How Much Information Is In Entangled Quantum States

How Much Information Is In Entangled Quantum States? | Scott Aaronson MIT

The Problem In quantum mechanics, a state of n entangled particles requires at least 2 n 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 Central problem: To do tomography on an entangled state of n particles, you need ~cn measurements Innsbruck group: 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?

Let’s 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 ~2 n 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: Limitations on the information content of quantum states [A. 2004], [A. 2006], [A. Drucker 2010] Lesson: “The linearity of QM helps tame the exponentiality of QM”

First, where does the exponentiality of quantum states manifest itself? Quantum Proofs and Advice: Let G be a finite (but exponentially-large) group G, and let H G be an exponentially[A. -Kuperberg 2007]: For the Group Nonlarge subgroup. Then the following highly-entangled state (if Membership therewhether might also be short you have it) can beproblem, used to decide a given element that are quickly verifiable byform a QCx H: x Gclassical is in H orproofs not, and to prove statements of the But at least “relative to a quantum oracle, ” there also cases where quantum proofs are provably exponentially more compact than proofsthe Theclassical trick: Measure qubit in Outstanding challenge to show suchfirst a separation relative to a classical oracle

The Absent-Minded Advisor Problem Can you give your graduate student a quantum state | with ~n qubits, 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 ALL MEASUREMENTS USING ≤n 2 QUANTUM GATES

How does theorem work? I 321 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… 1. Computing the classical “compressed representation” of quantum state seems astronomically hard 2. 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 entangled state of n particles Suppose you just want to be able to estimate the acceptance probabilities of most measurements E drawn “Quantum states are from some probability distribution D PAC-learnable” Then it suffices to do the following, for some m=O(n): 1. Choose m measurements independently from D 2. Go into your lab and estimate acceptance probabilities of all of them on | 3. Find any “hypothesis state” approximately consistent with all measurement outcomes

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 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

Combining My Postselection and Quantum Learning Results [A. -Drucker 2010]: Let | be an n-qubit state and let T be a complexity bound. Then there exists a local Hamiltonian H on poly(n, T) qubits, such that any 2 -outcome measurement on | performable by a circuit of size T can be simulated in poly(n, T) time by a suitable measurement M’ on the unique ground state of H Application: Trusted quantum advice is equivalent to trusted classical advice + untrusted quantum advice

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 [Aaronson-Gottesman, in preparation] Learnability of stabilizer states