## The Spectrum of the Partial Transpose of a Density Matrix

It is a simple fact that, given any density matrix (i.e., quantum state) \(\rho\in M_n\), the eigenvalues of \(\rho\) are the same as the eigenvalues of \(\rho^T\) (the transpose of \(\rho\)). However, strange things can happen if we instead only apply the transpose to one half of a quantum state. That is, if \(\rho\in M_m \otimes M_n\) then its eigenvalues in general will be very different from the eigenvalues of \((id_m\otimes T)(\rho)\), where \(id_m\) is the identity map on \(M_m\) and \(T\) is the transpose map on \(M_n\) (the map \(id_m\otimes T\) is called the *partial transpose*).

In fact, even though \(\rho\) is positive semidefinite (since it is a density matrix), the matrix \((id_m\otimes T)(\rho)\) in general can have negative eigenvalues. To see this, define \(p:={\rm min}\{m,n\}\) and let \(\rho=|\psi\rangle\langle\psi|\), where

\(|\psi\rangle=\displaystyle\frac{1}{\sqrt{p}}\sum_{j=1}^{p}|j\rangle\otimes|j\rangle\)

is the standard maximally-entangled pure state. It then follows that

\((id_m\otimes T)(\rho)=\displaystyle\frac{1}{p}\sum_{i,j=1}^{p}|i\rangle\langle j|\otimes|j\rangle\langle i|\),

which has \(p(p+1)/2\) eigenvalues equal to \(1/p\), \(p(p-1)/2\) eigenvalues equal to \(-1/p\), and \(p|m-n|\) eigenvalues equal to \(0\).

The fact that \((id_m\otimes T)(\rho)\) can have negative eigenvalues is another way of saying that the transpose map is positive but not completely positive, and thus plays a big role in entanglement theory. In this post we consider the question of how exactly the partial transpose map can transform the eigenvalues of \(\rho\):

**Question.** For which ordered lists \(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_{mn}\in\mathbb{R}\) does there exist a density matrix \(\rho\) such that \((id_m\otimes T)(\rho)\) has eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_{mn}\)?

### The Answer for Pure States

In the case when \(\rho\) is a pure state (i.e., has rank 1), we can completely characterize the eigenvalues of \((id_m\otimes T)(\rho)\) by making use of the Schmidt decomposition. In particular, we have the following:

**Theorem 1.** Let \(|\phi\rangle\) have Schmidt rank \(r\) and Schmidt coefficients \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_r>0\). Then the spectrum of \((id_m\otimes T)(|\phi\rangle\langle\phi|)\) is

\(\{\alpha_i^2 : 1\leq i\leq r\}\cup\{\pm\alpha_i\alpha_j:1\leq i<j\leq r\}\),

together with the eigenvalue \(0\) with multiplicity \(p|n-m|+p^2-r^2\).

*Proof.* If \(|\phi\rangle\) has Schmidt decomposition

\(\displaystyle|\phi\rangle=\sum_{i=1}^r\alpha_i|a_i\rangle\otimes|b_i\rangle\)

then

\(\displaystyle(id_m\otimes T)(|\phi\rangle\langle\phi|)=\sum_{i,j=1}^r\alpha_i\alpha_j|a_i\rangle\langle a_j|\otimes|b_j\rangle\langle b_i|.\)

It is then straightforward to verify, for all \(1\leq i<j\leq r\), that:

- \(|a_i\rangle\otimes|b_i\rangle\) is an eigenvector with eigenvalue \(\alpha_i^2\);
- \(|a_i\rangle\otimes|b_j\rangle\pm|a_j\rangle\otimes|b_i\rangle\) is an eigenvector with eigenvalue \(\pm\alpha_i\alpha_j\); and
- \({\rm rank}\big((id_m\otimes T)(|\phi\rangle\langle\phi|)\big)= r^2\), from which it follows that the remaining \(p|n-m|+p^2-r^2\) eigenvalues are \(0\).

Despite such a simple characterization in the case of rank-1 density matrices, there is no known characterization for general density matrices, since eigenvalues aren’t well-behaved under convex combinations.

### The Number of Negative Eigenvalues

Instead of asking for a complete characterization of the possible spectra of \((id_m\otimes T)(\rho)\), for now we focus on the simpler question that asks how many of the eigenvalues of \((id_m\otimes T)(\rho)\) can be negative. Theorem 1 answers this question when \(\rho=|\phi\rangle\langle\phi|\) is a pure state: the number of negative eigenvalues is \(r(r-1)/2\), where r is the Schmidt rank of \(|\phi\rangle\). Since \(r\leq p\), it follows that \((id_m\otimes T)(\rho)\) has at most \(p(p-1)/2\) negative eigenvalues when \(\rho\) is a pure state.

