Quantum discord measures nonclassical correlations that can remain even when a mixed state is separable. It asks whether information about one subsystem can be revealed by a local measurement without disturbing the joint state more than a genuinely classical correlation would require.
Guide 8 owns entropy and mutual information. Guide 31 owns LOCC and entanglement transformations. Guide 56 owns contextuality. This guide owns the asymmetry between two classically equivalent definitions of mutual information once local quantum measurements enter: classical–quantum states, measurement-induced conditional entropy, one-sided discord, zero-discord criteria and operational caveats.
Total correlations − maximum locally accessible classical correlations = one-sided quantum discord.
1. Classical mutual information has two equivalent forms
For classical random variables A and B,
I(A:B)=H(A)+H(B)−H(A,B).
Because H(A|B)=H(A,B)−H(B), the same quantity is
I(A:B)=H(A)−H(A|B).
These definitions are identical classically.
2. Quantum mutual information
For density matrix ρAB, define
I(A:B)=S(ρ_A)+S(ρ_B)−S(ρ_{AB}).
This quantifies total correlations between A and B.
But the classical-looking conditional-entropy version requires us to decide what it means to “know B”. In quantum theory, that means choosing a measurement on B.
3. Measurement-induced conditional entropy
Let {Πj} be a projective measurement on B. Outcome j occurs with
p_j=Tr[(I⊗Π_j)ρ_{AB}].
The conditional A state is
ρ_{A|j}=Tr_B[(I⊗Π_j)ρ_{AB}(I⊗Π_j)]/p_j.
Define measurement-induced conditional entropy
S(A|{Π_j})=Σ_j p_j S(ρ_{A|j}).
4. Classical correlation extracted by measuring B
For one chosen measurement, the information gained about A is
J_{A|B}({Π_j})=S(ρ_A)−Σ_jp_jS(ρ_{A|j}).
Optimise over allowed measurements on B:
J_{A|B}=max_{Π_B} J_{A|B}({Π_j}).
This is one standard measure of classically accessible correlation under a one-sided local measurement.
5. One-sided quantum discord
Define
D_{A|B}=I(A:B)−J_{A|B}.
It is generally nonnegative.
The notation matters: DA|B means B is measured. In general D_{A|B}≠D_{B|A}.
6. Product states have zero discord
If ρ_{AB}=ρ_A⊗ρ_B, then I(A:B)=0.
Therefore both one-sided discords vanish.
Discord measures correlation, so no correlation means no discord.
7. Classical–quantum states
A state with zero discord when A is measured can be written
ρ_{AB}=Σ_i p_i |i⟩⟨i|_A ⊗ ρ_i^B
for some orthonormal basis {|i⟩} on A.
Measuring A in that basis reveals the classical label i without disturbing the state.
Such states are called classical–quantum with respect to A.
8. Quantum–classical states
Zero discord when B is measured requires the analogous form
ρ_{AB}=Σ_j p_j ρ_j^A⊗|j⟩⟨j|_B.
A state can have zero discord in one direction and nonzero discord in the other.
This asymmetry is one of the first differences from entanglement, which is symmetric between the two subsystems.
9. Worked classical–quantum example
Let
ρ=(1/2)|0⟩⟨0|⊗|0⟩⟨0|+(1/2)|1⟩⟨1|⊗|+⟩⟨+|.
This state is classical on A because the label 0 or 1 is stored in orthogonal A states.
Measuring A in the computational basis leaves ρ unchanged. Therefore the discord associated with measuring A is zero.
But the two B states |0⟩ and |+⟩ are nonorthogonal. B does not contain a perfectly classical distinguishable label, so measuring B generally disturbs the correlations and the opposite discord can be nonzero.
10. Separable does not imply zero discord
A separable state has the form
ρ_{AB}=Σ_k p_k ρ_k^A⊗ρ_k^B.
If the local states in this mixture do not commute, there may be no local basis that reads the classical label without disturbance.
Therefore separable states can possess nonzero discord.
11. Every entangled state has nonzero discord
A zero-discord state is separable because it is explicitly a convex mixture of product states with an orthogonal local label.
Hence entanglement implies nonzero discord in both directions, while the converse is false.
12. Bell state
For |Φ+⟩=(|00⟩+|11⟩)/√2, the joint entropy is zero and each marginal entropy is one bit.
Therefore I(A:B)=2 bits.
Measuring B in the Schmidt basis reveals one classical bit and leaves conditional A states pure, so J_{A|B}=1.
Thus D_{A|B}=1 bit, equal to the entanglement entropy for a pure bipartite state.
13. Pure bipartite states
For any pure |ψ⟩AB, the discord equals the entanglement entropy when the measurement is optimised:
D_{A|B}=S(ρ_A)=S(ρ_B).
The distinction between discord and entanglement therefore becomes especially important for mixed states.
14. Measurement disturbance criterion
A state has zero discord with respect to a projective measurement on B if there exists a complete local projective measurement that leaves the joint state invariant:
ρ_{AB}=Σ_j(I⊗Π_j)ρ_{AB}(I⊗Π_j).
This gives a direct structural test: zero discord means the relevant classical information can be read without altering the joint state.
15. Geometric discord
Another family of measures quantifies the distance from ρ to the set of zero-discord states.
A Hilbert–Schmidt version is mathematically convenient but is not contractive under general quantum channels and can behave undesirably when an uncorrelated ancilla is added.
Contractive distances such as trace distance or Bures distance avoid some of these issues.
16. Relative-entropy discord
Define the relative entropy
S(ρ||σ)=Tr[ρ(logρ−logσ)].
Relative-entropy-based discord minimises this distance over classical–quantum states.
