Entanglement becomes a mathematical resource theory when we specify which operations are free, which states are valuable and which transformations are possible. For separated parties, the natural free operations are local quantum operations assisted by classical communication—LOCC.
If Alice and Bob share a pure bipartite state, can they convert it into another entangled state without transmitting quantum systems? Sometimes yes, sometimes only probabilistically, and sometimes not at all. The answer is encoded in the Schmidt coefficients and the theory of majorisation.
This guide develops deterministic conversion, probabilistic conversion, catalysis and asymptotic concentration. The aim is to understand entanglement not only as a number such as entropy, but as a partially ordered family of resources.
State → Schmidt spectrum → compare majorisation → choose LOCC protocol → track success probability and entanglement monotones.
1. What LOCC permits
Under LOCC, Alice may perform a local quantum channel or measurement on her subsystem. She may send the classical outcome to Bob. Bob may then choose a local operation based on that message, send a reply, and so on for a finite or suitably defined sequence of rounds.
They do not exchange quantum systems. Any increase in nonlocal resource must therefore come from entanglement already present in the shared state.
LOCC is a strict operational class. The set of separable quantum operations is larger: some separable maps cannot be implemented by LOCC. Distinguish the mathematical form of Kraus operators from the communication protocol actually available.
2. LOCC cannot create entanglement from a product state
Start with ρ_A⊗ρ_B. Alice’s local operation changes only her factor, possibly conditioned on a local outcome. Sending that outcome allows Bob to choose a local response.
Conditioned branches remain product states; forgetting the classical branch label produces a convex mixture of product states, which is separable.
Thus LOCC can redistribute or consume existing entanglement but cannot generate it from scratch across the Alice–Bob partition.
3. Schmidt decomposition is the pure-state coordinate system
Any bipartite pure state can be written
|ψ⟩=Σ_i √λ_i |i_A⟩|i_B⟩
with nonnegative coefficients λ_i summing to one.
The vector λ(ψ), sorted in decreasing order, contains the eigenvalues of either reduced density matrix and completely determines bipartite pure-state entanglement up to local unitaries.
A product state has Schmidt vector (1,0,…). A maximally entangled d-dimensional state has uniform vector (1/d,…,1/d).
4. Majorisation
For probability vectors x and y sorted decreasingly, x is majorised by y, written x≺y, when
Σ_{i=1}^k x_i ≤ Σ_{i=1}^k y_i
for every k before the final component, with equality for the total sum.
The uniform distribution is majorised by every more peaked distribution. Majorisation therefore orders probability vectors by spread.
5. Nielsen’s theorem
For bipartite pure states, a deterministic LOCC conversion
|ψ⟩ → |φ⟩
is possible if and only if
λ(ψ)≺λ(φ).
This direction is worth checking carefully. A maximally entangled state has a uniform Schmidt vector and can be converted by LOCC into a product state. Uniform is majorised by peaked, so the theorem has the correct orientation. [1]
6. Worked deterministic conversion
Let
λ(ψ)=(0.5,0.5)
and
λ(φ)=(0.8,0.2).
For k=1, 0.5≤0.8. The total sums are equal. Therefore λ(ψ)≺λ(φ), so the maximally entangled two-qubit state can be deterministically converted into the less-entangled target.
The reverse conversion is impossible deterministically because 0.8≤0.5 fails.
7. Entropy follows the correct direction
For a bipartite pure state, the entropy of entanglement is
E(ψ)=−Σ_i λ_i log₂λ_i.
Entropy is Schur-concave: if x≺y, then H(x)≥H(y). Therefore deterministic LOCC cannot increase pure-state entanglement entropy.
But entropy alone does not fully determine single-copy convertibility in dimensions above two. Two Schmidt vectors can have ordered entropies while still being incomparable by majorisation.
8. Incomparable pure states
Consider
x=(0.5,0.25,0.25)
and
y=(0.45,0.40,0.15).
At k=1, x₁=0.5>0.45, so x is not majorised by y. At k=2, y₁+y₂=0.85>0.75, so y is not majorised by x.
Neither deterministic conversion is possible by LOCC. Entanglement is therefore only partially ordered at the single-copy level.
9. Nielsen’s protocol intuition
If λ(ψ)≺λ(φ), majorisation theory says λ(ψ) can be obtained from λ(φ) by a doubly stochastic mixing operation, equivalently by a sequence of elementary T-transforms.
In the quantum protocol, Alice performs a carefully chosen local measurement whose outcomes produce states that differ from the target only by local basis permutations. She communicates the outcome, and local unitaries finish the conversion.
The theorem is therefore constructive: majorisation is not merely a necessary inequality test.
