In classical systems, comparing two pieces of data—like numbers or text—is straightforward. You can use simple distances like subtraction or Hamming distance.
But in quantum systems, the objects we’re comparing are quantum states. These are not just bits (0 or 1), but qubits—which can be in superpositions, entangled with others, or mixed (i.e., uncertain).
So, we need specialized ways to compare quantum states:
- How similar are they?
- How different are they?
- Did a quantum operation distort the state too much?
That’s where fidelity and distance measures come in.
2. What is Quantum Fidelity?
Fidelity tells us how close two quantum states are—essentially, how much overlap they have. Think of it like a “closeness score” between 0 and 1.
- If the score is 1, the states are identical.
- If it’s 0, the states are completely different (orthogonal).
Fidelity is especially useful when:
- You’re sending quantum information through a noisy channel and want to compare the original state and the received state.
- You’re building a quantum computer and want to measure how accurate your output is.
- You’re verifying entangled or mixed states to check how well you preserved entanglement.
Fidelity works for both:
- Pure states: Which are well-defined and not probabilistic.
- Mixed states: Which represent a statistical mixture of possibilities (common in real-world experiments).
3. Key Properties of Fidelity
Some intuitive characteristics of quantum fidelity:
- Symmetry: Comparing state A to state B gives the same result as comparing B to A.
- Normalization: It’s always between 0 and 1.
- Invariance under unitary transformations: If you rotate or evolve the quantum states in the same way, their fidelity remains unchanged.
This last point is crucial in quantum computing, where operations evolve states over time using unitary transformations.
4. What is a Quantum Distance Measure?
A distance measure quantifies how different two quantum states are. While fidelity measures closeness, distance focuses on how far apart they are.
In the quantum world, distance isn’t always as straightforward as “Euclidean distance” in classical math. Instead, quantum distances must handle:
- Superpositions
- Entanglement
- Uncertainty (probabilistic mixtures)
Some popular quantum distance measures include:
- Trace Distance
- Bures Distance
- Hilbert-Schmidt Distance
- Quantum Relative Entropy
Each has its strengths and is suited for different types of analysis.
5. Trace Distance – A Classical Feel in Quantum Form
Trace distance is one of the most intuitive quantum distance measures. It behaves similarly to total variation distance in classical probability.
It essentially tells you:
“How distinguishable are these two quantum states?”
- If two states are very different, the trace distance is high.
- If they’re almost indistinguishable, the trace distance is low.
Why does this matter?
- In quantum cryptography, trace distance can tell you how much information an eavesdropper might learn.
- In quantum computing, it measures how far your output is from what it should be.
6. Quantum Relative Entropy – Asymmetrical Distance
Relative entropy is a directional distance. It measures how one state diverges from another.
While it’s called a “distance,” it’s not symmetric—that is, comparing state A to state B might give a different value than B to A.
Relative entropy has many applications:
- In quantum thermodynamics, it represents a kind of “entropy cost.”
- In quantum hypothesis testing, it tells you how well you can distinguish a true state from a hypothesis.
- In quantum machine learning, it can act like a loss function to compare quantum data points.
It’s useful when we care about information loss or gain.
7. Bures Distance – Based on Fidelity
The Bures distance is built directly from fidelity. If fidelity measures closeness, Bures distance gives a true metric (a proper geometrical distance) based on that closeness.
- If two states are identical, Bures distance is zero.
- If they’re completely different, it’s at its maximum.
Bures distance is popular in quantum metrology and quantum statistical analysis, because it connects nicely with geometry and information theory.
8. Applications Across Quantum Technologies
Let’s look at how fidelity and distance measures are used in practice.
a) Quantum Communication
- Compare the sent and received states to see if the channel was reliable.
- Measure how much noise or decoherence affected the transmission.
b) Quantum Cryptography
- Use trace distance to check how much information an attacker might have.
- Ensure secure key distribution by keeping distances below certain thresholds.
c) Quantum Computing
- Use fidelity to benchmark gate performance (Did your quantum logic gate work correctly?).
- Track error propagation over circuits.
d) Quantum Error Correction
- Measure how close your corrected state is to the original.
- Use fidelity and relative entropy to evaluate code performance.
e) Quantum Machine Learning
- Quantum distances can act as loss functions.
- Help optimize models working with quantum data sets.
9. Why No Single Measure Is Enough
Different situations call for different metrics. Here’s why:
- Fidelity is great for measuring similarity but doesn’t satisfy all properties of a metric.
- Trace Distance is more interpretable in terms of distinguishability.
- Relative Entropy is suited for asymmetrical comparisons and statistical reasoning.
- Bures Distance is geometrically sound and fidelity-based.
So, researchers often choose based on:
- What they’re trying to achieve,
- How the quantum states are prepared,
- And whether precision, security, or stability is the main goal.
10. Conceptual Insights: What Do These Measures Really Tell Us?
Think of fidelity and distance as tools to navigate the quantum state space—a huge, complex landscape.
- Fidelity is like asking, “Am I on the same path or trajectory?”
- Distance is like asking, “How many steps (or how much effort) would it take to move from one state to the other?”
They don’t just give numbers—they provide diagnostic insights:
- Is my quantum algorithm stable?
- Is this quantum channel too noisy?
- Is my encryption scheme secure?
- Is my experimental output close enough to the theoretical target?