Quantum Information

Entanglement As Structural Property

How To See It Yourself

Entanglement is not spooky action at a distance. From an information science perspective, it simply means that information is not stored locally in a variable, but in only the correlation between two variables.

by Frank Zickert

November 6, 2025

You've probably been told that to understand a system, you just have to take it apart. Break it into smaller pieces, analyze each piece carefully, and then put it all back together. That mindset works beautifully in classical physics and Machine Learning.Machine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming.
Learn more about Machine Learning Because the whole is just the sum of its parts.

But in Quantum Systems,A quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results.
Learn more about Quantum System that rule quietly collapses. You can stare at each Qubit,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit measureIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement it perfectly, and still miss what's really going on. You’ll see noise where there’s actually order, and randomness where there’s structure. In Quantum Machine Learning,Quantum Machine Learning is the field of research that combines principles from quantum computing with traditional machine learning to solve complex problems more efficiently than classical approaches.
Learn more about Quantum Machine Learning that mistake can cripple your model before it even trains.

There’s a hidden wiring. There's a kind of structure that doesn’t live in the parts at all. It lives between them.

When the parts lie to you

Let’s see this for ourselves. ? builds two two-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit systems in Qiskit:

one (a normal, uncorrelated pair),
and one pair (a Bell StateA **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State).

We’ll then measure how each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit looks , and how they .

entanglement.py

from qiskit import QuantumCircuit

from qiskit.quantum_info import Statevector, SparsePauliOp

def expval(state, pauli_label):

    """Compute <state|P|state> for a Pauli operator like 'Z', 'ZI', 'ZZ'."""

    op = SparsePauliOp.from_list([(pauli_label, 1.0)])

    return float(state.expectation_value(op))

# --- 1) Product (separable) state |00| ---

prod = QuantumCircuit(2)

psi_prod = Statevector.from_instruction(prod)

print("Product |00|:")

print(

    "<Z0> =",

    expval(psi_prod, "ZI"),

    "<Z1> =",

    expval(psi_prod, "IZ"),

    "<Z0Z1> =",

    expval(psi_prod, "ZZ"),

)

# --- 2) Entangled (Bell) state |Φ+> = (|00> + |11>)/√2 ---

bell = QuantumCircuit(2)

bell.h(0)  # create superposition on qubit 0

bell.cx(0, 1)  # link qubit 0 and 1

psi_bell = Statevector.from_instruction(bell)

print("\nBell |Φ+> :")

print(

    "<Z0> =",

    expval(psi_bell, "ZI"),

    "<Z1> =",

    expval(psi_bell, "IZ"),

    "<Z0Z1> =",

    expval(psi_bell, "ZZ"),

)

Listing 1 Creating two exemplary circuits

? depicts the output when running the above code.

output

Product |00|:

 = 1.0  = 1.0  = 1.0

Bell |Φ+> :

 = 0.0  = 0.0  = 0.9999999999999998

Listing 2 Output of the two circuits

What does this tell us?

In the product state, everything behaves as expected: each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit gives a clean , and so does their joint measurement.

In the Bell state, the single-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit results look completely random on their own. B are . But the joint observable ZZ is again, perfectly correlated.

You’re looking at the same two QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit. Each one looks noisy. Together, they show perfect order.

That’s the first sign that the usual analyze-the-parts instinct doesn’t work here.

The whole is more than the parts

Let’s look deeper. What if we try to describe just one QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit from that Bell pair. As if we had access to only half the system?

The code in ? performs a partial trace. This is a standard method for examining a single subsystem (one QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit) of a larger entangled state such as a Bell pair. The results it produces illustrate a fundamental feature of entanglement: how global order can coexist with local randomness.

entanglement_structure.py

import numpy as np

from qiskit.quantum_info import DensityMatrix, partial_trace

rho_bell = DensityMatrix(psi_bell)

rho_A = partial_trace(rho_bell, [1])

rho_B = partial_trace(rho_bell, [0])

# Get eigenvalues using numpy

eigenvals = np.linalg.eigvals(rho_A.data)

print("Trace of rho_A =", rho_A.trace())

print("Spectrum of rho_A =", eigenvals)

Listing 3 Partial analysis of a Bell State

This code produces the output denoted in ?

Trace of rho_A = (0.9999999999999998+0j)

Spectrum of rho_A = [0.5+0.j 0.5+0.j]

Listing 4 Partial analysis of a Bell State

The Bell state we produce here is . It is a pure two-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit state. Its density matrix completely describes a perfectly ordered system. There is no uncertainty if you look at the whole thing. , rho_bell = DensityMatrix(psi_bell) constructs this full two-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit description.

The effectively ignores QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit and asks, what is the state of QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit alone? This operation traces out the degrees of freedom of QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit . The resulting object, rho_A, is a reduced density matrix describing QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit as viewed in isolation. Similarly, rho_B = partial_trace(rho_bell, [0]) the state of QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit alone.

The result is a maximally mixed single-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit state.Two things stand out:

The trace being confirms rho_A is a properly normalizedNormalization in quantum computing means that the total probability of all possible outcomes of a quantum state must equal 1. Mathematically, if a quantum state is written as a vector of complex amplitudes, the sum of the squares of their magnitudes must be 1. This ensures that when the quantum state is measured, one of the possible outcomes will definitely occur.
Learn more about Normalization Quantum State.A quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State
The eigenvalues [0.5, 0.5] reveal that QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit 0 is in a maximally mixed state. It has equal probabilities of being is a basis state.
Learn more about or is a basis state.
Learn more about , with no preference.