10. Probabilistic conversion
Even when deterministic conversion fails, a target can sometimes be reached with nonzero probability.
Define tail sums
E_k(ψ)=Σ_{i=k}^d λ_i(ψ).
Vidal showed that the maximum success probability for converting one bipartite pure state into another by LOCC is
P_max(ψ→φ)=min_k E_k(ψ)/E_k(φ)
with ratios taken where the target tail is nonzero. [2]
11. Worked probabilistic concentration
Let ψ have Schmidt vector (0.8,0.2) and let φ be a Bell state with vector (0.5,0.5).
The tail sums are
- k=1: E₁(ψ)=E₁(φ)=1;
- k=2: E₂(ψ)=0.2 and E₂(φ)=0.5.
Therefore
P_max=min(1,0.2/0.5)=0.4.
The partly entangled pair cannot become a Bell pair deterministically, but it can be concentrated into one with optimal success probability 40% for a single copy.
12. Procrustean filtering
For a two-qubit state
√λ|00⟩+√(1−λ)|11⟩
with λ≥1/2, Alice can locally attenuate the larger Schmidt component so that successful branches have equal amplitudes.
The success probability is 2(1−λ), matching Vidal’s formula.
Filtering illustrates the resource cost visibly: the successful state becomes more entangled, but only because unsuccessful branches are discarded. Average entanglement cannot increase under LOCC.
13. Entanglement monotones
An entanglement monotone is a quantity that does not increase on average under LOCC.
For pure bipartite states, the Vidal tail functions provide a complete family for optimal single-copy probabilistic conversion.
One scalar cannot generally capture every operational comparison. Different resource-conversion tasks can be constrained by different monotones. [3]
14. Entanglement catalysis
Sometimes ψ cannot be converted deterministically into φ, yet
ψ⊗c → φ⊗c
is possible for a suitable entangled catalyst state c that is returned unchanged.
The catalyst changes the majorisation relations of the combined Schmidt vectors without being consumed. Jonathan and Plenio demonstrated this entanglement-assisted transformation phenomenon. [4]
15. Catalysis does not violate monotonicity
The catalyst is part of the initial resource inventory and is present in the final inventory. The transformation is assessed on the combined system.
No entanglement resource is created from nowhere. Instead, tensoring with c changes the partial order so that a conversion previously blocked by majorisation becomes possible.
This is a striking reminder that resource comparison can depend on auxiliary resources even when those resources are returned.
16. Many-copy entanglement concentration
Suppose Alice and Bob share n identical copies of a partly entangled pure state with Schmidt probabilities λ.
Typical-sequence arguments show that for large n the Schmidt weight concentrates on roughly 2^{nH(λ)} typical terms.
By LOCC, the parties can asymptotically convert these n copies into approximately
nH(λ)
Bell pairs with vanishing fractional loss. [5]
17. Worked asymptotic yield
Take Schmidt vector (0.8,0.2). Its entropy is
H₂(0.2)=−0.8log₂0.8−0.2log₂0.2≈0.721928.
From 10,000 ideal identical copies, the asymptotic concentration rate predicts roughly 7,219 Bell pairs, up to finite-size corrections and protocol inefficiency.
Single-copy optimal success probability and many-copy asymptotic yield are different operational quantities.
18. Entanglement dilution
The reverse asymptotic task begins with Bell pairs and uses LOCC to create many copies of a desired bipartite pure state.
The asymptotic entanglement cost is also H(λ) ebits per copy for pure bipartite states.
Thus pure-state entanglement is asymptotically reversible: concentration and dilution have the same leading rate.
19. Mixed-state entanglement is more complicated
For mixed states, entanglement concentration generalises to distillation: extract nearly maximally entangled pairs from many noisy shared states using LOCC.
The distillable entanglement E_D is the asymptotic Bell-pair yield per copy under the allowed operations.
The entanglement cost E_C is the asymptotic Bell-pair rate required to create the state. Unlike the pure-state case, these can differ.
20. Bound entanglement
Some mixed entangled states have positive entanglement but zero distillable entanglement under standard LOCC: they are bound entangled.
A central class arises from positive-partial-transpose states in dimensions where PPT entanglement exists.
This shows again that “amount of entanglement” is not captured operationally by one universal number for all tasks.
21. Partial transpose as a diagnostic
For a bipartite state ρ, transpose one subsystem in a fixed product basis to obtain ρT_B.
Every separable state has positive partial transpose. Therefore a negative eigenvalue of ρT_B certifies entanglement.
For 2×2 and 2×3 systems, PPT is also sufficient for separability. In larger dimensions, PPT entangled states exist.
22. Negativity
The negativity can be defined as
N(ρ)=(||ρ^{T_B}||₁−1)/2.
