Local Transformations of Bipartite Entanglement Are Rigid

Let $|C⟩,|D⟩\in {ℂ}^d\otimes {ℂ}^d$ denote bipartite pure states; let $𝖠$ denote the first subsystem and $𝖡$ denote the second subsystem. What is the closest that one can get to $|D⟩$ by performing a unitary on subsystem $𝖡$ of the state $|C⟩$? Uhlmann’s theorem [18] quantifies the optimal overlap achievable:

where $\rho$ and $\sigma$ denote the reduced density matrices on subsystem $𝖠$ of $|C⟩$ and $|D⟩$ respectively, the function $\mathrm{F}(\rho ,\sigma )=\mathrm{Tr}(\sqrt{{\rho }^{1/2}\sigma {\rho }^{1/2}})$ denotes the fidelity between the two states, and the maximization is over all unitary transformations acting on subsystem $𝖡$. We call a unitary $U$ achieving equality in Equation 1.1 an Uhlmann transformation.

Given the ubiquity of Uhlmann’s theorem throughout quantum information science, it seems worthwhile to study the mathematical and computational properties of Uhlmann transforms. Many natural questions arise: how unique are Uhlmann transformations? How robust are they to perturbations of the underlying states $|C⟩,|D⟩$? What is the complexity of performing Uhlmann transformations on a quantum computer? Can difficult Uhlmann transformations be used for cryptography? The latter two questions were recently studied by Metger and Yuen [13] and Bostanci, et al. [2], who investigate a theory of state and unitary complexity, respectively.

This paper studies the first two questions concerning the uniqueness and robustness of Uhlmann transformations. At first glance, Uhlmann transformations are not generally unique. For example, suppose that the reduced density matrix of $|C⟩$ on subsystem $𝖡$ does not have full support. Then any Uhlmann transformation can behave arbitrarily on the orthogonal complement of the support, while remaining optimal. What if we disregard these trivial degrees of freedom, however – could Uhlmann transformations be unique in some other meaningful way?

We provide an answer via canonical Uhlmann transformations, first defined by Metger and Yuen [13]. For every pair of bipartite states $(|C⟩,|D⟩$), this is the operator

where ${\mathrm{Tr}}_{𝖠}$ denotes tracing out register $𝖠$ and $\mathrm{sgn}(\cdot )$ denotes the following function: for a matrix $X$ with singular value decomposition $X=U\mathrm{\Sigma }{V}^{\ast }$, we define $\mathrm{sgn}(X)≔U\mathrm{sgn}(\mathrm{\Sigma })V^{\ast }$ where $U,V$ are unitary operators and $\mathrm{sgn}(\mathrm{\Sigma })$ denotes the projection onto the eigenvectors of $\mathrm{\Sigma }$ with positive eigenvalues (i.e., the support of $\mathrm{\Sigma }$). The canonical transformation $W$ is a partial isometry¹¹1A partial isometry can be thought of as the restriction of a unitary to a subspace. More formally, an operator $W$ is a partial isometry if $W^{\ast }W$ is a projection.; it is unitary if and only if both reduced states $\rho ,\sigma$ (of $|C⟩,|D⟩$, respectively) are invertible.

The following was proven by Bostanci, et al. [2, Proposition 6.3] in their investigation of the computational complexity of implementing Uhlmann transformations:

In other words, the canonical Uhlmann transformation defined in Equation 1.2 achieves the optimal overlap between $|C⟩$ and $|D⟩$, and furthermore any other partial isometry achieving the optimal overlap must be supported on the domain of $W$. This is a rather weak statement, however: when $R$ is unitary, then $W^{\ast }W\le R^{\ast }R=\mathrm{𝟙}$ is satisfied for all partial isometries $W$.

A stronger statement is the following:

This says that any optimal Uhlmann transformation, when restricted to the support of $W$, must behave identically to $W$ on the state $|C⟩$. This provides some justification in calling the $W$ in Equation 1.2 the “canonical” Uhlmann transformation corresponding to $|C⟩,|D⟩$.

Claim 2 is in fact a special case of a more general robust rigidity theorem that we prove in this paper. Roughly speaking, the theorem (Theorem 6 below) states that any transformation $R$ that achieves approximately-optimal fidelity (meaning that $|⟨D|\mathrm{𝟙}\otimes R|C⟩|\ge \mathrm{F}(\rho ,\sigma )-ϵ$) must be approximately the canonical Uhlmann transformation $W$. This is analogous to rigidity results for some quantum information processing tasks such as nonlocal games [10, 11, 17] and superdense coding [15]. These results show that the only way for a quantum operation to achieve near-optimal performance according to some metric (e.g., winning probability in a nonlocal game, or decoding probability in superdense coding) is if, in fact, it is close to a canonical strategy or protocol.

First we give a way to quantify the rigidity of canonical Uhlmann transformations.

Thus, bounds on the function $\delta (ϵ)$ of an Uhlmann transformation quantifies the extent to which the exact rigidity statement of Claim 2 can be made robust.

Our main result is a general bound on the rigidity of Uhlmann transformations.

For readers who are not familiar with the (beautiful notion of the) matrix geometric mean we provide a brief introduction in Section 2.

