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

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

3. 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

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

5. 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

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

7. 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

8. 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

9. 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.

10. 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

11. 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.

12. 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

13. 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.

14. 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

15. 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

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

17. 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…

18. 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!

19. 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

20. 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

21. 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: [quant-ph]