arXiv subjects

K. R. Parthasarathy

Publications and source records attributed to K. R. Parthasarathy.

23 records · Page 2Linked to original sources

arXiv2004

On Estimating the State of a Finite Level Quantum System

We revisit the problem of mutually unbiased measurements in the context of estimating the unknown state of a $d$-level quantum system, first studied by W. K. Wootters and B. D. fields[7] in 1989 and later investigated by S. Bandyopadhyay et al [3] in 2001 and A. O. Pittenger and M. H. Rubin [6] in 2003. Our approach is based directly on the Weyl operators in the $L^2$-space over a finite field when $d=p^r$ is the power of a prime. When $d$ is not a prime power we sacrifice a bit of optimality and construct a recovery operator for reconstructing the unknown state from the probabilities of elementary events in different measurements.

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arXiv2004

On the maximal dimension of a completely entangled subspace for finite level quantum systems

Let $\mathcal{H}_i$ be a finite dimensional complex Hilbert space of dimension $d_i$ associated with a finite level quantum system $A_i$ for $i = i, 1,2, ..., k$. A subspace $S \subset \mathcal{H} = \mathcal{H}_{A_{1} A_{2}... A_{k}} = \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes ... \otimes \mathcal{H}_k $ is said to be {\it completely entangled} if it has no nonzero product vector of the form $u_1 \otimes u_2 \otimes ... \otimes u_k$ with $u_i$ in $\mathcal{H}_i$ for each $i$. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that $$\max_{S \in \mathcal{E}} \dim S = d_1 d_2... d_k - (d_1 + ... + d_k) + k - 1$$ where $\mathcal{E} $ is the collection of all completely entangled subspaces. When $\mathcal{H}_1 = \mathcal{H}_2 $ and $k = 2$ an explicit orthonormal basis of a maximal completely entangled subspace of $\mathcal{H}_1 \otimes \mathcal{H}_2$ is given. We also introduce a more delicate notion of a {\it perfectly entangled} subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.

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arXiv2003

Extremal Quantum States in Coupled Systems

Let ${\cal H}_1,$ ${\cal H}_2$ be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose $ρ_i$ is a state in ${\cal H}_i, i=1,2.$ Let ${\cal C} (ρ_1, ρ_2)$ be the convex set of all states $ρ$ in ${\cal H} = {\cal H}_1 \otimes {\cal H}_2$ whose marginal states in ${\cal H}_1$ and ${\cal H}_2$ are $ρ_1$ and $ρ_2$ respectively. Here we present a necessary and sufficient criterion for a $ρ$ in ${\cal C} (ρ_1, ρ_2)$ to be an extreme point. Such a condition implies, in particular, that for a state $ρ$ to be an extreme point of ${\cal C} (ρ_1, ρ_2)$ it is necessary that the rank of $ρ$ does not exceed $(d_1^2 + d_2^2 - 1)^{1/2},$ where $d_i = \dim {\cal H}_i, i=1,2.$ When ${\cal H}_1$ and ${\cal H}_2$ coincide with the 1-qubit Hilbert space $\mathbb{C}^2$ with its standard orthonormal basis $\{|0 >, |1> \}$ and $ρ_1 = ρ_2 = {1/2} I$ it turns out that a state $ρ\in {\cal C} ({1/2}I, {1/2}I)$ is extremal if and only if $ρ$ is of the form $|Ω>< Ω|$ where $| Ω> = \frac{1}{\sqrt{2}} (|0> | ψ_0 > + |1 > | ψ_1 >),$ $\{| ψ_0 >, | ψ_1> \}$ being an arbitrary orthonormal basis of $\mathbb{C}^2.$ In particular, the extremal states are the maximally entangled states.

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arXiv2002

Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group

This paper is motivated by the computer-generated nonadditive ((5,6,2)) code described in an article by Rains, Hardin, Shor and Sloane. We describe a theory of non-stabilizer codes of which the nonadditive code of Rains et al is an example. Furthermore, we give a general strategy of constructing good nonstabilizer codes from good stabilizer codes and give some explicit constructions and asymptotically good nonstabilizer codes. In fact, we explicitly construct a family of distance 2 non-stabilizer codes over all finite fields of which the ((5,6,2)) is an special example. More interestingly, using our theory, we are also able to explicitly construct examples of non-stablizer quantum codes of distance 3. Like in the case of stabilizer codes, we can design fairly efficient encoding and decoding procedures.

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arXiv2002

A Family of Quantum Stabilizer Codes Based on the Weyl Commutation Relations over a Finite Field

Using the Weyl commutation relations over a finite field we introduce a family of error-correcting quantum stabilizer codes based on a class of symmetric matrices over the finite field satisfying certain natural conditions. When the field is GF(2) the existence of a rich class of such symmetric matrices is demonstrated by a simple probabilistic argument depending on the Chernoff bound for i.i.d symmetric Bernoulli trials. If, in addition, these symmetric matrices are assumed to be circulant it is possible to obtain concrete examples by a computer program. The quantum codes thus obtained admit elegant encoding circuits.

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