1. Twirling states to simplify separability
Nathaniel Johnston – joint work with Jianxin Chen, Shane Grogan, Chi-Kwong Li,
and Sarah Plosker
Mount Allison University
Sackville, New Brunswick, Canada

2. Overview Separability versus Entanglement What is Twirling?
Coherence versus Entanglement

3. Overview Separability versus Entanglement What is Twirling?
How to prove a state is entangled?
How to prove a state is separable?
What is Twirling?
Coherence versus Entanglement

4. Separability versus Entanglement
A state is separable if we can write it in the form Otherwise, it is called entangled.
for some
positive semidefinite
A similar definition is used for separability and entanglement in

5. Separability versus Entanglement
Proving that a state is separable or entangled is hard in general, but there are some standard tricks that work in low dimensions.
Prove entangled?
The partial transpose (Størmer, Woronowicz, Peres, Horodecki3, ’63–’96):
If then ρ is entangled.
The realignment criterion (Chen and Rudolph, 2003).
Positive but not CP maps (e.g., the Choi map, Breuer–Hall map).
Symmetric extensions (Doherty–Parrilo–Spedalieri, 2004).

6. Separability versus Entanglement
Proving that a state is separable or entangled is hard in general, but there are some standard tricks that work in low dimensions.
Prove separable?
“Inner” symmetric extensions (Navascues–Owari–Plenio, 2009).
Some randomized numerical methods.
Look really hard at it and get good at guessing.

7. Overview Separability versus Entanglement What is Twirling?
Turn a state into a “more symmetric” one
Can prove separability of some states
Coherence versus Entanglement

8. What is Twirling? A twirling channel is a quantum channel of the form
where is a compact subgroup of the set of unitary matrices (and we integrate with respect to Haar measure).
For example, if then
If then

9. What is Twirling?
We can write down explicit formulas for twirling channels.
Let be the commutant of
Computing is not difficult.
The twirling channel just projects ρ onto
If then so Φ projects ρ onto all of Thus
If then so

10. Werner Twirling
As a less trivial example, consider the Werner twirling channel
In this example, is the set of local unitaries. The commutant is
“swap operator” defined by
For all ρ there exists a number α so that
“Werner state”

11. Werner Twirling
To determine when Werner states are separable/entangled…
Entangled: Partial transpose.
exactly when α < -1/n.
Thus entangled when α < -1/n.
Separable: Harder. No “general” tests work (to my knowledge).
Notice that is separable whenever ρ is.
Well, and
Thus separable when α ≥ -1/n.
(-1 ≤ α ≤ 1)

12. Werner Twirling
We can even consider Werner twirling in the multipartite setting, and the same ideas work. Briefly, if
then Φ is the projection onto the space spanned by the p! swap operators that act as
where π is a permutation of {1,2,…,p}.
(if p = 2 then the two swap operators are I and W from the previous slide)

13. Isotropic Twirling
A closely-related channel is the isotropic twirling channel
It is straightforward to check that
For all ρ there exists α so
Separable when α ≤ 1/(n+1)
Entangled when α > 1/(n+1)
“isotropic state”
(-1/(n2-1) ≤ α ≤ 1)

14. Real Orthogonal Twirling
We now know how to determine separability of:
Linear combinations of I and W, and
Linear combinations of I and
What about separability of linear combinations of I, W, and ?
Idea: Twirl over a group of unitaries contained in and
real orthogonal matrices

15. Real Orthogonal Twirling
Yes, the commutant in this case is Thus we can determine separability and entanglement of linear combinations of these three operators:
Reminder:
Fact
Let and
Then the state (0 ≤ x, y, 1-x-y ≤ 1) is separable if and only if x ≤ 1/n and y ≤ 1/2.

16. Real Orthogonal Twirling
If we consider real orthogonal twirling in the multipartite case, something strange happens:
This map is the projection onto the subspace spanned by:
The p! swap operators and
All possible partial transpositions of these swap operators.
for example,
There are exactly distinct such operators.

17. Real Orthogonal Twirling
Question: What is the dimension of the span of those operators? Equivalently, what is the dimension of the range of ? Clearly, for all n. It’s also not hard to show that if n ≥ p then those swap operators and their partial transposes are linearly independent, so However, if n < p then the answer is not known.
(as a function of n and p)
G. Lehrer and R. Zhang, The second fundamental theorem of invariant theory
for the orthogonal group, Ann. of Math. 176: (2012).

18. Overview Separability versus Entanglement What is Twirling?
Coherence versus Entanglement
How to measure coherence
Measuring entanglement
The connection between the two

19. What is Quantum Coherence?
A mixed quantum state is “useless” if it is diagonal:
Each of these pure basis 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 coherence
far from diagonal – lots of coherence

20. Measuring Coherence
How do we measure coherence? Lots of possible ways!
For this talk: how far (geometrically) is ρ from a diagonal 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
Can be computed via semidefinite programming.
No formula known for pure states.

21. Coherence of Pure States
One of our main results is an “almost” formula for
We give n different formulas, and determining the correct one is done by checking log2(n) inequalities.
Each formula is nasty-looking, but simple to compute.
Let’s have a (quick) look…

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

23. 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

24. 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:
This makes sense!
was already known, but follows easily in just two lines from our work

25. Measuring Entanglement
How do we measure entanglement? Lots of possible ways!
For this talk: how far (geometrically) is ρ from a separable state?
“E” stands for “entanglement”, and the “tr” refers to the trace norm
trace norm is the sum of singular values
Can be computed via semidefinite programming? Nope.
No formula (or SDP, or anything) known for pure states.

26. Entanglement of Pure States
Theorem
Let be a pure state with Schmidt coefficients
, and define Then
Now we can quickly compute on pure states!
The proof uses twirling!
(just leech off of our method of computing the trace distance of coherence)

27. Entanglement of Pure States
Idea of proof: If is the closest diagonal state to then a simple calculation shows that is at least as close to . This gives The opposite inequality is proved similarly, but needs fancier techniques. If σ is the closest separable state to , we want to find some diagonal state that is at least as close to .

28. Entanglement of Pure States
Idea of proof (cont.): Well, one way to make σ “more diagonal” without increasing its trace distance to is to perform the following twirl:
Φ preserves separability
diagonal unitary matrices

29. Entanglement of Pure States
Idea of proof (cont.): How does Φ affect general mixed states?

30. Entanglement of Pure States
Idea of proof (cont.): How does Φ affect general mixed states?

31. Quantum Coherence Thanks for your attention!
And then we do more magic to turn this “almost” diagonal state into a truly diagonal one. See the paper for details!
Thanks for your attention!
J. Chen, S. Grogan, N. J., C.-K. Li, S. Plosker. Quantifying the coherence of pure quantum
states. E-print: arXiv: [quant-ph]