How Much Information Is In A Quantum State?

| How Much Information Is In A Quantum State? Scott Aaronson MIT

The Problem In quantum mechanics, a state of n entangled particles requires at least 2n complex numbers to specify To a computer scientist, this is probably the central fact about quantum mechanics But why should we worry about it?

Answer 1: Quantum State Tomography Task: Given lots of copies of an unknown quantum state, produce an approximate classical description of it Not something I made up!“As seen in Science & Nature” Central problem: To do tomography on an entangled state of n particles, you need ~cn measurements Current record: 8 particles / ~656,000 experiments (!)

Answer 2: Quantum Computing Skepticism Levin Goldreich ‘t Hooft Davies Wolfram Some physicists and computer scientists believe quantum computers will be impossible for a fundamental reason For many of them, the problem is that a quantum computer would “manipulate an exponential amount of information” using only polynomial resources But is it really an exponential amount?

Today we’ll tame the exponential beast Idea: “Shrink quantum states down to reasonable size” by viewing them operationally Analogy: A probability distribution over n-bit strings also takes ~2n bits to specify. But that fact seems to be “more about the map than the territory” Holevo’s Theorem (1973): By sending an n-qubit quantum state, Alice can transmit no more than n classical bits to Bob This talk: New limitations on the information content of quantum states [A. 2004], [A. 2006], [A.-Dechter 2008], [A. Drucker 2009] Lesson: “The linearity of QM helps tame the exponentiality of QM”

The Absent-Minded Advisor Problem Can you give your graduate student a quantum state  with n qubits (or 10n, or n3, …)—such that by measuring  in a suitable basis, the student can learn your answer to any one yes-or-no question of size n? NO [Ambainis, Nayak, Ta-Shma, Vazirani 1999] Indeed, quantum communication is no better than classical for this problem as n

On the Bright Side… Suppose Alice wants to describe an n-qubit quantum state | to Bob … well enough that, for any 2-outcome measurement M in some finite set S, Bob can estimate the probability that M accepts | Theorem (A. 2004): In that case, it suffices for Alice to send Bob only ~n log n  log|S| classical bits

| ALL MEASUREMENTS PERFORMABLE USING ≤n2 QUANTUM GATES ALL MEASUREMENTS

How does the theorem work? 1 2 3 I Alice is trying to describe the quantum state  to Bob In the beginning, Bob knows nothing about , so he guesses it’s the maximally mixed state 0=I Then Alice helps Bob improve, by repeatedly telling him a measurement EtS on which his current guess t-1 badly fails Bob lets t be the state obtained by starting from t-1, then performing Et and postselecting on getting the right outcome

Just two tiny problems with this compression theorem… • Computing the classical “compressed representation” of quantum state seems astronomically hard • Given the compressed representation, computing the probability some measurement on the state accepts also seems astronomically hard • The “Quantum Occam’s Razor Theorem” [A. 2006] at least addresses the first problem…

Quantum Occam’s Razor Theorem • Let | be an unknown quantum state of n particles • Suppose you just want to be able to estimate the acceptance probabilities of most measurements E drawn from some probability distribution D • Then it suffices to do the following, for some m=O(n): • Choose m measurements independently from D • Go into your lab and estimate acceptance probabilities of all of them on | • Find any “hypothesis state” approximately consistent with all measurement outcomes “Quantum states are PAC-learnable”

Numerical Simulation[A.-Dechter 2008] We implemented this “pretty-good quantum state tomography” method in MATLAB, using a fast convex programming method developed specifically for this application [Hazan 2008] We then tested it (on simulated data) using an MIT cluster We studied how the number of sample measurements m needed for accurate predictions scales with the number of qubits n, for n≤10 Result of experiment: My theorem appears to be true

Recap: Given an unknown n-qubit quantum state , and a set S of two-outcome measurements… Learning theorem: “Any hypothesis state  consistent with a small number of sample points behaves like  on most measurements in S” Postselection theorem: “A particular state T (produced by postselection) behaves like  on all measurements in S” Dream theorem: “Any state  that passes a small number of tests behaves like  on all measurements in S” [A.-Drucker 2009]: The dream theorem holds Caveat:  will have more qubits than , and in general be a very different state Implication: Can get a quantum state that satisfies exp(n) conditions by imposing only poly(n) constraints

Summary In many natural scenarios, the “exponentiality” of quantum states is an illusion That is, there’s a short (though possibly cryptic) classical string that specifies how the quantum state behaves, on any measurement you could actually perform Applications: Pretty-good quantum state tomography, characterization of quantum computers with “magic initial states”… Biggest open problem: Find special classes of quantum states that can be learned in a computationally efficient way “Experimental demonstration” would be nice too

How Much Information Is In A Quantum State?