This connects discord to information loss under local dephasing measurements and to broader resource theories of quantum correlations.
17. Local broadcasting theorem connection
Classical correlations can be broadcast locally because orthogonal classical labels can be copied.
General quantum correlations cannot be locally broadcast.
States that can be locally broadcast on both sides are classical–classical states. One-sided variants connect closely to zero-discord structure.
This is another operational way to see why discord measures something beyond entanglement alone.
18. Discord and quantum Darwinism
Guide 81 studies environmental records. Total system–fragment mutual information can be decomposed conceptually into classically accessible pointer information plus genuinely quantum correlations that are not locally readable from one fragment.
As a Darwinistic plateau develops, small fragments can become rich in classical information while much of the discord-like remainder remains delocalised across large portions of the environment.
19. Discord consumption in state merging
Quantum information theory gives discord operational meanings connected with the extra entanglement or communication cost caused when one subsystem is measured or decohered before a protocol such as state merging.
These interpretations show that discord is not merely an arbitrary entropy difference, even though different discord-like measures can serve different tasks.
20. Discord does not automatically imply quantum computational speedup
Some mixed-state computational models can exhibit little or no entanglement while retaining discord.
This motivated the idea that discord might help identify nonclassical computational resources.
However, nonzero discord alone is not sufficient to prove an algorithmic speedup. Complexity, input model, circuit family and classical simulation methods must be analysed separately.
21. Discord is generic in mixed states
The set of exactly zero-discord states occupies a highly restricted subset of density-operator space.
Small perturbations can therefore create nonzero discord.
This means detecting tiny positive discord is not automatically evidence of a practically important quantum resource. Magnitude, robustness and operational task matter.
22. Optimisation difficulty
Computing discord requires an optimisation over local measurements.
For general mixed states this can be nontrivial. Closed formulas exist for selected families such as some two-qubit X states under restricted conditions, but generic problems require numerical optimisation.
One should not report a value obtained from one convenient measurement basis as “the discord” unless optimality has been established.
23. POVMs versus projective measurements
Some discord definitions optimise over rank-one projective measurements; others allow general POVMs.
Allowing a larger measurement class can change the optimum.
Always state the convention before comparing numerical discord values from different sources.
24. Thermal states can have discord
At nonzero temperature, entanglement can vanish while noncommuting local correlations survive.
Hence separable thermal states may have nonzero discord.
This does not mean every thermal correlation is useful for quantum information processing; it means the joint state is not classical–quantum in the relevant local basis.
25. Common misconceptions
“Discord is just weak entanglement.” No. Separable mixed states can have nonzero discord.
“Discord is symmetric.” One-sided discord depends on which subsystem is measured.
“Any local measurement basis gives the discord.” The relevant classical correlation must be optimised over the specified measurement class.
“Nonzero discord proves quantum speedup.” It does not; algorithmic advantage requires a separate complexity argument.
26. Worked synthesis problem
Take the Bell state |Φ+⟩.
S(AB)=0.S(A)=S(B)=1.- Thus
I(A:B)=2bits. - Measuring B in the Schmidt basis leaves pure conditional states of A, so the measurement-induced conditional entropy is zero.
- Hence
J_{A|B}=1andD_{A|B}=1.
The total two bits of correlation decompose into one bit accessible as a classical outcome correlation and one bit of irreducibly quantum correlation under this entropy definition.
27. Practice set
- Write quantum mutual information.
- Why does the second classical definition require a measurement in quantum theory?
- Define J_{A|B}.
- Define D_{A|B}.
- What form has zero discord when A is measured?
- Why can separable states have discord?
- What is the discord of a pure Bell state?
- Give an invariant-state test for zero one-sided discord.
- Why can geometric discord depend on the chosen norm?
- Why does nonzero discord not prove computational advantage?
Answers
S(A)+S(B)−S(AB).- There is no unique pre-existing classical value of B to condition on; one must choose a local measurement.
- The maximum reduction in A entropy obtainable by measuring B under the chosen measurement class.
I−J.Σ_ip_i|i⟩⟨i|_A⊗ρ_i^B.- A mixture of noncommuting local states can be separable while no local measurement reads its label without disturbance.
- One bit.
- There exists a local complete projective measurement that leaves the joint state unchanged.
- Noncontractive distances can change under irrelevant local ancilla operations and therefore distort resource interpretation.
- Discord is a correlation property; computational speedup depends on the full algorithm and classical simulation complexity.
Sources and further study
[1] Harold Ollivier and Wojciech H. Zurek, Quantum Discord: A Measure of the Quantumness of Correlations, Physical Review Letters 88, 017901 (2001). Foundational discord definition.
[2] L. Henderson and V. Vedral, Classical, quantum and total correlations, Journal of Physics A 34, 6899 (2001). Classical-correlation optimisation.
[3] Kavan Modi and colleagues, The classical-quantum boundary for correlations: Discord and related measures, Reviews of Modern Physics 84, 1655 (2012). Comprehensive review.
[4] Marco Piani and colleagues, No-local-broadcasting theorem for multipartite quantum correlations, Physical Review Letters 100, 090502 (2008).
[5] Madhok and Datta, Interpreting quantum discord through quantum state merging, Physical Review A 83, 032323 (2011).
Continue through Quantum Mathematics — Batch 21
Guide 81: Quantum Darwinism, Pointer States, Environment as Witness, Redundancy and Decoherence develops redundant environmental records. Guide 83: Weak Measurements, Weak Values, Postselection and Pointer Shifts develops gentle pre/postselected measurements. Guide 84: Leggett–Garg Inequalities, Macrorealism and Temporal Quantum Correlations develops temporal macrorealism tests.