Fidelity of quantum states

In quantum mechanics, notably 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 density operators and , the fidelity is generally defined as the quantity . In the special case where and represent pure quantum states, namely, and , the definition reduces to the squared overlap between the states: . While not obvious from the general definition, the fidelity is symmetric: .

MotivationEdit

Given two random variables   with values   (categorical random variables) and probabilities   and  , the fidelity of   and   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]

DefinitionEdit

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 may be written, using the trace norm

 

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 [2][3]

 

Qubit state means that   and   are represented by two-dimensional matrices. This result follows noticing that   is a positive semidefinite operator, hence  , where   and   are the (nonnegative) eigenvalues of  . If   (or  ) is pure, this result is simplified further to   since   for pure states.

Alternative definitionEdit

Some authors use an alternative definition   and call this quantity fidelity.[4] The definition of   however is more common.[5][6][7] 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 propertiesEdit

Unitary invarianceEdit

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

 

for any unitary operator  .

Uhlmann's theoremEdit

We saw that for two pure states, their fidelity coincides with the overlap. Uhlmann's theorem[8] generalizes this statement to mixed states, in terms of their purifications:

Theorem Let ρ and σ be density matrices acting on Cn. Let ρ12 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 proofEdit

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

 

and σ12 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)12). 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 decompositionsEdit

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
 

ConsequencesEdit

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,[4] so this angle is a metric on the state space: the Fubini–Study metric.[9]

Relationship with the fidelity between the corresponding probability distributionsEdit

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 inequalityEdit

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 operationsEdit

The fidelity between two states can be shown to never decrease when a non-selective quantum operation   is applied to the states:[10]

 
for any trace-preserving completely positive map  .

Relationship to trace distanceEdit

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,[11]

 

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.

 

ReferencesEdit

  1. ^ C. A. Fuchs, C. M. Caves: "Ensemble-Dependent Bounds for Accessible Information in Quantum Mechanics", Physical Review Letters 73, 3047(1994)
  2. ^ a b R. Jozsa, Fidelity for Mixed Quantum States, J. Mod. Opt. 41, 2315--2323 (1994). DOI: http://doi.org/10.1080/09500349414552171
  3. ^ M. Hübner, Explicit Computation of the Bures Distance for Density Matrices, Phys. Lett. A 163, 239--242 (1992). DOI: https://doi.org/10.1016/0375-9601%2892%2991004-B
  4. ^ a b Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. doi:10.1017/CBO9780511976667. ISBN 978-0521635035.
  5. ^ 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.
  6. ^ Walls, D. F.; Milburn, G. J. (2008). Quantum Optics. Berlin: Springer. ISBN 978-3-540-28573-1.
  7. ^ Jaeger, Gregg (2007). Quantum Information: An Overview. New York London: Springer. ISBN 978-0-387-35725-6.
  8. ^ 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.
  9. ^ K. Życzkowski, I. Bengtsson, Geometry of Quantum States, Cambridge University Press, 2008, 131
  10. ^ Nielsen, M. A. (1996-06-13). "The entanglement fidelity and quantum error correction". arXiv:quant-ph/9606012. Bibcode:1996quant.ph..6012N. Cite journal requires |journal= (help)
  11. ^ 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