This is not noise or error; it is precisely what we expect from an entangled system.

Each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit alone carries no definite information, because the meaningful information resides entirely in their correlation.

If you measure QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit 0 by itself, you get random outcomes: half is a basis state.
Learn more about , half is a basis state.
Learn more about .

But if you measure both QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit together, you always obtain perfectly correlated results. You either get 00 or 11. So, while each part appears random, the pair behaves in a completely ordered way.

This is the hallmark of entanglement as structure:

The global state is pure and well defined.
Each local subsystem appears noisy and uncertain.
The actual information is encoded in the pattern of correlation between subsystems.

Information lives in the correlations

If you only measure local quantities, you’ll see nothing useful. But if you measure correlations, suddenly everything makes sense.

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The code in ? shows exactly why single-qubit measurements can be misleading and why correlations are the real source of structure in entangled systems. It computes both local and joint observables for the Bell state to reveal the difference between looking at qubits individually versus together.

entanglement_structure.py

# Local vs joint observables

x0_bell = expval(psi_bell, "XI")  # qubit 0 only

y1_bell = expval(psi_bell, "IY")  # qubit 1 only

xx_bell = expval(psi_bell, "XX")  # joint X⊗X

yy_bell = expval(psi_bell, "YY")  # joint Y⊗Y

print("Local observables: <X0> =", x0_bell, "<Y1> =", y1_bell)

print("Joint observables: <X0X1> =", xx_bell, "<Y0Y1> =", yy_bell)

Listing 5 Local versus joint observables

In this example, the function expval(state, "…") evaluates the Expectation ValueThe expectation value is the average result you'd get if you repeated a measurement of a quantity many times under identical conditions. Mathematically, it’s the weighted average of all possible outcomes, where each outcome is weighted by its probability. In quantum mechanics, it represents the average value of an observable calculated from the wavefunction.
Learn more about Expectation Value of a given Pauli Operator.A Pauli operator is one of three 2×2 complex matrices — **σₓ, σᵧ, σ_z** — that represent the basic quantum spin operations on a single qubit. They correspond to rotations or measurements along the x, y, and z axes of the Bloch sphere. Together with the identity matrix, they form a basis for all single-qubit operations in quantum mechanics.
Learn more about Pauli Operator You use:

measures on qubit only (ignoring qubit 1).
measures on qubit only.
measures , the joint correlation between qubits and .
measures , another joint correlation.

For our Bell StateA **Bell state** is one of four specific quantum states in which two qubits are **maximally entangled**, meaning their measurement outcomes are perfectly correlated no matter how far apart they are. Each Bell state represents a different pattern of correlation between the qubits. These states are fundamental in quantum information for testing **nonlocality** (violations of Bell’s inequality) and for protocols like **quantum teleportation**.
Learn more about Bell State , ? shows the results.

Local observables:  = 0.0  = 0.0

Joint observables:  = 0.9999999999999998  = -0.9999999999999998

Listing 6 Result of the observables for the Bell state

and imply that the local ObservablesIn quantum computing, an **observable** is a physical quantity (like energy, spin, or position) that can be **measured** from a quantum state. Mathematically, it’s represented by a **Hermitian operator**, whose eigenvalues correspond to the possible measurement outcomes. When you measure an observable, the quantum state **collapses** into one of its eigenstates, yielding one of those eigenvalues as the result.
Learn more about Observable show no signal, as if each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit were random.
and imply that the joint ObservablesIn quantum computing, an **observable** is a physical quantity (like energy, spin, or position) that can be **measured** from a quantum state. Mathematically, it’s represented by a **Hermitian operator**, whose eigenvalues correspond to the possible measurement outcomes. When you measure an observable, the quantum state **collapses** into one of its eigenstates, yielding one of those eigenvalues as the result.
Learn more about Observable reveal perfect correlation.

This tells us that while each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit individually appears to behave like a fair coin toss, their outcomes are tightly linked. When you measure both together, the relationship is deterministic: the two always produce matching results along the and axes.

What this means

In classical terms, this is like two dice that always roll the same number. Yet each die alone looks random. Only by checking them together do you see the pattern and therefore, get the stored information. EntanglementEntanglement is a quantum phenomenon where two or more particles become correlated so that measuring one instantly determines the state of the other, no matter how far apart they are. This correlation arises because their quantum states are linked as a single system, not as independent parts. It doesn’t allow faster-than-light communication but shows that quantum systems can share information in ways classical physics can’t explain.
Learn more about Entanglement works the same way: the pair carries information that doesn’t exist in either component separately.

So when you only measure single Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit you cannot predict its outcome. Any observable acting on a single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit gives zero expectation value. Therefore, each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit alone looks like a coin toss: no bias, no structure.

But when you measure both Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit you see that they are perfectly correlated. Joint MeasurementsIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement reveal structure. Observables that probe the system as a whole expose the hidden order — correlations that define the entangled state. In the Bell pair, every axis () shows perfect agreement between Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit even though neither qubit has a fixed orientation individually.

This is the paradox of entangled systems. While the whole can be in a pure, orderly state while each part looks like noise.

It’s not a trick of Measurement;In quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement it’s a structural property of how quantum systems store and process information.