Quantum Coherence and Quantum Entanglement Nathaniel Johnston – with Jianxin Chen, Shane Grogan, Chi-Kwong Li, and Sarah Plosker Mount Allison University Sackville, New Brunswick, Canada

Overview What is Quantum Coherence? How do we Measure Coherence? What is Quantum Entanglement? Measuring Entanglement via Coherence

Overview What is Quantum Coherence? How do we Measure Coherence? Physics: How “useful” a quantum state is (superpositions) Math: Linear combinations of vectors How do we Measure Coherence? What is Quantum Entanglement? Measuring Entanglement via Coherence

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? How do we Measure Coherence? Many different methods have been proposed Different methods useful in different contexts What is Quantum Entanglement? Measuring Entanglement via Coherence

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 How far (geometrically) is ρ from a useless state? Method 2: 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. Geometrically nice, physically maybe not. Can be computed efficiently via semidefinite programming, but no analytic formula is known.

Measuring Coherence Reminder: The trace distance of coherence is Much less is known about this measure. For example… Formula on pure states? Well, If then What states give the largest value? formula for the ℓ1-norm of coherence

Measuring Coherence We derive 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…

Measuring Coherence 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!

Measuring Coherence 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

Measuring Coherence 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 of coherence was already known, but follows easily in just two lines from our work

Measuring Coherence Methods 3+: Other Measures There are also numerous other proposed methods for measuring coherence, such as… …and many others.

Overview What is Quantum Coherence? How do we Measure Coherence? What is Quantum Entanglement? Physics: How “useful” a quantum state is (again) Math: Some matrices can have rank larger than 1 Measuring Entanglement via Coherence

What is Quantum Entanglement? A pure state is separable if it has the form Otherwise, it is called entangled. for some A mixed state is separable if it has the form Otherwise, it is called entangled. for some

Measuring Entanglement Again, we think of separable states as “useless”, and the farther a state is from the set of separable states, the more “useful” it is. How do we quantify this? Method 1: Trace Distance How far (geometrically) is ρ from a useless state? “E” stands for “entanglement” trace norm is the sum of singular values Very hard to compute No formula, semidefinite program, or any other reasonable computational method known

Overview What is Quantum Coherence? How do we Measure Coherence? What is Quantum Entanglement? Measuring Entanglement via Coherence Strong link between measures of entanglement and coherence Can use coherence results to solve entanglement problems

Entanglement via Coherence Measures of entanglement are hard to compute. However, coherence can help us for pure states. Recall the Schmidt decomposition − we can write every in the form: Where are orthonormal sets and are non-negative scalars called the Schmidt coefficients of

Entanglement via Coherence Theorem Let be a pure state with Schmidt coefficients , and define . Then Now we can quickly compute on pure states! Neat! The entanglement of a pure state equals the coherence of its Schmidt coefficients. (just leech off of our method of computing the trace distance of coherence)

Entanglement via Coherence The same result works for pretty much any “good” measure of entanglement: when restricted to pure states, entanglement measures are coherence measures. For example… The cross norm of entanglement (or projective tensor norm): reduces to the ℓ1-norm of coherence on pure states. The robustness of entanglement: reduces to the robustness of coherence on pure states.

Quantum Coherence/Entanglement Thanks for your attention! J. Chen, S. Grogan, N. J., C.-K. Li, S. Plosker. Quantifying the coherence of pure quantum states. Physical Review A 94:042313, 2016. E-print: arXiv:1601.06269 [quant-ph]