Fidelity of quantum states

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In quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.

Given two states and , the fidelity is generally defined as the quantity . In the special case of and pure states, the definition reduces to the squared overlap between the states: if and . While not obvious from the general definition, the fidelity is symmetric: .

Motivation[edit]

Given two random variables X, Y with values (1, ..., n) (categorical random variables) and probabilities p = (p1...pn) and q = (q1...qn). The fidelity of X and Y is defined to be the quantity

.

The fidelity deals with the marginal distribution of the random variables. It says nothing about the joint distribution of those variables. In other words, the fidelity F(X,Y) is the square of the inner product of and viewed as vectors in Euclidean space. Notice that F(X,Y) = 1 if and only if p = q. In general, . The measure is known as the Bhattacharyya coefficient.

Given a classical measure of the distinguishability of two probability distributions, one can motivate a measure of distinguishability of two quantum states as follows. If an experimenter is attempting to determine whether a quantum state is either of two possibilities or , the most general possible measurement they can make on the state is a POVM, which is described by a set of Hermitian positive semidefinite operators . If the state given to the experimenter is , they will witness outcome with probability , and likewise with probability for . Their ability to distinguish between the quantum states and is then equivalent to their ability to distinguish between the classical probability distributions and . Naturally, the experimenter will choose the best POVM he can find, so this motivates defining the quantum fidelity as the squared Bhattacharyya coefficient when extremized over all possible POVMs :

It was shown by Fuchs and Caves that this manifestly symmetric definition is equivalent to the simple asymmetric formula given in the next section.[1]

Definition[edit]

Given two density matrices ρ and σ, the fidelity is defined by[2]

where, for a positive semidefinite matrix , denotes its unique positive square root, as given by the spectral theorem. The Euclidean inner product from the classical definition is replaced by the Hilbert–Schmidt inner product.

Some of the important properties of the quantum state fidelity are:

  • Symmetry. .
  • Bounded values. For any and , , and .
  • Consistency with fidelity between probability distributions. If and commute, the definition simplifies to
    where are the eigenvalues of , respectively. To see this, remember that if then they can be diagonalized in the same basis:
    so that
  • Simplified expressions for pure states. If is pure, , then . This follows from
    If both and are pure, and , then . This follows immediately from the above expression for pure.
  • Equivalent expression. An equivalent expression for the fidelity is
    where the absolute value of an operator is here defined as .
  • Explicit expression for qubits. If and are both qubit states, the fidelity can be computed as

Alternative definition[edit]

Some authors use an alternative definition and call this quantity fidelity.[3] The definition of however is more common.[4][5][6] To avoid confusion, could be called “square root fidelity”. In any case it is advisable to clarify the adopted definition whenever the fidelity is employed.

Other properties[edit]

Unitary invariance[edit]

Direct calculation shows that the fidelity is preserved by unitary evolution, i.e.

for any unitary operator .

Uhlmann's theorem[edit]

We saw that for two pure states, their fidelity coincides with the overlap. Uhlmann's theorem[7] generalizes this statement to mixed states, in terms of their purifications:

Theorem Let ρ and σ be density matrices acting on Cn. Let ρ½ be the unique positive square root of ρ and

be a purification of ρ (therefore is an orthonormal basis), then the following equality holds:

where is a purification of σ. Therefore, in general, the fidelity is the maximum overlap between purifications.

Sketch of proof[edit]

A simple proof can be sketched as follows. Let denote the vector

and σ½ be the unique positive square root of σ. We see that, due to the unitary freedom in square root factorizations and choosing orthonormal bases, an arbitrary purification of σ is of the form

where Vi's are unitary operators. Now we directly calculate

But in general, for any square matrix A and unitary U, it is true that |Tr(AU)| ≤ Tr ((A*A)½). Furthermore, equality is achieved if U* is the unitary operator in the polar decomposition of A. From this follows directly Uhlmann's theorem.

Proof with explicit decompositions[edit]

We will here provide an alternative, explicit way to prove Uhlmann's theorem.

Let and be purifications of and , respectively. To start, let us show that .