It measures the total absolute weight of negative eigenvalues of the partial transpose and is an entanglement monotone under suitable operation classes.
Zero negativity does not imply separability in all dimensions because PPT bound entanglement can have zero negativity.
23. Distillation protocols need noisy local operations too
Ideal resource theory treats LOCC operations as free. Real quantum networks perform those local gates and measurements imperfectly.
A practical repeater therefore balances raw-pair fidelity, memory decoherence, purification yield, local-gate error and classical signalling delay.
Resource-theory possibility is the first layer; fault-tolerant or hardware-level feasibility is a second layer.
24. Common misconception: more entropy always means every conversion is possible
At the single-copy level in dimensions above two, entropy does not fully order pure states. Majorisation can make two states incomparable even when one entropy is larger.
25. Common misconception: successful filtering creates entanglement
Conditioned successful branches can be more entangled, but unsuccessful branches occur too. The average resource cannot increase under LOCC.
26. Common misconception: distillation and error correction are the same task
Entanglement distillation consumes multiple shared noisy states to produce fewer cleaner shared states. Quantum error correction protects encoded information against ongoing physical errors. They can be combined in a network but are operationally different.
27. Worked synthesis problem
Let
λ(ψ)=(0.6,0.25,0.15)
and
λ(φ)=(0.7,0.2,0.1).
Step 1: Deterministic test. k=1 gives 0.6≤0.7. k=2 gives 0.85≤0.9. Therefore λ(ψ)≺λ(φ), so ψ→φ is possible deterministically by LOCC.
Step 2: Reverse test. 0.7≤0.6 fails, so φ→ψ is not deterministic.
Step 3: Probabilistic reverse probability. Tail ratios for φ→ψ are 1, 0.3/0.4=0.75 and 0.1/0.15=2/3.
Thus P_max=2/3.
Interpretation. The more peaked target can be reached freely from the more entangled source, while recovering the more uniform Schmidt spectrum succeeds only probabilistically.
28. Practice set
- What operations are allowed under LOCC?
- Why can LOCC not create entanglement from a product state?
- What information does a Schmidt vector contain?
- Define x≺y.
- State Nielsen’s deterministic-conversion criterion.
- Why does the maximally entangled state majorise in the correct direction for conversion to a product state?
- Why is entropy insufficient for every single-copy comparison?
- State Vidal’s optimal probabilistic conversion formula.
- What is entanglement catalysis?
- What is the asymptotic pure-state concentration rate?
- What is bound entanglement?
- What does a negative partial transpose certify?
Answers
- Local quantum operations plus classical communication between separated parties.
- Each branch remains locally generated from separable inputs, and mixtures of product branches are separable.
- The reduced-state eigenvalues and all bipartite pure-state entanglement up to local unitaries.
- All ordered partial sums of x are no larger than those of y, with equal totals.
|ψ⟩→|φ⟩deterministically iffλ(ψ)≺λ(φ).- The uniform vector is majorised by a peaked vector, so highly entangled states can deterministically lose entanglement.
- Majorisation is a partial order containing more information than one scalar entropy.
min_k E_k(ψ)/E_k(φ).- An auxiliary entangled state enables a conversion while being returned unchanged.
- The entropy of entanglement H(λ) ebits per copy.
- Entangled mixed states from which no Bell pairs can be distilled under the specified LOCC setting.
- Entanglement.
Sources and further study
[1] M. A. Nielsen, Conditions for a Class of Entanglement Transformations, Physical Review Letters 83, 436 (1999). The majorisation criterion for deterministic bipartite pure-state LOCC conversion.
[2] Guifré Vidal, Entanglement of Pure States for a Single Copy, Physical Review Letters 83, 1046 (1999). Optimal probabilistic pure-state conversion.
[3] Guifré Vidal, Entanglement monotones. A resource-theoretic treatment of monotonic quantities under local transformations.
[4] Daniel Jonathan and Martin B. Plenio, Entanglement-Assisted Local Manipulation of Pure Quantum States, Physical Review Letters 83, 3566 (1999). Entanglement catalysis.
[5] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu and Benjamin Schumacher, Concentrating partial entanglement by local operations, Physical Review A 53, 2046 (1996). Asymptotic pure-state entanglement concentration.
Continue through Quantum Mathematics
Guide 29: Bell Inequalities, CHSH, Tsirelson Bounds and No-Signalling distinguishes entanglement from Bell nonlocality. Guide 30: Quantum Teleportation, Superdense Coding and Entanglement Swapping uses shared entanglement operationally. Guide 32: Quantum Shannon Theory, Holevo Bounds, Channel Capacities and Coding Theorems asks how rapidly information can be transmitted in the asymptotic limit.