Thus, the canonical Uhlmann transformation is indeed robustly rigid, up to some blow-up that depends on two parameters $\eta$ and $\kappa$ (called the spectral gap and obliqueness, respectively) of the reduced density matrices $\rho ,\sigma$. Intuitively, the obliqueness parameter $\kappa$ is a measure of a combination of non-commutativity and non-invertibility of $\rho ,\sigma$ and the spectral gap parameter $\eta$ is a measure of how “well-conditioned” the matrix geometric mean ${\rho }^{-1}\mathrm{\#}\sigma$ is (which one can think of as a notion of “ratio” between $\sigma$ and $\rho$).

In [3, Sections 4.2 and 4.4] respectively we further show that some dependence on the spectral gap parameter $\eta$ and the obliqueness parameter $\kappa$ in Theorem 6 is necessary.

The matrix geometric mean is a noncommutative generalization of the geometric mean $\sqrt{ab}$ of two nonnegative numbers $a,b$. If $A,B$ are commuting positive semidefinite matrices, then the matrix geometric mean $A\mathrm{\#}B$ is defined as $A^{1/2}{B}^{1/2}$. For general positive definite (i.e., all eigenvalues are strictly positive) matrices $A,B$, the matrix geometric mean $A\mathrm{\#}B$ is defined as

For positive definite matrices $A,B$ the matrix geometric mean enjoys many pleasant properties, including

1. 1.

   $A\mathrm{\#}B$ is positive definite.

2. 2.

   $A\mathrm{\#}B=B\mathrm{\#}A$.

3. 3.

   $A\mathrm{\#}B$ is the unique positive solution to the equation $X{A}^{-1}X=B$.

4. 4.

   If $X$ is invertible, then $X(A\mathrm{\#}B)X^{-1}=(XA{X}^{-1})\mathrm{\#}(XB{X}^{-1})$.

5. 5.

   $A\mathrm{\#}B\le \frac{1}{2}(A+B)$, a noncommutative analogue of the arithmetic-geometric meaninequality.

6. 6.

   $\mathrm{\Phi }(A)\mathrm{\#}\mathrm{\Phi }(B)\le \mathrm{\Phi }(A\mathrm{\#}B)$ for all positive maps $\mathrm{\Phi }$.

Proofs of these properties can be found in [1, Chapter 4]. For more applications of the matrix geometric mean in quantum information theory, see [4, 5, 9].

For noninvertible $A,B$ (but still positive semidefinite), the matrix geometric mean $A\mathrm{\#}B$ is typically defined as a limit of geometric means of sequences of strictly positive matrices converging to $A,B$; however, in this case not all of the properties listed above are satisfied. For example, the symmetry property $A\mathrm{\#}B=B\mathrm{\#}A$ need not hold.

In this paper we do not use this limit definition, and instead stick to Equation 2.1 as the definition for the matrix geometric mean for all positive semidefinite matrices $A,B$, with the inverses now being Moore-Penrose pseudoinverses. Although it does not satisfy all the properties listed above, it satisfies a few important properties that are needed for the proof of Theorem 6; for example, as we will show in [3, Claim 3.1], when the density matrices $\rho ,\sigma$ are real, the canonical Uhlmann transformation can equivalently be expressed in terms of a matrix geometric mean:

for some unitary operators $X,Y$ (see the start of [3, Section 3] for their definitions). Furthermore, the fidelity between $\rho$ and $\sigma$ can also be written as $\mathrm{F}(\rho ,\sigma )=\mathrm{Tr}(({\rho }^{-1}\mathrm{\#}\sigma )\rho )$ (see e.g. [4]).

Given the centrality of Uhlmann transformations, the rigidity statement in Theorem 6 may be of interest in its own right, but it also turns out to be a useful technical tool for other applications. To illustrate this, we briefly discuss applications of our robust rigidity theorem to unitary complexity theory and approximate representation theory.

Uhlmann transformations, which are local transformations of bipartite (pure state) entanglement, are fundamental in quantum information theory. In this paper we showed that Uhlmann transformations possess a robust form of rigidity: near-optimal entanglement transformations must close (in a well-defined sense) to a unique optimal transformation. This unique optimal transformation is the canonical Uhlmann transformation introduced by [13], and our result gives further justification to calling it “canonical.”

We showed that the robustness of the rigidity theorem inherently depends on two parameters called the spectral gap and obliqueness, which are functions of the underlying pure states $|C⟩,|D⟩$. An interesting open question is whether there is a general “rounding” procedure that converts any pair of states $(|C⟩,|D⟩)$ into a nearby pair $(|\tilde{C}⟩,|\tilde{D}⟩)$ with controlled spectral gap and controlled obliqueness. In [3, Section 4.3], we provide a rounding lemma that only controls the spectral gap.

Finally, we presented two applications of our robust rigidity theorem. The first is to unitary complexity theory, where we can improve the round complexity required for the task of synthesizing canonical Uhlmann transformations in the regime where the fidelity between the reduced states is not $1$. Second, we demonstrate that the robust rigidity theorem is a very general form of robustness that can be used to derive other stability theorems. As an example, we re-derive the stability of approximate group representations by reducing to the robust rigidity of the canonical Uhlmann transformation between a specific pair of states. Given the ubiquity of Uhlmann transformations in protocols across quantum information theory and quantum complexity theory, developing an understanding of their rigidity properties should unlock further applications and deeper insight into the ways that bipartite entanglement can be locally transformed.