The general form of the purifications of the states is:

were are the eigenvectors of , and are arbitrary orthonormal bases. The overlap between the purifications is
where the unitary matrix is defined as
The conclusion is now reached via using the inequality :
Note that this inequality is the triangle inequality applied to the singular values of the matrix. Indeed, for a generic matrix and unitary , we have
where are the (always real and non-negative) singular values of , as in the singular value decomposition. The inequality is saturated and becomes an equality when , that is, when and thus . The above shows that when the purifications and are such that . Because this choice is possible regardless of the states, we can finally conclude that

Consequences[edit]

Some immediate consequences of Uhlmann's theorem are

  • Fidelity is symmetric in its arguments, i.e. F (ρ,σ) = F (σ,ρ). Note that this is not obvious from the original definition.
  • F (ρ,σ) lies in [0,1], by the Cauchy–Schwarz inequality.
  • F (ρ,σ) = 1 if and only if ρ = σ, since Ψρ = Ψσ implies ρ = σ.

So we can see that fidelity behaves almost like a metric. This can be formalized and made useful by defining

As the angle between the states and . It follows from the above properties that is non-negative, symmetric in its inputs, and is equal to zero if and only if . Furthermore, it can be proved that it obeys the triangle inequality,[3] so this angle is a metric on the state space: the Fubini–Study metric.[8]

Relationship with the fidelity between the corresponding probability distributions[edit]

Let be an arbitrary positive operator-valued measure (POVM); that is, a set of operators satisfying , , and . Then, for any pair of states and , we have

where in the last step we denoted with the probability distributions obtained by measuring with the POVM .

This shows that the square root of the fidelity between two quantum states is upper bounded by the Bhattacharyya coefficient between the corresponding probability distributions in any possible POVM. Indeed, it is more generally true that

where , and the minimum is taken over all possible POVMs.

Proof of inequality[edit]

As was previously shown, the square root of the fidelity can be written as which is equivalent to the existence of a unitary operator such that

Remembering that holds true for any POVM, we can then write
where in the last step we used Cauchy-Schwarz inequality as in .

Behavior under quantum operations[edit]

The fidelity between two states can be shown to never decrease when a non-selective quantum operation is applied to the states:[9]

for any trace-preserving completely positive map .

Relationship to trace distance[edit]

We can define the trace distance between two matrices A and B in terms of the trace norm by

When A and B are both density operators, this is a quantum generalization of the statistical distance. This is relevant because the trace distance provides upper and lower bounds on the fidelity as quantified by the Fuchs–van de Graaf inequalities,[10]

Often the trace distance is easier to calculate or bound than the fidelity, so these relationships are quite useful. In the case that at least one of the states is a pure state Ψ, the lower bound can be tightened.

References[edit]

  1. ^ C. A. Fuchs, C. M. Caves: "Ensemble-Dependent Bounds for Accessible Information in Quantum Mechanics", Physical Review Letters 73, 3047(1994)
  2. ^ Jozsa, Richard (1994). "Fidelity for Mixed Quantum States". Journal of Modern Optics. 41 (12): 2315–2323. Bibcode:1994JMOp...41.2315J. doi:10.1080/09500349414552171.
  3. ^ a b Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. doi:10.1017/CBO9780511976667. ISBN 978-0521635035.
  4. ^ Bengtsson, Ingemar (2017). Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge, United Kingdom New York, NY: Cambridge University Press. ISBN 978-1-107-02625-4.
  5. ^ Walls, D. F.; Milburn, G. J. (2008). Quantum Optics. Berlin: Springer. ISBN 978-3-540-28573-1.
  6. ^ Jaeger, Gregg (2007). Quantum Information: An Overview. New York London: Springer. ISBN 978-0-387-35725-6.
  7. ^ Uhlmann, A. (1976). "The "transition probability" in the state space of a ∗-algebra" (PDF). Reports on Mathematical Physics. 9 (2): 273–279. Bibcode:1976RpMP....9..273U. doi:10.1016/0034-4877(76)90060-4. ISSN 0034-4877.
  8. ^ K. Życzkowski, I. Bengtsson, Geometry of Quantum States, Cambridge University Press, 2008, 131
  9. ^ Nielsen, M. A. (1996-06-13). "The entanglement fidelity and quantum error correction". arXiv:quant-ph/9606012. Bibcode:1996quant.ph..6012N.
  10. ^ C. A. Fuchs and J. van de Graaf, "Cryptographic Distinguishability Measures for Quantum Mechanical States", IEEE Trans. Inf. Theory 45, 1216 (1999). arXiv:quant-ph/9712042