Quantifying quantum coherence Nathaniel Johnston – joint work with Jianxin Chen, Chi-Kwong Li, and Sarah Plosker Mount Allison University Sackville, New Brunswick, Canada

Overview What is Quantum Coherence? How Do We Measure Coherence? Coherence of Pure States

Overview What is Quantum Coherence? Measuring Coherence Physics: How “useful” a quantum state is Math #1: How far away from the standard basis a vector is Math #2: How far from diagonal a matrix is Measuring Coherence Coherence of Pure States

What is Quantum Coherence? Pure quantum state: with i.e., with Dual (row) vector: Inner product:

What is Quantum Coherence? A pure quantum state is “useless” if it is a standard basis vector: The farther a state is from these basis states, the more “useful” it is. close to (1,0)T – only a little “useful” far away from (1,0)T and (0,1)T – “most useful” state

What is Quantum Coherence? Mixed quantum state: Trace 1 Positive semidefinite equivalent Pure state (again): Rank 1 Trace 1 Positive semidefinite equivalent

What is Quantum Coherence? A mixed quantum state is “useless” if it is diagonal: Each of these pure states are “useless”, so the mixed state is too. The farther a state is from diagonal, the more “useful” it is. off-diagonal entries are small – only a little “useful” far from diagonal – “most useful” state

Overview What is Quantum Coherence? Measuring Coherence Many different methods have been proposed Different methods useful in different contexts Still very undeveloped Coherence of Pure States

Measuring Coherence Several ways of measuring coherence have been proposed. Method 1: The ℓ1-norm Just add up all of the off-diagonal entries of the mixed state ρ: “C” stands for “coherence”, and the “ℓ1” refers to how much this looks like the 1-norm of a vector Trivial to compute. Okay-ish physical properties.

Measuring Coherence Some nice mathematical properties include: Reminder: The ℓ1-norm of coherence is Some nice mathematical properties include: A simple formula when restricted to pure states: is maximized exactly when

Measuring Coherence becoming useless? Method 2: Robustness How much does ρ have to be mixed with another state before becoming useless? “C” stands for “coherence”, and the “R” refers to “robustness” – this measures how much noise/interference a state can tolerate before becoming useless Harder to compute. Nicer physical properties. Can be computed efficiently via semidefinite programming, but no analytic formula is known.

Measuring Coherence Some nice mathematical properties include: Reminder: The robustness of coherence is Some nice mathematical properties include: A simple formula when restricted to pure states: is maximized exactly when same formula as for the ℓ1-norm same condition as for the ℓ1-norm

Measuring Coherence How far (geometrically) is ρ from a useless state? Method 3: Trace Distance How far (geometrically) is ρ from a useless state? “C” stands for “coherence”, and the “tr” refers to the trace norm, which is typically the “right” norm to use on quantum states trace norm is the sum of singular values Harder to compute. Nice physical properties. Again, can be computed efficiently via semidefinite programming, but no analytic formula is known.

Measuring Coherence Some nice mathematical properties include: Reminder: The trace distance of coherence is Some nice mathematical properties include: A simple formula when restricted to pure states? Wait a minute… If then What states give the largest value? formula for the ℓ1-norm and robustness of coherence

Overview What is Quantum Coherence? Measuring Coherence Coherence of Pure States “Almost” formula for trace distance of coherence Our method is fast Also answers which states maximize trace distance of coherence

Coherence of Pure States One of our results didn’t make the cut…

Coherence of Pure States Our main result is an “almost” formula for We give m different formulas, and determining the correct one is done by checking log2(m) inequalities. Each formula is nasty-looking, but simple to compute. Let’s get started…

Coherence of Pure States Step 0: assume WLOG that each vi is real and v1 ≥ v2 ≥ … ≥ vm ≥ 0. Step 1: for ℓ = 1, 2, …, m, compute: Step 2: find the largest index k such that vk ≥ qk. Step 3: This can be done via binary search in log2(m) steps. Yay! Done!

Coherence of Pure States Some notes are in order: Our method also tells us how to construct the (unique) closest diagonal state D such that Even though our method is not an explicit formula, it’s fast: Known SDP methods: 1.5 minutes for Our method: 0.5 seconds for formula for D is also ugly

Coherence of Pure States Our method is also useful analytically, and has several straightforward corollaries: is maximized exactly when Easy to compute exact value of , whereas SDP methods just give a numerical approximation. In the m = 2 case, our method simplifies to an explicit formula: same condition as for the ℓ1-norm and robustness of coherence was already known, but follows easily in just two lines from our work

Quantum Coherence Thanks for your attention! J. Chen, N. J., C.-K. Li, S. Plosker. Quantifying the coherence of pure quantum states. E-print: arXiv:1601.06269 [quant-ph]