Non-mathematical preview of the notions of quantum entanglement and its measures.

1. Basic Concepts of
Entanglement Measures
Ryohei Suzuki

2. Quantum Entanglement
for pure state: state which is not a product state
for mixed state: state which is not a separable state
Definition
Maximally entangled state
(Bell/EPR state)
Measure 1st qubit (Alice)
0 1
Pure state →
Prob. mix of with
→
Review: density matrix
50% 50%

3. Entanglement (non-local) vs.
Classical (local) correlation
Maximally entangled state
Measure 1st qubit (Alice)
0 1
Classically correlated state
Measure 1st qubit (Alice)
10
CHSH ineq.
is broken for Bell state under certain measurement angles
How to distinguish?
50% 50% 50% 50%
see Shimizu-san’s book for details

4. Entanglement Entropy
Definition
for
von Neumann entropy
Shannon entropy
product state
entangled state
EE is useless for mixed state

5. Usages of Entanglement
Alice
Bob
State is transferred from A to B
using an EPR state and 2-bit
classical communication
Quantum Teleportation
2-bits of classical information
can be transferred by sending
a single entangled qubit
Dense Coding
Bennet & Wiesner (1992) PRL 69, 2881.Bennet et al., (1993) PRL 70, 1895.
Alice
Bob

6. Entanglement as Resource
1. We can utilize prepared entanglement for something special
2. Any operations by Alice or Bob only decrease entanglement
Observations
→ We are consuming entanglement as a resource
No entanglement left after teleportation or decoding

7. Formulation of Allowed
Operations (LOCC)
Eisert (2001) Ph.D Thesis.
Resource properties depend on the allowed operations:
Local Operation + Classical Communication
LOCC
Generic local operation (local unitary, ancilla, measurement)
Kraus operators (CPTP)
One-way classical communication
Measurement by Alice Feedback by Bob
see Sagawa-san’s book for Kraus formalism

8. Distillation and Dilution by
LOCC Protocol
Transforming entangled
states into maximally
entangled states (Bell singlet)
Distillation
Transforming maximally
entangled states into a
specific entangled states
Dilution
Bennet et al., (1995) PRL 76, 722.
1. random bilateral SU(2) rotation
Uniformly mixed by random rot.
2. bilateral CNOT → measurement
Use symmetry & intra-party entanglement Prepare locally then transport
Bennet et al., (1996) PRA 53, 2046.
1. Initially share an |EPR>
2. Alice prepare desired pair by
local unitary transformation
3. transport the 2nd qubit of

9. Entanglement Monotone
How can we measure the entanglement of mixed states?
Question
We want a functional which satisfies
for separable state and:
1. Monotonicity under LOCC
2. Convexity
3. (Computability)
Axioms for “Entanglement Monotone”
state after a
measurement
(ignorance degrades entanglement)

10. Operational Measure:
Distillable Entanglement
Optimal rate of maximally entangled states
that can be distilled from a state σ by LOCC
Definition
We don’t know an optimal distillation protocol for general states!
Upper bound by relative entropy
all PPT states
all separable states
Rains, (1999) PRA 60, 179.Peres, (1996) PRL 77, 1413.

11. Entanglement Cost &
Entanglement of Formation
How many maximally entangled states
are needed to create a state σ by LOCC
Entanglement Cost Ec
Entanglement of Formation
minimization over decompositions
Bennet et al., (1996) PRA 54, 3824. Hayden et al., (2001) J. Phys. A 34(35).

12. Computable Measures
Concurrence
Negativity
Only computable measure for general mixed state (as far as I know)
Wootters, (1997) PRL 80(10) 2245.
Vidal & Werner, (2002) PRA 65, 032314.
Życzkowski et al., (1998) PRA 58(2), 883.
: eigenvalues of
EF(σ) in explicit formula
(only for qubit-pair system)
: trace norm: partial transpose w.r.t. B
satisfies monotone axioms,
but not reduce to EE for pure states

13. Quantum Correlations and
Maxwell’s Demon
1. Observe the system and obtain some information
2. Use the correlation (mutual information) to extract work
What Demons Do
Image taken from Sagawa & Ueda, arXiv:1111.5769
Szilard Engine
Questions
1. How is quantum correlation
related to extractable work?
2. What is the relationship of
quantum correlation and
mutual information?

14. Work Deficit
Loss of extractable work by restricting operations to LOCC
Definition
Demon with global freedom
2bits work
extraction
demon knows the
state to be EPR
LOCC demon (example)
(sequence of LOCC)
only 1bit extractable
disturbed
qubit
Oppenheim, et al., 2002, PRL, 89, 180402

15. Quantum Discord
Setup
System(S) Apparatus(A)
Measurement
by Demon
Mutual information
Measurement on A may
disturb the state of S
/
Quantum discord
Zurek, 2003, PRA, 23(1)
(no correlation)
(strong correlation)
No entanglement,
but quantum correlation exists
(haven’t calculated the exact value, sorry)
best measurement

16. Thermodynamic
Entanglement Detection
Discord & Deficit: require knowledge of global state ρAB
→ Can we detect entanglement only by local observables?
Problem Maruyama et al., 2005, PRA, 71, 012108
1. Alice measures by angle Aθ
2. Bob extract work by angle Bθ’
using the information by Alice
3. Repeat it with sweeping θ, θ’
cf. Bell tests
mean extractable work by sweep
extractable work by a single trial
Proposition
if is separable,
implies entanglement

17. Interesting Topics in
Entanglement Measures
• Analysis of computational complexity for evaluating
entanglement / quantum correlation measures
• Computing discord, EF, etc. are NP-complete/hard (Huang 2014)
• Experimentally measurable entanglement evidence
without tomography of ρ (entanglement witness)
• Bounds for negativity, etc. expressed by observable values
have been proposed (Eisert 2007)
• Unification of measurable values and abstract measures?

18. References (reviews)
Eisert, 2001, “Entanglement in Quantum Information Theory,” Ph.D thesis
(Univ. Potsdam).
Vedral, 2002, “The Role of Relative Entropy in Quantum Information Theory,”
Rev. Mod. Phys., 74, 197.
Horodecki, et al., 2009, “Quantum Entanglement,” Rev. Mod. Phys., 81, 865.
Horodecki & Oppenheim, 2012, “(Quantumness in the context of) Resource
Theories,” Int. J. Mod. Phys. B, 27, 134.
Modi et al., 2012, “The Classical-Quantum Boundary for Correlation: Discord
and Related Measures,” Rev. Mod. Phys., 84, 4.
Goold, et al., 2016, “The Role of Quantum Information in Thermodynamics
- A topical review,” J. Phys. A, 49, 14.