It was conjectured in [1] that a similar fact holds for general (not necessarily pure) density matrices \(\rho\) as well. In particular, they conjectured that if \(\rho\in M_n\otimes M_n\) then \((id_n\otimes T)(\rho)\) has at most \(n(n-1)/2\) negative eigenvalues. However, this conjecture is easily shown to be false just by randomly-generating many density matrices \(\rho\) and then counting the number of negative eigenvalues of \((id_n\otimes T)(\rho)\); density matrices whose partial transposes have more than \(n(n-1)/2\) negative eigenvalues are very common.

In [2,3], it was shown that if \(\rho\in M_m\otimes M_n\) then \((id_m\otimes T)(\rho)\) can not have more than \((m-1)(n-1)\) negative eigenvalues. In [4], this bound was shown to be tight when \({\rm min}\{m,n\}=2\) by explicitly constructing density matrices \(\rho\in M_2\otimes M_n\) such that \((id_2\otimes T)(\rho)\) has \(n-1\) negative eigenvalues. Similarly, this bound was shown to be tight via explicit construction when \(m=n=3\) in [3]. Finally, it was shown in [5] that this bound is tight in general. That is, we have the following result:

**Theorem 2.** The maximum number of negative eigenvalues that \((id_m\otimes T)(\rho)\) can have when \(\rho\in M_m\otimes M_n\) is \((m-1)(n-1)\).

It is worth pointing out that the method used in [5] to prove that this bound is tight is not completely analytic. Instead, a numerical method was presented that is proved to always generate a density matrix \(\rho\in M_m\otimes M_n\) such that \((id_m\otimes T)(\rho)\) has \((m-1)(n-1)\) negative eigenvalues. Code that implements this numerical procedure in MATLAB is available here, but no general analytic form for such density matrices is known.

### Other Bounds on the Spectrum

Unfortunately, not a whole lot more is known about the spectrum of \((id_m\otimes T)(\rho)\). Here are some miscellaneous other results that impose certain restrictions on its maximal and minimal eigenvalues (which we denote by \(\lambda_\textup{max}\) and \(\lambda_\textup{min}\), respectively):

**Theorem 3 [3].** \(1\geq\lambda_\textup{max}\geq\lambda_\textup{min}\geq -1/2\).

**Theorem 4 [2].** \(\lambda_\textup{min}\geq\lambda_\textup{max}(1-{\rm min}\{m,n\})\).

**Theorem 5 [6].** If \((id_m\otimes T)(\rho)\) has \(q\) negative eigenvalues then

\(\displaystyle\lambda_\textup{min}\geq\lambda_\textup{max}\Big(1-\big\lceil\tfrac{1}{2}\big(m+n-\sqrt{(m-n)^2+4q-4}\big)\big\rceil\Big)\) and

\(\displaystyle\lambda_\textup{min}\geq\lambda_\textup{max}\Big(1-\frac{mn\sqrt{mn-1}}{q\sqrt{mn-1}+\sqrt{mnq-q^2}}\Big)\).

However, these bounds in general are fairly weak and the question of what the possible spectra of \((id_m\otimes T)(\rho)\) are is still far beyond our grasp.

**Update [August 21, 2017]:** Everett Patterson and I have now written a paper about this topic.

**References**

- R. Xi-Jun, H. Yong-Jian, W. Yu-Chun, and G. Guang-Can. Partial transposition on bipartite system.
*Chinese Phys. Lett.*, 25:35, 2008. - N. Johnston and D. W. Kribs. A family of norms with applications in quantum information theory.
*J. Math. Phys.*, 51:082202, 2010. E-print: arXiv:0909.3907 [quant-ph] - S. Rana. Negative eigenvalues of partial transposition of arbitrary bipartite states.
*Phys. Rev. A*, 87:054301, 2013. E-print: arXiv:1304.6775 [quant-ph] - L. Chen, D. Z. Djokovic. Qubit-qudit states with positive partial transpose.
*Phys. Rev. A*, 86:062332, 2012. E-print: arXiv:1210.0111 [quant-ph] - N. Johnston. Non-positive-partial-transpose subspaces can be as large as any entangled subspace.
*Phys. Rev. A*, 87:064302, 2013. E-print: arXiv:1305.0257 [quant-ph] - N. Johnston.
*Norms and Cones in the Theory of Quantum Entanglement*. PhD thesis, University of Guelph, 2012.

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