeness • 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. Proof: NP-hardness + algorithm for solving embedding problems using integer semi-definite programming. • 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)]

Conclusions • Both the quantum and classical embedding problemsare 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?

Laying the Quantum and Classical Embedding Problems to Rest